ISM_T12_PRE_VII Solution Manual
User Manual:
Open the PDF directly: View PDF 
.
Page Count: 439
| Download | |
| Open PDF In Browser | View PDF |
INSTRUCTOR’S
SOLUTIONS MANUAL
MULTIVARIABLE
WILLIAM ARDIS
Collin County Community College
THOMAS’ CALCULUS
TWELFTH EDITION
BASED ON THE ORIGINAL WORK BY
George B. Thomas, Jr.
Massachusetts Institute of Technology
AS REVISED BY
Maurice D. Weir
Naval Postgraduate School
Joel Hass
University of California, Davis
The author and publisher of this book have used their best efforts in preparing this book. These efforts
include the development, research, and testing of the theories and programs to determine their
effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to
these programs or the documentation contained in this book. The author and publisher shall not be liable in
any event for incidental or consequential damages in connection with, or arising out of, the furnishing,
performance, or use of these programs.
Reproduced by Addison-Wesley from electronic files supplied by the author.
Copyright © 2010, 2005, 2001 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or
transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-60072-1
ISBN-10: 0-321-60072-X
1 2 3 4 5 6 BB 14 13 12 11 10
PREFACE TO THE INSTRUCTOR
This Instructor's Solutions Manual contains the solutions to every exercise in the 12th Edition of THOMAS' CALCULUS
by Maurice Weir and Joel Hass, including the Computer Algebra System (CAS) exercises. The corresponding Student's
Solutions Manual omits the solutions to the even-numbered exercises as well as the solutions to the CAS exercises (because
the CAS command templates would give them all away).
In addition to including the solutions to all of the new exercises in this edition of Thomas, we have carefully revised or
rewritten every solution which appeared in previous solutions manuals to ensure that each solution
ì conforms exactly to the methods, procedures and steps presented in the text
ì is mathematically correct
ì includes all of the steps necessary so a typical calculus student can follow the logical argument and algebra
ì includes a graph or figure whenever called for by the exercise, or if needed to help with the explanation
ì is formatted in an appropriate style to aid in its understanding
Every CAS exercise is solved in both the MAPLE and MATHEMATICA computer algebra systems. A template showing
an example of the CAS commands needed to execute the solution is provided for each exercise type. Similar exercises within
the text grouping require a change only in the input function or other numerical input parameters associated with the problem
(such as the interval endpoints or the number of iterations).
For more information about other resources available with Thomas' Calculus, visit http://pearsonhighered.com.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
TABLE OF CONTENTS
10 Infinite Sequences and Series 569
10.1 Sequences 569
10.2 Infinite Series 577
10.3 The Integral Test 583
10.4 Comparison Tests 590
10.5 The Ratio and Root Tests 597
10.6 Alternating Series, Absolute and Conditional Convergence 602
10.7 Power Series 608
10.8 Taylor and Maclaurin Series 617
10.9 Convergence of Taylor Series 621
10.10 The Binomial Series and Applications of Taylor Series 627
Practice Exercises 634
Additional and Advanced Exercises 642
11 Parametric Equations and Polar Coordinates 647
11.1
11.2
11.3
11.4
11.5
11.6
11.7
Parametrizations of Plane Curves 647
Calculus with Parametric Curves 654
Polar Coordinates 662
Graphing in Polar Coordinates 667
Areas and Lengths in Polar Coordinates 674
Conic Sections 679
Conics in Polar Coordinates 689
Practice Exercises 699
Additional and Advanced Exercises 709
12 Vectors and the Geometry of Space 715
12.1
12.2
12.3
12.4
12.5
12.6
Three-Dimensional Coordinate Systems 715
Vectors 718
The Dot Product 723
The Cross Product 728
Lines and Planes in Space 734
Cylinders and Quadric Surfaces 741
Practice Exercises 746
Additional Exercises 754
13 Vector-Valued Functions and Motion in Space 759
13.1
13.2
13.3
13.4
13.5
13.6
Curves in Space and Their Tangents 759
Integrals of Vector Functions; Projectile Motion 764
Arc Length in Space 770
Curvature and Normal Vectors of a Curve 773
Tangential and Normal Components of Acceleration 778
Velocity and Acceleration in Polar Coordinates 784
Practice Exercises 785
Additional Exercises 791
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
14 Partial Derivatives 795
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
14.9
14.10
Functions of Several Variables 795
Limits and Continuity in Higher Dimensions 804
Partial Derivatives 810
The Chain Rule 816
Directional Derivatives and Gradient Vectors 824
Tangent Planes and Differentials 829
Extreme Values and Saddle Points 836
Lagrange Multipliers 849
Taylor's Formula for Two Variables 857
Partial Derivatives with Constrained Variables 859
Practice Exercises 862
Additional Exercises 876
15 Multiple Integrals 881
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
Double and Iterated Integrals over Rectangles 881
Double Integrals over General Regions 882
Area by Double Integration 896
Double Integrals in Polar Form 900
Triple Integrals in Rectangular Coordinates 904
Moments and Centers of Mass 909
Triple Integrals in Cylindrical and Spherical Coordinates 914
Substitutions in Multiple Integrals 922
Practice Exercises 927
Additional Exercises 933
16 Integration in Vector Fields 939
16.1
16.2
16.3
16.4
16.5
16.6
16.7
16.8
Line Integrals 939
Vector Fields and Line Integrals; Work, Circulation, and Flux 944
Path Independence, Potential Functions, and Conservative Fields 952
Green's Theorem in the Plane 957
Surfaces and Area 963
Surface Integrals 972
Stokes's Theorem 980
The Divergence Theorem and a Unified Theory 984
Practice Exercises 989
Additional Exercises 997
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
CHAPTER 10 INFINITE SEQUENCES AND SERIES
10.1 SEQUENCES
1. a" œ
1 1
1#
2. a" œ
1
1!
3.
a" œ
"2
##
œ 0, a# œ
œ 1, a# œ
(1)#
#1
"
#!
œ
œ 1, a# œ
œ 4" , a$ œ
13
3#
"
2
1
6
, a$ œ
(")$
41
œ
1
3!
, a% œ
œ "3 , a$ œ
14
4#
œ 92 , a% œ
œ
1
4!
(1)%
61
œ
"
5
3
œ 16
1
24
(1)&
81
, a% œ
œ 7"
4. a" œ 2 (1)" œ 1, a# œ 2 (1)# œ 3, a$ œ 2 (1)$ œ 1, a% œ 2 (1)% œ 3
5. a" œ
2
##
6. a" œ
2"
#
"
#
œ
, a# œ
œ
"
#
a( œ
, a) œ
8. a" œ 1, a# œ
a* œ
"
362,880
"
#
œ
"
#
255
128
"
#
œ
3
#
œ
511
256
, a$ œ
3
#
"
#
œ
"
##
, a"! œ
ˆ #" ‰
"
3 œ 6
"
3,628,800
, a$ œ
, a"! œ
3
4
, a$ œ
, a* œ
2$
#%
, a$ œ
2# 1
2#
, a# œ
7. a" œ 1, a# œ 1
127
64
2#
2$
, a% œ
, a% œ
2$ 1
2$
œ
7
4
œ
2%
2&
7
8
œ
"
#
, a% œ
, a% œ
7
4
2% "
2%
"
#$
œ
a' œ
,
15
8
ˆ "6 ‰
4
œ
"
#4
, a& œ
ˆ #"4 ‰
5
œ
$
(1)% ˆ "# ‰
(1)# (2)
œ 1, a$ œ (1)2 (1) œ "# , a% œ
#
#
"
"
a( œ 3"# , a) œ 64
, a* œ 1#"8 , a"! œ 256
1†(2)
œ 1, a$ œ 2†(31) œ 32 , a%
#
a) œ "4 , a* œ 29 , a"! œ "5
10. a" œ 2, a# œ
a( œ 27 ,
15
16
, a& œ
15
8
"
#%
œ
œ
31
16 , a'
63
32
,
1023
512
9. a" œ 2, a# œ
"
16
œ
œ
3†ˆ 23 ‰
4
"
1 #0
, a' œ
"
7 #0
œ 4" , a& œ
œ "# , a& œ
, a( œ
"
5040
(1)& ˆ "4 ‰
#
4†ˆ "# ‰
5
, a) œ
œ
"
8
"
40,320
,
,
œ 52 , a' œ 3" ,
11. a" œ 1, a# œ 1, a$ œ 1 1 œ 2, a% œ 2 1 œ 3, a& œ 3 2 œ 5, a' œ 8, a( œ 13, a) œ 21, a* œ 34, a"! œ 55
12. a" œ 2, a# œ 1, a$ œ "# , a% œ
ˆ "# ‰
1
œ
"
#
, a& œ
ˆ "# ‰
ˆ "# ‰
œ 1, a' œ 2, a( œ 2, a) œ 1, a* œ "# , a"! œ
13. an œ (1)n1 , n œ 1, 2, á
14. an œ (1)n , n œ 1, 2, á
15. an œ (1)n1 n# , n œ 1, 2, á
16. an œ
(")n
n#
1
, n œ 1, 2, á
18. an œ
2n 5
nan 1b
, n œ 1, 2, á
17. an œ
2n 1
3an 2b ,
n œ 1, 2, á
19. an œ n# 1, n œ 1, 2, á
20. an œ n 4 , n œ 1, 2, á
21. an œ 4n 3, n œ 1, 2, á
22. an œ 4n 2 , n œ 1, 2, á
23. an œ
3n 2
n! ,
n œ 1, 2, á
24. an œ
n3
5n 1
, n œ 1, 2, á
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
#
570
Chapter 10 Infinite Sequences and Series
25. an œ
1 (1)n
#
1
, n œ 1, 2, á
26. an œ
27. n lim
2 (0.1)n œ 2 Ê converges
Ä_
n (")n
n
29. n lim
Ä_
" 2n
1 #n
30. n lim
Ä_
2n "
1 3È n
œ n lim
Ä_
31. n lim
Ä_
" 5n%
n% 8n$
œ n lim
Ä_
32. n lim
Ä_
n3
n# 5n 6
œ n lim
Ä_
n3
(n 3)(n 2)
œ n lim
Ä_
33. n lim
Ä_
n# 2n 1
n1
œ n lim
Ä_
(n 1)(n 1)
n1
œ n lim
(n 1) œ _ Ê diverges
Ä_
34 n lim
Ä_
" n$
70 4n#
ˆ "n ‰ 2
ˆ "n ‰ 2
œ n lim
Ä_
œ 1 Ê converges
2Èn Š È"n ‹
Š È"n 3‹
1 ˆ 8n ‰
"
‹n
n#
70
Š #‹4
n
Š
œ n lim
Ä_
2
#
œ n lim
Ä_
Š n"% ‹ 5
œ 1 Ê converges
œ _ Ê diverges
œ 5 Ê converges
"
n#
œ 0 Ê converges
œ _ Ê diverges
36. n lim
(1)n ˆ1 "n ‰ does not exist Ê diverges
Ä_
35. n lim
a1 (1)n b does not exist Ê diverges
Ä_
ˆ n #n " ‰ ˆ1 "n ‰ œ lim ˆ "#
37. n lim
Ä_
nÄ_
ˆ2
38. n lim
Ä_
" ‰ˆ
3
#n
"‰
#n
ˆ "# ‰n œ lim
40. n lim
Ä_
nÄ_
É n 2n
41. n lim
1 œ É n lim
Ä_
Ä_
42. n lim
Ä_
"
(0.9)n
" ‰ˆ
1
#n
n" ‰ œ
œ 6 Ê converges
(")n
#n
œ Ú n# Û, n œ 1, 2, á
(Theorem 5, #4)
28. n lim
Ä_
œ n lim
1
Ä_
(1)n
n
n "# (1)n ˆ "# ‰
#
"
#
Ê converges
39. n lim
Ä_
(")nb1
#n 1
œ 0 Ê converges
œ 0 Ê converges
2n
n1
œ Ên lim
Š 2 ‹ œ È2 Ê converges
Ä _ 1 "
n
ˆ "0 ‰n œ _ Ê diverges
œ n lim
Ä_ 9
ˆ 1 n" ‰‹ œ sin
43. n lim
sin ˆ 1# n" ‰ œ sin Šn lim
Ä_
Ä_ #
1
#
œ 1 Ê converges
44. n lim
n1 cos (n1) œ n lim
(n1)(1)n does not exist Ê diverges
Ä_
Ä_
45. n lim
Ä_
sin n
n
46. n lim
Ä_
sin# n
#n
47. n lim
Ä_
n
#n
œ 0 because n" Ÿ
œ 0 because 0 Ÿ
œ n lim
Ä_
"
#n ln 2
sin n
n
sin# n
#n
Ÿ
Ÿ
"
n
"
#n
Ê converges by the Sandwich Theorem for sequences
Ê converges by the Sandwich Theorem for sequences
^
œ 0 Ê converges (using l'Hopital's
rule)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.1 Sequences
48. n lim
Ä_
3n
n$
49. n lim
Ä_
ln (n ")
Èn
50. n lim
Ä_
ln n
ln 2n
œ n lim
Ä_
3n ln 3
3n#
œ n lim
Ä_
œ n lim
Ä_
œ n lim
Ä_
ˆn " 1‰
" ‹
Š #È
n
ˆ "n ‰
2 ‰
ˆ 2n
3n (ln 3)#
6n
œ n lim
Ä_
œ n lim
Ä_
2È n
n1
3n (ln 3)$
6
œ n lim
Ä_
^
œ _ Ê diverges (using l'Hopital's
rule)
Š È2n ‹
1 Š n" ‹
œ 0 Ê converges
œ 1 Ê converges
51. n lim
81În œ 1 Ê converges
Ä_
(Theorem 5, #3)
52. n lim
(0.03)1În œ 1 Ê converges
Ä_
(Theorem 5, #3)
ˆ1 7n ‰n œ e( Ê converges
53. n lim
Ä_
ˆ1 "n ‰n œ lim ’1
54. n lim
Ä_
nÄ_
(")
n “
(Theorem 5, #5)
n
œ e" Ê converges
(Theorem 5, #5)
n
È
55. n lim
10n œ n lim
101În † n1În œ 1 † 1 œ 1 Ê converges
Ä_
Ä_
#
n
n
È
ˆÈ
56. n lim
n# œ n lim
n‰ œ 1# œ 1 Ê converges
Ä_
Ä_
ˆ 3 ‰1În œ nÄ_ 1În œ
57. n lim
lim n
Ä_ n
nÄ_
lim 31În
"
1
œ 1 Ê converges
(Theorem 5, #3 and #2)
(Theorem 5, #2)
(Theorem 5, #3 and #2)
58. n lim
(n 4)1ÎÐn4Ñ œ x lim
x1Îx œ 1 Ê converges; (let x œ n 4, then use Theorem 5, #2)
Ä_
Ä_
59. n lim
Ä_
ln n
n1În
lim
Ä_ ln1Înn œ
œ nlim
n
n
Ä_
_
1
œ _ Ê diverges
(Theorem 5, #2)
60. n lim
cln n ln (n 1)d œ n lim
ln ˆ n n 1 ‰ œ ln Šn lim
Ä_
Ä_
Ä_
n
n
È
61. n lim
4n n œ n lim
4È
n œ 4 † 1 œ 4 Ê converges
Ä_
Ä_
n
n1‹
œ ln 1 œ 0 Ê converges
(Theorem 5, #2)
n
È
62. n lim
32n1 œ n lim
32 a1Înb œ n lim
3# † 31În œ 9 † 1 œ 9 Ê converges
Ä_
Ä_
Ä_
œ n lim
Ä_
"†2†3â(n 1)(n)
n†n†nân†n
63. n lim
Ä_
n!
nn
64. n lim
Ä_
(4)n
n!
65. n lim
Ä_
n!
106n
œ n lim
Ä_
"
'n
Š (10n! ) ‹
66. n lim
Ä_
n!
2n 3n
œ n lim
Ä_
"
ˆ 6n!n ‰
œ 0 Ê converges
ˆ " ‰ œ 0 and
Ÿ n lim
Ä_ n
n!
nn
0 Ê n lim
Ä_
n!
nn
(Theorem 5, #3)
œ 0 Ê converges
(Theorem 5, #6)
œ _ Ê diverges
œ _ Ê diverges
(Theorem 5, #6)
(Theorem 5, #6)
ˆ " ‰1ÎÐln nÑ œ lim exp ˆ ln"n ln ˆ n" ‰‰ œ lim exp ˆ ln 1lnnln n ‰ œ e" Ê converges
67. n lim
Ä_ n
nÄ_
nÄ_
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
571
572
Chapter 10 Infinite Sequences and Series
n
ˆ1 n" ‰n ‹ œ ln e œ 1 Ê converges
68. n lim
ln ˆ1 "n ‰ œ ln Šn lim
Ä_
Ä_
(Theorem 5, #5)
" ‰‰
ˆ 3n " ‰n œ lim exp ˆn ln ˆ 3n
69. n lim
œ n lim
exp Š ln (3n 1) " ln (3n 1) ‹
3n 1
Ä _ 3n 1
nÄ_
Ä_
n
3
3
6n
#Î$
ˆ6‰
œ n lim
exp 3n 1 "3n 1 œ n lim
exp Š (3n 1)(3n
Ê converges
1) ‹ œ exp 9 œ e
Ä_
Ä_
Š ‹
#
n#
"
"
ˆ n ‰n œ lim exp ˆn ln ˆ n n 1 ‰‰ œ lim exp Š ln n ln" (n 1) ‹ œ lim exp n n 1
70. n lim
ˆn‰
Ä _ n1
nÄ_
nÄ_
nÄ_
Š "# ‹
n
œ n lim
exp Š
Ä_
n#
n(n 1) ‹
"
œe
Ê converges
1)
ˆ x ‰1În œ lim x ˆ #n " 1 ‰1În œ x lim exp ˆ "n ln ˆ #n " 1 ‰‰ œ x lim exp Š ln (2n
71. n lim
‹
n
Ä _ 2n 1
nÄ_
nÄ_
nÄ_
2
!
œ x n lim
exp ˆ 2n1 ‰ œ xe œ x, x 0 Ê converges
Ä_
n
ˆ1
72. n lim
Ä_
" ‰n
n#
œ n lim
exp ˆn ln ˆ1
Ä_
" ‰‰
n#
œ n lim
exp
Ä_
ln Š1 n"# ‹
exp –
œ n lim
Ä_
ˆ n" ‰
Š n2$ ‹‚Š1 n"# ‹
Š n"# ‹
—
œ n lim
exp ˆ n# 2n1 ‰ œ e! œ 1 Ê converges
Ä_
73. n lim
Ä_
3 n †6 n
2cn †n!
œ n lim
Ä_
36n
n!
œ 0 Ê converges
ˆ 10 ‰n
ˆ 12
‰n ˆ 10
‰n
11
11
12 ‰n ˆ 9 ‰n
12 ‰n ˆ 11 ‰n
ˆ 11
ˆ
11
10
12
11
74. n lim
lim
n
11 ‰n œ
Ä _ ˆ 109 ‰ ˆ 12
nÄ_
(Theorem 5, #4)
75. n lim
tanh n œ n lim
Ä_
Ä_
en e
en e
76. n lim
sinh (ln n) œ n lim
Ä_
Ä_
77. n lim
Ä_
n# sin ˆ n" ‰
2n 1
œ n lim
Ä_
(Theorem 5, #6)
n
n
œ n lim
Ä_
eln n e
2
ln n
sin ˆ "n ‰
Èn sinŠ È1 ‹ œ lim
79. n lim
n
Ä_
nÄ_
ˆ" cos "n ‰
ˆ "n ‰
sinŠ È1n ‹
Èn
1
œ n lim
Ä_
n ˆ n" ‰
#
œ n lim
Ä_
œ n lim
Ä_
Š 2n n"# ‹
78. n lim
n ˆ1 cos "n ‰ œ n lim
Ä_
Ä_
e2n "
e2n 1
ˆ 120
‰n
121
n
ˆ 108
‰
1
110
œ n lim
Ä_
2e2n
2e2n
Š n2# n2$ ‹
œ n lim
Ä_
œ n lim
" œ 1 Ê converges
Ä_
œ _ Ê diverges
ˆcos ˆ "n ‰‰ Š n"# ‹
œ n lim
Ä_
œ 0 Ê converges
œ n lim
Ä_
sin ˆ "n ‰‘ Š "# ‹
n
Š n"# ‹
cos Š È1n ‹Š
1
2n3Î2
1
‹
2n3Î2
cos ˆ n" ‰
# ˆ 2n ‰
œ
"
#
Ê converges
œ n lim
sin ˆ "n ‰ œ 0 Ê converges
Ä_
œ n lim
cos Š È1n ‹ œ cos 0 œ 1 Ê converges
Ä_
80. n lim
a3n 5n b1În œ n lim
exp’lna3n 5n b1În “ œ n lim
exp’ lna3 n 5 b “ œ n lim
exp–
Ä_
Ä_
Ä_
Ä_
n
n
œ n lim
exp’
Ä_
Š 35n ‹ln 3 ln 5
81. n lim
tan" n œ
Ä_
ˆ 35nn ‰ 1
1
#
exp’
“ œ n lim
Ä_
Ê converges
ˆ 35 ‰n ln 3 ln 5
ˆ 35 ‰n 1 “
n
3n ln 3 b 5n ln 5
3n b 5n
1
—
œ expaln 5b œ 5
82. n lim
Ä_
"
Èn
tan" n œ 0 †
1
#
œ 0 Ê converges
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.1 Sequences
ˆ " ‰n
83. n lim
Ä_ 3
"
È 2n
573
n
n
œ n lim
Šˆ 3" ‰ Š È"2 ‹ ‹ œ 0 Ê converges
Ä_
(Theorem 5, #4)
#
n
1‰
!
È
84. n lim
n# n œ n lim
exp ’ ln ann nb “ œ n lim
exp ˆ 2n
n# n œ e œ 1 Ê converges
Ä_
Ä_
Ä_
85. n lim
Ä_
(ln n)#!!
n
86. n lim
Ä_
(ln n)&
Èn
œ n lim
Ä_
200 (ln n)"**
n
œ n lim
Ä_
200†199 (ln n)"*)
n
œ á œ n lim
Ä_
200!
n
œ 0 Ê converges
%
œ n lim
Ä_ –
Š 5(lnnn) ‹
"
Š #Èn ‹
— œ n lim
Ä_
10(ln n)%
Èn
œ n lim
Ä_
È
80(ln n)$
Èn
œ á œ n lim
Ä_
#
87. n lim
Šn Èn# n‹ œ n lim
Šn Èn# n‹ Š n Èn# n ‹ œ n lim
Ä_
Ä_
Ä_
n n n
œ
"
#
88. n lim
Ä_
œ 0 Ê converges
œ n lim
Ä_
"
1 É1
"
n
Ê converges
"
È n# 1 È n# n
œ n lim
Š
Ä_ È
É1 n"# É1 "n
œ n lim
Ä_
89. n lim
Ä_
n
n È n# n
3840
Èn
ˆ n" 1‰
'
90. n lim
Ä_ 1
n
"
xp
œ n lim
Ä_
È n# 1 È n# n
1 n
œ 2 Ê converges
'1n x" dx œ n lim
Ä_
"
n
È #
È #
"
‹ Š Èn# 1 Èn# n ‹
n# 1 È n# n
n 1 n n
ln n
n
dx œ n lim
’ "
Ä _ 1 p
œ n lim
Ä_
n
"
xpc1 “ 1
"
n
œ 0 Ê converges
œ n lim
Ä_
"
1 p
ˆ np"c1 1‰ œ
(Theorem 5, #1)
"
p 1
if p 1 Ê converges
72
91. Since an converges Ê n lim
a œ L Ê n lim
a
œ n lim
ÊLœ
Ä_ n
Ä _ n1
Ä _ 1 an
Ê L œ 9 or L œ 8; since an 0 for n 1 Ê L œ 8
72
1L
Ê La1 Lb œ 72 Ê L2 L 72 œ 0
an 6
92. Since an converges Ê n lim
a œ L Ê n lim
a
œ n lim
ÊLœ
Ä_ n
Ä _ n1
Ä _ an 2
Ê L œ 3 or L œ 2; since an 0 for n 2 Ê L œ 2
L6
L2
Ê LaL 2b œ L 6 Ê L2 L 6 œ 0
È8 2an Ê L œ È8 2L Ê L2 2L 8 œ 0 Ê L œ 2
93. Since an converges Ê n lim
a œ L Ê n lim
a
œ n lim
Ä_ n
Ä _ n1
Ä_
or L œ 4; since an 0 for n 3 Ê L œ 4
È8 2an Ê L œ È8 2L Ê L2 2L 8 œ 0 Ê L œ 2
94. Since an converges Ê n lim
a œ L Ê n lim
a
œ n lim
Ä_ n
Ä _ n1
Ä_
or L œ 4; since an 0 for n 2 Ê L œ 4
È5an Ê L œ È5L Ê L2 5L œ 0 Ê L œ 0 or L œ 5; since
95. Since an converges Ê n lim
a œ L Ê n lim
a
œ n lim
Ä_ n
Ä _ n1
Ä_
an 0 for n 1 Ê L œ 5
ˆ12 Èan ‰ Ê L œ Š12 ÈL‹ Ê L2 25L 144 œ 0
96. Since an converges Ê n lim
a œ L Ê n lim
a
œ n lim
Ä_ n
Ä _ n1
Ä_
Ê L œ 9 or L œ 16; since 12 Èan 12 for n 1 Ê L œ 9
97. an 1 œ 2
1, a1 œ 2. Since an converges Ê n lim
a œ L Ê n lim
a
œ n lim
Š2
Ä_ n
Ä _ n1
Ä_
Ê L2 2L 1 œ 0 Ê L œ 1 „ È2; since an 0 for n 1 Ê L œ 1 È2
1
an ,
n
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1
an ‹
ÊLœ2
1
L
574
Chapter 10 Infinite Sequences and Series
98. an 1 œ È1 an , n
È1 an Ê L œ È1 L
1, a1 œ È1. Since an converges Ê n lim
a œ L Ê n lim
a
œ n lim
Ä_ n
Ä _ n1
Ä_
1 „ È5
2 ;
Ê L2 L 1 œ 0 Ê L œ
since an 0 for n
1ÊLœ
1 È5
2
99. 1, 1, 2, 4, 8, 16, 32, á œ 1, 2! , 2" , 2# , 2$ , 2% , 2& , á Ê x" œ 1 and xn œ 2nc2 for n
2
100. (a) 1# 2(1)# œ 1, 3# 2(2)# œ 1; let f(aß b) œ (a 2b)# 2(a b)# œ a# 4ab 4b# 2a# 4ab 2b#
œ 2b# a# ; a# 2b# œ 1 Ê f(aß b) œ 2b# a# œ 1; a# 2b# œ 1 Ê f(aß b) œ 2b# a# œ 1
#
‰ 2œ
(b) r#n 2 œ ˆ aa2b
b
a# 4ab 4b# 2a# 4ab 2b#
(a b)#
In the first and second fractions, yn
for n a positive integer
lim rn œ È2.
n. Let
a
b
œ
aa# 2b# b
(a b)#
œ
„"
y#n
#
Ê rn œ Ê2 „ Š y"n ‹
represent the (n 1)th fraction where
3. Now the nth fraction is
a 2b
ab
and a b
2b
a
b
2n 2
1 and b
n Ê yn
n1
n. Thus,
nÄ_
101. (a) f(x) œ x# 2; the sequence converges to 1.414213562 ¸ È2
(b) f(x) œ tan (x) 1; the sequence converges to 0.7853981635 ¸
1
4
(c) f(x) œ ex ; the sequence 1, 0, 1, 2, 3, 4, 5, á diverges
102. (a) n lim
nf ˆ n" ‰ œ lim b f(??xx) œ lim b f(0??x)x f(0) œ f w (0), where ?x œ
Ä_
?x Ä !
?x Ä !
"
" ˆ " ‰
w
"
(b) n lim
n
tan
œ
f
(0)
œ
x
# œ 1, f(x) œ tan
n
1
0
Ä_
"
n
(c) n lim
n ae1În 1b œ f w (0) œ e! œ 1, f(x) œ ex 1
Ä_
(d) n lim
n ln ˆ1 2n ‰ œ f w (0) œ 1 22(0) œ 2, f(x) œ ln (1 2x)
Ä_
#
103. (a) If a œ 2n 1, then b œ Ú a# Û œ Ú 4n
#
4n 1
Û
#
#
#
œ Ú2n# 2n "# Û œ 2n# 2n, c œ Ü a# Ý œ Ü2n# 2n "# Ý
#
œ 2n# 2n 1 and a# b# œ (2n 1) a2n# 2nb œ 4n# 4n 1 4n% 8n$ 4n#
#
œ 4n% 8n$ 8n# 4n 1 œ a2n# 2n 1b œ c# .
(b) a lim
Ä_
#
Ú a# Û
#
Ü a# Ý
œ a lim
Ä_
2n# 2n
2n# 2n 1
œ 1 or a lim
Ä_
#
Ú a# Û
#
Ü a# Ý
œ a lim
sin ) œ
Ä_
2n1 ‰
104. (a) n lim
(2n1)1Î a2nb œ n lim
exp ˆ ln2n
œ n lim
exp
Ä_
Ä_
Ä_
21
Š 2n
1‹
#
(b)
n
40
50
60
15.76852702
19.48325423
23.19189561
sin ) œ 1
exp ˆ #"n ‰ œ e! œ 1;
œ n lim
Ä_
n
n
n! ¸ ˆ ne ‰ È
2n1 , Stirlings approximation Ê È
n! ¸ ˆ ne ‰ (2n1)1Î a2nb ¸
n
È
n!
lim
) Ä 1 Î2
n
e
for large values of n
n
e
14.71517765
18.39397206
22.07276647
ˆ"‰
ln n
"
n
105. (a) n lim
œ n lim
œ n lim
œ0
Ä _ nc
Ä _ cncc1
Ä _ cnc
Ðln %ÑÎc
(b) For all % 0, there exists an N œ e
such that n eÐln %ÑÎc Ê ln n lnc % Ê ln nc ln ˆ "% ‰
Ê nc "% Ê n"c % Ê ¸ n"c 0¸ % Ê lim n"c œ 0
nÄ_
106. Let {an } and {bn } be sequences both converging to L. Define {cn } by c2n œ bn and c2nc1 œ an , where
n œ 1, 2, 3, á . For all % 0 there exists N" such that when n N" then kan Lk % and there exists N#
such that when n N# then kbn Lk %. If n 1 2max{N" ß N# }, then kcn Lk %, so {cn } converges to L.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.1 Sequences
575
107. n lim
n1În œ n lim
exp ˆ "n ln n‰ œ n lim
exp ˆ n" ‰ œ e! œ 1
Ä_
Ä_
Ä_
108. n lim
x1În œ n lim
exp ˆ "n ln x‰ œ e! œ 1, because x remains fixed while n gets large
Ä_
Ä_
109. Assume the hypotheses of the theorem and let % be a positive number. For all % there exists a N" such that
when n N" then kan Lk % Ê % an L % Ê L % an , and there exists a N# such that when
n N# then kcn Lk % Ê % cn L % Ê cn L %. If n max{N" ß N# }, then
L % an Ÿ bn Ÿ cn L % Ê kbn Lk % Ê n lim
b œ L.
Ä_ n
110. Let % !. We have f continuous at L Ê there exists $ so that kx Lk $ Ê kf(x) f(L)k %. Also, an Ä L Ê there
exists N so that for n N kan Lk $ . Thus for n N, kf(an ) f(L)k % Ê f(an ) Ä f(L).
an Ê
111. an1
3(n 1) 1
(n 1) 1
3n 1
n1
3n 4
n#
Ê
3n 1
n1
Ê 3n# 3n 4n 4 3n# 6n n 2
Ê 4 2; the steps are reversible so the sequence is nondecreasing;
3n "
n1
3 Ê 3n 1 3n 3
Ê 1 3; the steps are reversible so the sequence is bounded above by 3
an Ê
112. an1
(2(n 1) 3)!
((n 1) 1)!
(2n 3)!
(n 1)!
Ê
(2n 5)!
(n 2)!
(2n 3)!
(n 1)!
Ê
(2n 5)!
(2n 3)!
(n 2)!
(n 1)!
Ê (2n 5)(2n 4) n 2; the steps are reversible so the sequence is nondecreasing; the sequence is not
bounded since
113. an1 Ÿ an Ê
(2n 3)!
(n 1)!
œ (2n 3)(2n 2)â(n 2) can become as large as we please
2nb1 3nb1
(n 1)!
Ÿ
2n 3n
n!
2nb1 3nb1
2n 3n
Ê
(n 1)!
n!
Ÿ
Ê 2 † 3 Ÿ n 1 which is true for n
5; the steps are
reversible so the sequence is decreasing after a& , but it is not nondecreasing for all its terms; a" œ 6, a# œ 18,
a$ œ 36, a% œ 54, a& œ 324
5 œ 64.8 Ê the sequence is bounded from above by 64.8
an Ê 2
114. an1
2
n 1
"
#nb1
2
2
n
"
#n
Ê
reversible so the sequence is nondecreasing; 2
115. an œ 1
"
n
converges because
116. an œ n
"
n
diverges because n Ä _ and
117. an œ
2 n 1
2n
œ1
"
#n
and 0
"
#n
"
n
2
n
2
n
2
n 1
"
#n Ÿ
"
#nb1
"
#n
Ê
2
n(n 1)
#n"b1 ; the steps are
2 Ê the sequence is bounded from above
Ä 0 by Example 1; also it is a nondecreasing sequence bounded above by 1
"
n
; since
"
n
"
n
Ä 0 by Example 1, so the sequence is unbounded
Ä 0 (by Example 1) Ê
"
#n
Ä 0, the sequence converges; also it is
a nondecreasing sequence bounded above by 1
118. an œ
2 n 1
3n
n
œ ˆ 23 ‰
"
3n
; the sequence converges to ! by Theorem 5, #4
119. an œ a(1)n 1b ˆ nn 1 ‰ diverges because an œ 0 for n odd, while for n even an œ 2 ˆ1 n" ‰ converges to 2; it
diverges by definition of divergence
120. xn œ max {cos 1ß cos 2ß cos 3ß á ß cos n} and xn1 œ max {cos 1ß cos 2ß cos 3ß á ß cos (n 1)}
so the sequence is nondecreasing and bounded above by 1 Ê the sequence converges.
121. an
and
an1 Í
1 È2n
Èn
1 È2n
Èn
" È2(n 1)
Èn 1
Í Èn 1 È2n# 2n
xn with xn Ÿ 1
Èn È2n# 2n Í Èn 1
È2 ; thus the sequence is nonincreasing and bounded below by È2 Ê it converges
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Èn
576
Chapter 10 Infinite Sequences and Series
122. an
a n 1 Í
n1
n
(n 1) "
n1
Í n# 2n 1
n# 2n Í 1
0 and
n1
n
1; thus the sequence is
nonincreasing and bounded below by 1 Ê it converges
123.
4nb1 3n
œ
4n
3 ‰n
ˆ
4 4
n
4 ˆ 34 ‰ so an
an1 Í 4 ˆ 34 ‰
n
4 ˆ 34 ‰
n"
n
Í ˆ 34 ‰
ˆ 34 ‰n1 Í 1
3
4
and
4; thus the sequence is nonincreasing and bounded below by 4 Ê it converges
124. a" œ 1, a# œ 2 3, a$ œ 2(2 3) 3 œ 2# a22 "b † 3, a% œ 2 a2# a22 "b † 3b 3 œ 2$ a2$ 1b 3,
a& œ 2 c2$ a2$ 1b 3d 3 œ 2% a2% 1b 3, á , an œ 2n" a2n" 1b 3 œ 2n" 3 † 2n1 3
œ 2n1 (1 3) 3 œ 2n 3; an an1 Í 2n 3 2n1 3 Í 2n 2n1 Í 1 Ÿ 2
so the sequence is nonincreasing but not bounded below and therefore diverges
125. Let 0 M 1 and let N be an integer greater than
Ê n M nM Ê n M(n 1) Ê
n
n1
M
1M
. Then n N Ê n
M.
M
1M
Ê n nM M
126. Since M" is a least upper bound and M# is an upper bound, M" Ÿ M# . Since M# is a least upper bound and M"
is an upper bound, M# Ÿ M" . We conclude that M" œ M# so the least upper bound is unique.
127. The sequence an œ 1
(")n
#
is the sequence
"
#
,
3
#
,
"
#
,
3
#
, á . This sequence is bounded above by
3
#
,
but it clearly does not converge, by definition of convergence.
128. Let L be the limit of the convergent sequence {an }. Then by definition of convergence, for
corresponds an N such that for all m and n, m N Ê kam Lk
kam an k œ kam L L an k Ÿ kam Lk kL an k
%
#
%
#
%
#
%
#
there
and n N Ê kan Lk #% . Now
œ % whenever m N and n N.
129. Given an % 0, by definition of convergence there corresponds an N such that for all n N,
kL" an k % and kL# an k %. Now kL# L" k œ kL# an an L" k Ÿ kL# an k kan L" k % % œ 2%.
kL# L" k 2% says that the difference between two fixed values is smaller than any positive number 2%.
The only nonnegative number smaller than every positive number is 0, so kL" L# k œ 0 or L" œ L# .
130. Let k(n) and i(n) be two order-preserving functions whose domains are the set of positive integers and whose
ranges are a subset of the positive integers. Consider the two subsequences akÐnÑ and aiÐnÑ , where akÐnÑ Ä L" ,
aiÐnÑ Ä L# and L" Á L# . Thus ¸akÐnÑ aiÐnÑ ¸ Ä kL" L# k 0. So there does not exist N such that for all m, n N
Ê kam an k %. So by Exercise 128, the sequence Öan × is not convergent and hence diverges.
131. a2k Ä L Í given an % 0 there corresponds an N" such that c2k N" Ê ka2k Lk %d . Similarly,
a2k1 Ä L Í c2k 1 N# Ê ka2k1 Lk %d . Let N œ max{N" ß N# }. Then n N Ê kan Lk % whether
n is even or odd, and hence an Ä L.
132. Assume an Ä 0. This implies that given an % 0 there corresponds an N such that n N Ê kan 0k %
Ê kan k % Ê kkan kk % Ê kkan k 0k % Ê kan k Ä 0. On the other hand, assume kan k Ä 0. This implies that
given an % 0 there corresponds an N such that for n N, kkan k 0k % Ê kkan kk % Ê kan k %
Ê kan 0k % Ê an Ä 0.
133. (a) f(x) œ x# a Ê f w (x) œ 2x Ê xn1 œ xn
x#n a
#xn
Ê x n 1 œ
2x#n ax#n ab
2xn
œ
x#n a
2xn
œ
ˆxn xa ‰
n
#
(b) x" œ 2, x# œ 1.75, x$ œ 1.732142857, x% œ 1.73205081, x& œ 1.732050808; we are finding the positive
number where x# 3 œ 0; that is, where x# œ 3, x 0, or where x œ È3 .
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.2 Infinite Series
577
134. x" œ 1, x# œ 1 cos (1) œ 1.540302306, x$ œ 1.540302306 cos (1 cos (1)) œ 1.570791601,
x% œ 1.570791601 cos (1.570791601) œ 1.570796327 œ 1# to 9 decimal places. After a few steps, the
arc axnc1 b and line segment cos axnc1 b are nearly the same as the quarter circle.
135-146. Example CAS Commands:
Mathematica: (sequence functions may vary):
Clear[a, n]
a[n_]; = n1 / n
first25= Table[N[a[n]],{n, 1, 25}]
Limit[a[n], n Ä 8]
Mathematica: (sequence functions may vary):
Clear[a, n]
a[n_]; = n1 / n
first25= Table[N[a[n]],{n, 1, 25}]
Limit[a[n], n Ä 8]
The last command (Limit) will not always work in Mathematica. You could also explore the limit by enlarging your table
to more than the first 25 values.
If you know the limit (1 in the above example), to determine how far to go to have all further terms within 0.01 of the
limit, do the following.
Clear[minN, lim]
lim= 1
Do[{diff=Abs[a[n] lim], If[diff < .01, {minN= n, Abort[]}]}, {n, 2, 1000}]
minN
For sequences that are given recursively, the following code is suggested. The portion of the command a[n_]:=a[n] stores
the elements of the sequence and helps to streamline computation.
Clear[a, n]
a[1]= 1;
a[n_]; = a[n]= a[n 1] (1/5)(n1)
first25= Table[N[a[n]], {n, 1, 25}]
The limit command does not work in this case, but the limit can be observed as 1.25.
Clear[minN, lim]
lim= 1.25
Do[{diff=Abs[a[n] lim], If[diff < .01, {minN= n, Abort[]}]}, {n, 2, 1000}]
minN
10.2 INFINITE SERIES
1. sn œ
a a 1 rn b
(1 r)
œ
n
2 ˆ1 ˆ "3 ‰ ‰
"
1 ˆ3‰
2. sn œ
a a 1 rn b
(1 r)
œ
" ‰n ‰
9 ‰ˆ
ˆ 100
1 ˆ 100
"
1 ˆ 100 ‰
3. sn œ
a a 1 rn b
(1 r)
œ
1 ˆ "# ‰
1 ˆ "# ‰
4. sn œ
1 (2)n
1 (2)
, a geometric series where krk 1 Ê divergence
5.
"
(n 1)(n #)
œ
"
n1
n
Ê n lim
s œ
Ä_ n
Ê n lim
s œ
Ä_ n
Ê n lim
s œ
Ä_ n
"
n#
2
1 ˆ "3 ‰
"
ˆ #3 ‰
œ3
9 ‰
ˆ 100
" ‰
1 ˆ 100
œ
œ
"
11
2
3
Ê sn œ ˆ #" 3" ‰ ˆ 3" 4" ‰ á ˆ n " 1
" ‰
n#
œ
"
#
"
n#
Ê n lim
s œ
Ä_ n
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
#
578
6.
Chapter 10 Infinite Sequences and Series
œ
5
n(n 1)
5
n
5
n1
Ê sn œ ˆ5 25 ‰ ˆ 25 35 ‰ ˆ 35 45 ‰ á ˆ n 5 1 n5 ‰ ˆ n5
5 ‰
n1
œ 5
5
n1
Ê n lim
s œ5
Ä_ n
7. 1
8.
"
16
9.
7
4
"
4
10. 5
"
16
"
64
7
16
5
4
"
256
"
64
7
64
5
16
á , the sum of this geometric series is
á , the sum of this geometric series is
á , the sum of this geometric series is
5
64
5
1 ˆ "# ‰
"
1 ˆ "3 ‰
œ 10
œ
3
#
"
1 ˆ "3 ‰
œ 10
œ
3
#
14. 2
4
5
"
1 ˆ "5 ‰
8
25
œ2
16
125
5
6
" ‰
25
œ 17
6
œ
4
5
"
1#
7
3
5
1 ˆ "4 ‰
œ4
" ‰
#7
á , is the sum of two geometric series; the sum is
" ‰
#7
á , is the difference of two geometric series; the sum is
ˆ 18
á œ 2 ˆ1
15. Series is geometric with r œ
œ
"
1 ˆ "4 ‰
17
#
13. (1 1) ˆ 1# "5 ‰ ˆ 14
1
1 ˆ "# ‰
ˆ 74 ‰
1 ˆ "4 ‰
œ
œ
23
#
12. (5 1) ˆ 5# "3 ‰ ˆ 54 9" ‰ ˆ 58
5
1 ˆ "# ‰
" ‰
ˆ 16
1 ˆ "4 ‰
á , the sum of this geometric series is
11. (5 1) ˆ 5# "3 ‰ ˆ 54 9" ‰ ˆ 58
"
1 ˆ "4 ‰
2
5
" ‰
1#5
4
25
á , is the sum of two geometric series; the sum is
8
125
á ‰ ; the sum of this geometric series is 2 Š 1 "ˆ 2 ‰ ‹ œ
5
Ê ¹ 25 ¹ 1 Ê Converges to
2
5
1
1 25
œ
5
3
1
8
œ
1
7
16. Series is geometric with r œ 3 Ê ¹3¹ 1 Ê Diverges
17. Series is geometric with r œ
Ê ¹ 18 ¹ 1 Ê Converges to
1
8
1 18
18. Series is geometric with r œ 23 Ê ¹ 23 ¹ 1 Ê Converges to
_
19. 0.23 œ !
n œ0
_
21. 0.7 œ !
nœ0
23
100
7
10
ˆ 10" # ‰n œ
" ‰n
ˆ 10
œ
23
Š 100
‹
"
1 ˆ 100 ‰
7
Š 10
‹
1
"
Š 10
‹
œ
œ
_
nœ0
n œ0
_
n œ0
414
1000
n œ0
22. 0.d œ !
"
1 Š 10
‹
24. 1.414 œ 1 !
_
7
9
6
Š 100
‹
ˆ 10" $ ‰n œ 1
œ 25
20. 0.234 œ !
23
99
_
1 ‰ ˆ 6 ‰ ˆ " ‰n
23. 0.06 œ ! ˆ 10
œ
10
10
23
1 ˆ 23 ‰
œ
6
90
414
Š 1000
‹
"
1 Š 1000
‹
œ
d
10
234
1000
ˆ 10" $ ‰n œ
" ‰n
ˆ 10
œ
234
Š 1000
‹
"
1 Š 1000
‹
d
Š 10
‹
"
1 Š 10
‹
œ
d
9
"
15
œ1
414
999
œ
"413
999
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
œ
234
999
10
3
Section 10.2 Infinite Series
25. 1.24123 œ
124
100
_
!
123
10&
n œ0
_
26. 3.142857 œ 3 !
n œ0
œ lim
124
100
28.
lim nan 1b
nÄ_ an 2ban 3b
29.
lim 1
nÄ_ n 4
œ 0 Ê test inconclusive
30.
lim 2 n
nÄ_ n 3
œ lim
33.
34.
10
1Š
"
‹
10$
Š 142,857
' ‹
10
1Š
"
‹
10'
œ
124
100
œ3
123
10& 10#
142,857
10' 1
œ
œ
124
100
3,142,854
999,999
123
99,900
œ
œ
123,999
99,900
œ
41,333
33,300
116,402
37,037
œ 1 Á 0 Ê diverges
lim n
nÄ_ n 10
32.
Š 123& ‹
ˆ 10" ' ‰n œ 3
142,857
10'
27.
31.
1
nÄ_ 1
ˆ 10" $ ‰n œ
579
n2 n
2
nÄ_ n 5n 6
2n 1
nÄ_ 2n 5
œ lim
œ lim
œ lim
2
nÄ_ 2
œ 1 Á 0 Ê diverges
œ 0 Ê test inconclusive
1
nÄ_ 2n
lim cos 1n œ cos 0 œ 1 Á 0 Ê diverges
nÄ_
n
lim ne
nÄ_ e n
œ
n
lim n e
nÄ_ e 1
en
n
nÄ_ e
œ lim
œ lim
1
nÄ_ 1
œ 1 Á 0 Ê diverges
lim ln 1n œ _ Á 0 Ê diverges
nÄ_
lim cos n 1 œ does not exist Ê diverges
nÄ_
35. sk œ ˆ1 "2 ‰ ˆ "2 "3 ‰ ˆ "3 "4 ‰ á ˆ k " 1 k" ‰ ˆ k"
œ lim ˆ1
kÄ_
" ‰
k1
kÄ_
œ 1
"
k1
Ê
œ 1, series converges to 1
36. sk œ ˆ 31 34 ‰ ˆ 34 39 ‰ ˆ 39
œ lim Š3
" ‰
k1
3
‹
ak 1b2
3 ‰
16
á Š ak 3 1b2
3
k2 ‹
Š k32
3
‹
ak 1b2
œ 3
lim sk
kÄ_
3
ak 1b2
Ê
lim sk
kÄ_
œ 3, series converges to 3
37. sk œ ŠlnÈ2 lnÈ1‹ ŠlnÈ3 lnÈ2‹ ŠlnÈ4 lnÈ3‹ á ŠlnÈk lnÈk 1‹ ŠlnÈk 1 lnÈk‹
œ lnÈk 1 lnÈ1 œ lnÈk 1 Ê
lim sk œ lim lnÈk 1 œ _; series diverges
kÄ_
kÄ_
38. sk œ atan 1 tan 0b atan 2 tan 1b atan 3 tan 2b á atan k tan ak 1bb atan ak 1b tan kb
œ tan ak 1b tan 0 œ tan ak 1b Ê lim sk œ lim tan ak 1b œ does not exist; series diverges
kÄ_
kÄ_
39. sk œ ˆcos1 ˆ 12 ‰ cos1 ˆ 13 ‰‰ ˆcos1 ˆ 13 ‰ cos1 ˆ 14 ‰‰ ˆcos1 ˆ 14 ‰ cos1 ˆ 15 ‰‰ á
ˆcos1 ˆ 1k ‰ cos1 ˆ k 1 1 ‰‰ ˆcos1 ˆ k 1 1 ‰ cos1 ˆ k 1 # ‰‰ œ 13 cos1 ˆ k 1 # ‰
Ê
lim sk œ lim ’ 13 cos1 ˆ k 1 # ‰“ œ
kÄ_
kÄ_
1
3
1
2
œ 16 , series converges to
1
6
40. sk œ ŠÈ5 È4‹ ŠÈ6 È5‹ ŠÈ7 È6‹ á ŠÈk 3 Èk 2‹ ŠÈk 4 Èk 3‹
œ Èk 4 2 Ê
lim sk œ lim ’Èk 4 2“ œ _; series diverges
kÄ_
kÄ_
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
580
41.
42.
Chapter 10 Infinite Sequences and Series
4
"
"
"‰
" ‰
ˆ
ˆ" "‰ ˆ"
(4n 3)(4n 1) œ 4n 3 4n 1 Ê sk œ 1 5 5 9 9 13
ˆ 4k " 3 4k " 1 ‰ œ 1 4k " 1 Ê lim sk œ lim ˆ1 4k " 1 ‰ œ 1
kÄ_
kÄ_
œ
6
(2n 1)(2n 1)
A
2n 1
A(2n 1) B(2n 1)
(2n 1)(2n 1)
œ
B
2n 1
á ˆ 4k " 7
" ‰
4k 3
Ê A(2n 1) B(2n 1) œ 6 Ê (2A 2B)n (A B) œ 6
k
k
2A 2B œ 0
ABœ0
6
Ê œ
ʜ
Ê 2A œ 6 Ê A œ 3 and B œ 3. Hence, ! (2n 1)(2n
œ 3 ! ˆ #n " 1
1)
A Bœ6
ABœ6
n œ1
nœ1
œ 3 Š "1
"
3
lim 3 ˆ1
kÄ_
43.
40n
(2n1)# (2n1)#
"
3
"
5
" ‰
#k 1
œ
A
(2n1)
"
5
"
7
á
"
#(k 1) 1
"
2k 1
"
#k 1 ‹
œ
A(2n1)(2n1)# B(2n1)# C(2n1)(2n1)# D(2n1)#
(2n1)# (2n1)#
#
#
œ 3 ˆ1
" ‰
#k 1
" ‰
#n 1
Ê the sum is
œ3
B
(2n1)#
C
(2n1)
#
D
(2n1)#
Ê A(2n 1)(2n 1)# B(2n 1) C(2n 1)(2n 1) D(2n 1) œ 40n
Ê A a8n$ 4n# 2n 1b B a4n# 4n 1b C a8n$ 4n# 2n 1b œ D a4n# 4n 1b œ 40n
Ê (8A 8C)n$ (4A 4B 4C 4D)n# (2A 4B 2C 4D)n (A B C D) œ 40n
Ú
Ú
8A 8C œ 0
8A 8C œ 0
Ý
Ý
Ý
Ý
B Dœ 0
4A 4B 4C 4D œ 0
A BC Dœ 0
Ê œ
Ê 4B œ 20 Ê B œ 5
Ê Û
Ê Û
œ
œ
2D œ 20
2A
4B
2C
4D
40
A
2
B
C
2D
20
2B
Ý
Ý
Ý
Ý
Ü A B C D œ 0
Ü A B C D œ 0
k
ACœ0
Ê C œ 0 and A œ 0. Hence, ! ’ (#n1)40n
and D œ 5 Ê œ
# (2n1)# “
A 5 C 5 œ 0
n œ1
k
œ 5 ! ’ (#n" 1)#
n œ1
44.
"
(#n1)# “
œ 5 Š1
"
(2k1)# ‹
2n 1
n# (n 1)#
"
n#
Ê
œ
45. sk œ Š1
Ê
Š È"2
kÄ_
" ‰
#"Î#
"
ˆ #"Î#
lim sk œ
kÄ_
47. sk œ ˆ ln"3
œ ln"#
" ‰
ln #
"
#
"
1
œ
"
9
"
#5
"
#5
á
"
(2k1)# ‹
Ê
"
(#k1)#
"
(#k1)# ‹
œ5
" ‰
16
á ’ (k " 1)#
"
k# “
’ k"#
"
(k 1)# “
"
È4 ‹
á ŠÈ "
k1
"
Èk ‹
Š È"
k
"
Èk 1 ‹
œ1
"
Èk 1
œ1
"
ˆ #"Î$
" ‰
ln 3
"
(2(k1) 1)#
œ1
Š È"3
"
Èk 1 ‹
" ‰
#"Î$
"#
ˆ ln"4
"
ln (k 2)
"
(k 1)# “
"
È3 ‹
lim sk œ lim Š1
kÄ_
46. sk œ ˆ "#
Ê
kÄ_
"
È2 ‹
Ê sk œ ˆ1 4" ‰ ˆ 4" 9" ‰ ˆ 9"
lim sk œ lim ’1
kÄ_
"
9
Ê the sum is n lim
5 Š1
Ä_
"
(n 1)#
œ 5 Š 1"
" ‰
#"Î%
ˆ ln"5
á ˆ #1ÎÐ"k
" ‰
ln 4
1Ñ
" ‰
#1Îk
á Š ln (k" 1)
ˆ #1"Îk
"
ln k ‹
" ‰
#1ÎÐk1Ñ
Š ln (k" 2)
œ
"
#
"
#1ÎÐk1Ñ
"
ln (k 1) ‹
lim sk œ ln"#
kÄ_
48. sk œ ctan" (1) tan" (2)d ctan" (2) tan" (3)d á ctan" (k 1) tan" (k)d
ctan" (k) tan" (k 1)d œ tan" (1) tan" (k 1) Ê lim sk œ tan" (1)
kÄ_
49. convergent geometric series with sum
"
1 Š È" ‹
2
50. divergent geometric series with krk œ È2 1
œ
È2
È 2 1
1
#
œ
1
4
1
#
œ 14
œ 2 È2
51. convergent geometric series with sum
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Š 3# ‹
1 Š "# ‹
œ1
Section 10.2 Infinite Series
52. n lim
(1)n1 n Á 0 Ê diverges
Ä_
53. n lim
cos (n1) œ n lim
(1)n Á 0 Ê diverges
Ä_
Ä_
54. cos (n1) œ (1)n Ê convergent geometric series with sum
"
1Š
55. convergent geometric series with sum
56. n lim
ln
Ä_
"
3n
"
‹
e#
2
"
1 Š 10
‹
58. convergent geometric series with sum
"
1 Š "x ‹
59. difference of two geometric series with sum
ˆ1 "n ‰n œ lim ˆ1
60. n lim
Ä_
nÄ_
_
63. !
n œ1
n!
1000n
2n 3n
4n
since r œ
_
!
n œ1
64.
2n 3n
4n
_
n œ1
5
6
" ‰n
n
2œ
œ
Ê
2n
4n
_
!
¹ 12 ¹
nœ1
3n
4n
_
20
9
œ
18
9
2
9
x
x1
"
1 Š 23 ‹
"
1 Š 3" ‹
œ3
œ
3
#
3
#
œ e" Á 0 Ê diverges
62. n lim
Ä_
_
n
_
n
nn
n!
œ n lim
Ä_
_
n
n †n â n
1†#ân
n lim
n œ _ Ê diverges
Ä_
n
œ ! ˆ 21 ‰ ! ˆ 43 ‰ ; both œ ! ˆ 21 ‰ and ! ˆ 43 ‰ are geometric series, and both converge
nœ1
1 and r œ
nœ1
3
4
Ê
¹ 34 ¹
n œ1
1, respectivley Ê
n œ1
_
! ˆ 1 ‰n
2
n œ1
œ
1
2
1 12
_
n
œ 1 and ! ˆ 34 ‰ œ
nœ1
3
4
1 34
œ3Ê
œ 1 3 œ 4 by Theorem 8, part (1)
2n 4n
n
n
nÄ_ 3 4
lim
œ
e#
e # 1
œ _ Á 0 Ê diverges
œ!
1
2
œ
"
1 Š "5 ‹
œ _ Á 0 Ê diverges
57. convergent geometric series with sum
61. n lim
Ä_
581
œ
lim
nÄ_
_
_
n œ1
n œ1
2n
4n
3n
4n
"
"
ˆ 12 ‰n "
3 n
nÄ_ ˆ 4 ‰ "
œ lim
œ
1
1
œ 1 Á 0 Ê diverges by nth term test for divergence
65. ! ln ˆ n n 1 ‰ œ ! cln (n) ln (n 1)d Ê sk œ cln (1) ln (2)d cln (2) ln (3)d cln (3) ln (4)d á
cln (k 1) ln (k)d cln (k) ln (k 1)d œ ln (k 1) Ê
lim sk œ _, Ê diverges
kÄ_
66. n lim
a œ n lim
ln ˆ 2n n 1 ‰ œ ln ˆ "# ‰ Á 0 Ê diverges
Ä_ n
Ä_
67. convergent geometric series with sum
68. divergent geometric series with krk œ
_
_
n œ0
n œ0
"
1 ˆ 1e ‰
e1
1e
¸
œ
23.141
22.459
1
1e
1
69. ! (1)n xn œ ! (x)n ; a œ 1, r œ x; converges to
_
_
n œ0
n œ0
"
1 (x)
n
70. ! (1)n x2n œ ! ax# b ; a œ 1, r œ x# ; converges to
œ
"
1 x#
"
1x
for kxk 1
for kxk 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
582
Chapter 10 Infinite Sequences and Series
71. a œ 3, r œ
_
72. !
n œ0
œ
(1)n
#
x1
#
; converges to
_
ˆ 3 "sin x ‰n œ !
n œ0
3 sin x
2(4 sin x)
œ
3 sin x
8 2 sin x
3
1 Šx
ˆ 3 "sin x ‰n ; a œ
"
#
"
1 2x
74. a œ 1, r œ x"# ; converges to
for k2xk 1 or kxk
"
#
; converges to
77. a œ 1, r œ sin x; converges to
_
79. (a) !
nœ2
_
80. (a) !
nœ1
"
1 (x 1)
"
1 Š3
x
# ‹
"
1 sin x
œ
œ
"
#x
for kx 1k 1 or 2 x 0
for kln xk 1 or e" x e
_
5
(n 2)(n 3)
(b) !
n œ0
_
n œ3
"
4
(b) one example is 3#
(c) one example is 1
"
#
for all x‰
for x Á (2k 1) 1# , k an integer
(b) !
"
#
" ‹
1 Š 3 sin
x
for ¸ 3 # x ¸ 1 or 1 x 5
2
x1
"
(n 4)(n 5)
81. (a) one example is
ˆ "# ‰
#
"
1 ln x
78. a œ 1, r œ ln x; converges to
"
3 sin x
Ÿ
; converges to
x
¸1¸
" ‹ œ x# 1 for x# 1 or kxk 1.
#
x
75. a œ 1, r œ (x 1)n ; converges to
3x
#
"
3 sin x
,rœ
"
#
"
1Š
"
4
"
#
Ÿ
for all x ˆsince
73. a œ 1, r œ 2x; converges to
76. a œ 1, r œ
6
x"
" œ 3 x for 1 # 1 or 1 x 3
# ‹
"
8
"
16
á œ
3
4
3
8
3
16
"
4
"
8
Š "# ‹
1 Š "# ‹
á œ
"
16
_
"
(n 2)(n 3)
(c) !
5
(n 2)(n 1)
(c) !
n œ5
_
nœ20
"
(n 3)(n #)
5
(n 19)(n 18)
œ1
Š 3# ‹
1 Š "# ‹
á œ 1
œ 3
Š "# ‹
1 Š "# ‹
œ 0.
_
Š k# ‹
n œ0
1 Š "# ‹
n 1
82. The series ! kˆ 12 ‰
is a geometric series whose sum is
œ k where k can be any positive or negative number.
_
_
_
_
_
nœ1
nœ1
nœ1
nœ1
nœ1
_
_
_
_
_
nœ1
nœ1
nœ1
nœ1
nœ1
n
n
83. Let an œ bn œ ˆ "# ‰ . Then ! an œ ! bn œ ! ˆ "# ‰ œ 1, while ! Š bann ‹ œ ! (1) diverges.
n
n
n
84. Let an œ bn œ ˆ "# ‰ . Then ! an œ ! bn œ ! ˆ "# ‰ œ 1, while ! aan bn b œ ! ˆ 4" ‰ œ
n
n
_
85. Let an œ ˆ 4" ‰ and bn œ ˆ "# ‰ . Then A œ ! an œ
n œ1
"
3
_
_
_
nœ1
nœ1
nœ1
"
3
Á AB.
n
, B œ ! bn œ 1 and ! Š bann ‹ œ ! ˆ "# ‰ œ 1 Á
86. Yes: ! Š a"n ‹ diverges. The reasoning: ! an converges Ê an Ä 0 Ê
"
an
A
B
.
Ä _ Ê ! Š a"n ‹ diverges by the
nth-Term Test.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.3 The Integral Test
87. Since the sum of a finite number of terms is finite, adding or subtracting a finite number of terms from a series
that diverges does not change the divergence of the series.
88. Let An œ a" a# á an and n lim
A œ A. Assume ! aan bn b converges to S. Let
Ä_ n
Sn œ (a" b" ) (a# b# ) á (an bn ) Ê Sn œ (a" a# á an ) (b" b# á bn )
Ê b" b# á bn œ Sn An Ê n lim
ab" b# á bn b œ S A Ê ! bn converges. This
Ä_
contradicts the assumption that ! bn diverges; therefore, ! aan bn b diverges.
89. (a)
(b)
2
1r
œ5 Ê
Š 13
2 ‹
1r
2
5
œ5 Ê
œ1r Ê rœ
13
10
90. 1 eb e2b á œ
#
; 2 2 ˆ 35 ‰ 2 ˆ 35 ‰ á
3
5
3
œ 1 r Ê r œ 10
;
"
1 e b
"
9
œ9 Ê
13
2
13
#
3 ‰
ˆ 10
œ 1 eb Ê eb œ
13
#
3 ‰#
ˆ 10
13
#
3 ‰$
ˆ 10
á
Ê b œ ln ˆ 89 ‰
8
9
91. sn œ 1 2r r# 2r$ r% 2r& á r2n 2r2n1 , n œ 0, 1, á
Ê sn œ a1 r# r% á r2n b a2r 2r$ 2r& á 2r2n1 b Ê n lim
s œ
Ä_ n
1 2r
œ 1 r# , if kr# k 1 or krk 1
92. L sn œ
a
1r
a a1 rn b
1r
œ
#
#
#
94. (a) L" œ 3, L# œ 3 ˆ 43 ‰ , L$ œ 3 ˆ 43 ‰ , á , Ln œ 3 ˆ 43 ‰
nc1
"
#
á œ
4
1
"
#
An œ
lim
È3
4
È3
ˆ " ‰2
4 ‹ 3
nÄ_
! 3a4bk2 Š
È3 8
ˆ5‰
4
œ
kœ2
An œ
È3
4
œ
È3
ˆ " ‰2
4 ‹ 33
n
2r
1 r#
È3
1#
, A$ œ A# 3a4bŠ
, A5 œ A4 3a4b3 Š
È3
ˆ " ‰k1
4 ‹ 32
È3
lim
nÄ_ Œ 4
œ
n
3 È 3 Œ!
kœ2
È3
4
È3
ˆ " ‰2
4 ‹ 32
È3
ˆ " ‰2
4 ‹ 34 ,
œ
n
k 1
œ
œ_
È3 2
4 s , we see that
È3
È3
È3
4 12 #7 ,
È3
4 ,
A" œ
n
4kc$
.
9k 1
œ
È3
4
3È3Œ!
3È 3Œ 1 4 œ
È3
4
1 ‰
3È3ˆ 20
œ
kœ2
È3
4
nc1
...,
! 3È3a4bk$ ˆ "9 ‰
4kc$
9k 1
œ 8 m#
Ê n lim
L œ n lim
3 ˆ 43 ‰
Ä_ n
Ä_
(b) Using the fact that the area of an equilateral triangle of side length s is
A% œ A$ 3a4b2 Š
arn
1 r
93. area œ 2# ŠÈ2‹ (1)# Š È"2 ‹ á œ 4 2 1
A# œ A" 3Š
"
1 r#
1
36
9
kœ2
È3
ˆ
4 1
53 ‰
œ 85 A"
10.3 THE INTEGRAL TEST
1. faxb œ
1
x2
œ lim
bÄ_
2. faxb œ
1
x0.2
œ lim
bÄ_
_1
1; '1
is positive, continuous, and decreasing for x
_1
ˆ 1b 1‰ œ 1 Ê '
1
_
x2
dx converges Ê !
n œ1
1
n2
_
ˆ 54 b0.8 54 ‰ œ _ Ê '
1
1
x0.2
_
dx œ lim
bÄ_
'1b x1
2
dx œ lim
bÄ_
b
’ 1x “
1
converges
1; '1
is positive, continuous, and decreasing for x
_
x2
1
x0.2
dx œ lim
bÄ_
'1b x1
0.2
dx œ lim
bÄ_
dx diverges Ê ! n10.2 diverges
n œ1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
b
’ 54 x0.8 “
1
583
584
Chapter 10 Infinite Sequences and Series
3. faxb œ
œ
1
x2 4
lim ˆ 1 tan1 b2
bÄ_ 2
4. faxb œ
1
x4
_
1; '1
is positive, continuous, and decreasing for x
1
1 1 ‰
2 tan
2
œ
1
4
_
2
_
1; '1
is positive, continuous, and decreasing for x
œ lim
ˆ 2e12b
bÄ_
1
xaln xb2
œ lim
_
'3
x
x2 4
x
x2 4
_
œ
1 ‰
2e2
1
2e2
_
œ
1 ‰
ln 2
1
ln 2
_ ln x2
'3
dx œ lim
bÄ_
ln x2
x
_
Ê '2
'3
x
x2 4
dx converges Ê !
n œ3
n œ1
bÄ_
'3
ln x
x
2
_
'7
2
x
exÎ3
n œ3
n œ2
2
n
enÎ3
n œ1
10. faxb œ
œ
1
e1Î3
x4
x2 2x 1
decreasing for x
œ lim
bÄ_
_
Ê !
nœ8
2
x
exÎ3
18b
Š 3a6b
‹
ebÎ3
bÄ_
Ê !
'7
b
1
œ
4
e2Î3
x4
a x 1 b2
_
8; '8
’lnlx 1l
bÄ_
9
e1
bÄ_
n4
n2 2n 1 diverges
16
e4Î3
x4
ax 1b2
_
Ê !
nœ2
36
e2
bÄ_
18x
exÎ3
!
n œ7
"
10
’ ln1x “
bÄ_
2
converges
3;
_
ˆ 12 lnab2 4b 12 lna13b‰ œ _ Ê '
3
_ ln x2
a2aln bb 2aln 3bb œ _ Ê '3
b
x
x2 4
dx
x
3;
dx
54
“
exÎ3 7
327
e7Î3
2
n
enÎ3
0 for x 6, thus f is decreasing for x
œ lim
bÄ_
_
Ê '7
x2
exÎ3
Š 3b
2
18b 54
ebÎ3
b
x1
ax 1b2
œ 2
1
4
b
dx converges Ê !
n œ7
13. diverges; by the nth-Term Test for Divergence, n lim
Ä_
3
ax 1b2
œ
n2
converges
enÎ3
1
16
2
25
3
36
7x
ax 1 b 3
dx• œ lim ”'8
bÄ_
b
_
ln 7 37 ‰ œ _ Ê '8
0
1
327
‹
e7Î3
7;
converges
dx '8
3
b1
_
2, f is positive for x 4, and f w axb œ
ˆlnlb 1l
n4
n2 2n 1
b
dx œ lim
2
0 for x e, thus f is decreasing for x
x a x 6 b
3exÎ3
œ
_
”'8
bÄ_
dx œ lim
11. converges; a geometric series with r œ
2 ln x2
x2
bÄ_
3
327
e7Î3
e
bÄ_
'2b xaln1xb
n œ3
1, f w axb œ
’ e3xxÎ3
25
e5Î3
œ lim
_
dx œ lim
n œ3
is continuous for x
b
1
2
! lnnn diverges
ln 4
2
ˆ b54
‰
Î3
œ lim
3
x 1 “8
b
’ 12 e2x “
lim
! n2 n 4 diverges
_
2
dx œ lim
327
e7Î3
2
8
2, f w axb œ
is positive and continuous for x
bÄ_
œ lim
1
5
’2aln xb“ œ lim
bÄ_
_
dx œ lim
_
bÄ_
b
’lnlx 4l“
lim
0 for x 2, thus f is decreasing for x
bÄ_
3
b
dx œ lim
_
4 x2
ax2 4b2
’ 21 lnax2 4b“ œ lim
is positive and continuous for x
b
1
naln nb2
n œ2
1, f w axb œ
1
xaln xb2
b
dx œ lim
_
bÄ_
x2
exÎ3
_
2; '2
_
1
xaln xb2
2
2
diverges Ê ! lnnn diverges Ê ! lnnn œ
9. faxb œ
1
n œ1
_
dx œ lim
x
'1b e2x dx œ
Ê '1 e2x dx converges Ê ! e2n converges
is positive and continuous for x
b
bÄ_
_
diverges Ê ! n2 n 4 diverges Ê ! n2 n 4 œ
8. faxb œ
bÄ_
bÄ_
1; '1 e2x dx œ lim
is positive, continuous, and decreasing for x
ˆ ln1b
bÄ_
7. faxb œ
’ 21 tan1 x2 “
n œ1
5. faxb œ e2x is positive, continuous, and decreasing for x
6. faxb œ
bÄ_
'1b x 1 4 dx œ
dx œ lim
1
x4
b
lim
2
n œ1
_
bÄ_
2
bÄ_
_
alnlb 4l ln 5b œ _ Ê '1 x 1 4 dx diverges Ê ! n 1 4 diverges
œ lim
'1b x 1 4 dx œ
dx œ lim
_
Ê '1 x 1 4 dx converges Ê ! n 1 4 converges
1
1 1
2 tan
2
1
x2 4
_
!
n œ8
x4
ax 1b2
n4
n2 2n 1
0 for x 7, thus f is
1
x1
dx '8
3
ax 1b2
dx diverges
diverges
12. converges; a geometric series with r œ
n
n1
b
œ1Á0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
e
1
dx•
Section 10.3 The Integral Test
14. diverges by the Integral Test; '1
n
_
15. diverges; !
n œ1
3
Èn
_
16. converges; !
n œ1
_
"
Èn
œ3!
nœ1
2
nÈ n
_
dx œ 5 ln (n 1) 5 ln 2 Ê '1
5
x1
_
n œ1
8
n
_
œ 2 !
n œ1
_
œ 8 !
nœ1
dx Ä _
, which is a divergent p-series (p œ #" )
"
n$Î#
, which is a convergent p-series (p œ 3# )
17. converges; a geometric series with r œ
18. diverges; !
5
x1
"
8
1
_
and since !
1
n
nœ1
19. diverges by the Integral Test:
_
"
n
diverges, 8 !
n œ1
'2n lnxx dx œ "# aln# n ln 2b
Ê
t œ ln x ×
dt œ dx
Ä
x
Õ dx œ et dt Ø
œ lim 2ebÎ2 (b 2) 2eÐln 2ÑÎ2 (ln 2 2)‘ œ _
20. diverges by the Integral Test:
'2_ lnÈxx dx; Ô
1
n
diverges
'2_ lnxx dx
'ln_2 tetÎ2 dt œ
Ä _
b
lim 2tetÎ2 4etÎ2 ‘ ln 2
bÄ_
bÄ_
21. converges; a geometric series with r œ
22. diverges; n lim
Ä_
_
23. diverges; !
n œ0
2
n 1
5n
4n 3
_
œ 2 !
n œ0
"
n1
1
ˆ ln 5 ‰ ˆ 54 ‰n œ _ Á 0
œ n lim
Ä _ ln 4
5n ln 5
4n ln 4
œ n lim
Ä_
2
3
, which diverges by the Integral Test
24. diverges by the Integral Test:
'1n 2xdx 1 œ "# ln (2n 1)
25. diverges; n lim
a œ n lim
Ä_ n
Ä_
2n
n1
26. diverges by the Integral Test:
'1n Èx ˆÈdxx 1‰ ; – u œ
27. diverges; n lim
Ä_
Èn
ln n
œ n lim
Ä_
œ n lim
Ä_
2n ln 2
1
œ_Á0
Èx "
du œ
"
Š 2È
‹
n
Š "n ‹
œ n lim
Ä_
Èn
#
Ä _ as n Ä _
dx
Èx
Ènb1 du
'
—Ä 2
u
œ ln ˆÈn 1‰ ln 2 Ä _ as n Ä _
œ_Á0
ˆ1 n" ‰n œ e Á 0
28. diverges; n lim
a œ n lim
Ä_ n
Ä_
29. diverges; a geometric series with r œ
"
ln #
30. converges; a geometric series with r œ
31. converges by the Integral Test:
¸ 1.44 1
"
ln 3
¸ 0.91 1
'3_ (ln x) ÈŠ(ln‹x) 1 dx; ”
"
x
#
u œ ln x
Ä
du œ "x dx •
'ln_3
"
uÈ u# 1
du
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
585
586
Chapter 10 Infinite Sequences and Series
b
œ lim csec" kukd ln 3 œ lim csec" b sec" (ln 3)d œ lim cos" ˆ "b ‰ sec" (ln 3)‘
bÄ_
bÄ_
œ cos" (0) sec" (ln 3) œ
1
#
32. converges by the Integral Test:
'1_ x a1 "ln xb dx œ '1_ 1 Š(ln‹x)
"
x
#
œ lim ctan" ud 0 œ lim atan" b tan" 0b œ
b
bÄ_
bÄ_
sec" (ln 3) ¸ 1.1439
bÄ_
1
#
0œ
dx; ”
#
'0_ 1"u
u œ ln x
Ä
du œ "x dx •
#
du
1
#
33. diverges by the nth-Term Test for divergence; n lim
n sin ˆ "n ‰ œ n lim
Ä_
Ä_
sin ˆ "n ‰
ˆ "n ‰
œ lim
34. diverges by the nth-Term Test for divergence; n lim
n tan ˆ "n ‰ œ n lim
Ä_
Ä_
tan ˆ "n ‰
ˆ "n ‰
œ n lim
Ä_
xÄ0
œ1Á0
sin x
x
Š n"# ‹ sec# ˆ n" ‰
Š n"# ‹
œ n lim
sec# ˆ "n ‰ œ sec# 0 œ 1 Á 0
Ä_
35. converges by the Integral Test:
'1_ 1 e e
x
1
#
œ lim atan" b tan" eb œ
bÄ_
36. converges by the Integral Test:
œ lim 2 ln
bÄ_
u ‘b
u1 e
dx; ”
2x
'e_
u œ ex
Ä
du œ ex dx •
"
1 u#
ctan" ud e
du œ n lim
Ä_
b
tan" e ¸ 0.35
_
'1
u œ ex ×
_
_
dx; du œ ex dx Ä 'e u(1 2 u) du œ 'e ˆ 2u
Õ dx œ " du Ø
u
Ô
2
1 ex
2 ‰
u1
du
œ lim 2 ln ˆ b b 1 ‰ 2 ln ˆ e e 1 ‰ œ 2 ln 1 2 ln ˆ e e 1 ‰ œ 2 ln ˆ e e 1 ‰ ¸ 0.63
bÄ_
37. converges by the Integral Test:
38. diverges by the Integral Test:
'1_ 81tancx x dx; ” u œ tan dx x •
"
"
#
du œ
1 x#
'1_ x x1 dx; ” u œ x
39. converges by the Integral Test:
#
#
1
Ä
du œ 2x dx •
'1_ sech x dx œ 2
Ä
x
x #
bÄ_
#
'2_ du4 œ
"
#
'1b 1 eae b
lim
'11ÎÎ42 8u du œ c4u# d 11ÎÎ24 œ 4 Š 14
b
lim #" ln u‘ 2 œ lim
1#
16 ‹
"
bÄ_ #
bÄ_
œ
31 #
4
(ln b ln 2) œ _
dx œ 2 lim ctan" ex d 1
b
bÄ_
œ 2 lim atan" eb tan" eb œ 1 2 tan" e ¸ 0.71
bÄ_
40. converges by the Integral Test:
'1_ sech# x dx œ
œ 1 tanh 1 ¸ 0.76
41.
'1_ ˆ x a 2 x " 4 ‰ dx œ
a
lim (bb2)4
bÄ_
lim
bÄ_
'1b sech# x dx œ
lim ca ln kx 2k ln kx 4kd 1 œ lim ln
b
bÄ_
œ a lim (b 2)
bÄ_
bÄ_
a 1
lim ctanh xd b1 œ lim (tanh b tanh 1)
bÄ_
(b 2)a
b4
bÄ_
ln ˆ 35 ‰ ;
a
_, a 1
œœ
Ê the series converges to ln ˆ 53 ‰ if a œ 1 and diverges to _ if
1, a œ 1
a 1. If a 1, the terms of the series eventually become negative and the Integral Test does not apply. From
that point on, however, the series behaves like a negative multiple of the harmonic series, and so it diverges.
42.
'3_ ˆ x " 1 x 2a 1 ‰ dx œ
"
2ac1
b Ä _ #a(b 1)
œ lim
b
lim ’ln ¹ (xx1)12a ¹“ œ lim ln
bÄ_
œ
3
bÄ_
b1
(b 1)2a
b"
ln ˆ 422a ‰ ; lim
2a
b Ä _ (b 1)
1, a œ "#
Ê the series converges to ln ˆ #4 ‰ œ ln 2 if a œ
_, a "#
"
#
and diverges to _ if
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.3 The Integral Test
if a
"
#
. If a
"
#
587
, the terms of the series eventually become negative and the Integral Test does not apply.
From that point on, however, the series behaves like a negative multiple of the harmonic series, and so it diverges.
43. (a)
(b) There are (13)(365)(24)(60)(60) a10* b seconds in 13 billion years; by part (a) sn Ÿ 1 ln n where
n œ (13)(365)(24)(60)(60) a10* b Ê sn Ÿ 1 ln a(13)(365)(24)(60)(60) a10* bb
œ 1 ln (13) ln (365) ln (24) 2 ln (60) 9 ln (10) ¸ 41.55
_
44. No, because !
n œ1
"
nx
œ
"
x
_
!
n œ1
"
n
_
and !
n œ1
"
n
diverges
_
_
_
nœ1
nœ1
nœ1
45. Yes. If ! an is a divergent series of positive numbers, then ˆ "# ‰ ! an œ ! ˆ a#n ‰ also diverges and
an
#
an .
_
There is no “smallest" divergent series of positive numbers: for any divergent series ! an of positive numbers
n œ1
_
! ˆ an ‰ has smaller terms and still diverges.
#
n œ1
_
_
_
nœ1
nœ1
nœ1
46. No, if ! an is a convergent series of positive numbers, then 2 ! an œ ! 2an also converges, and 2an
an .
There is no “largest" convergent series of positive numbers.
47. (a) Both integrals can represent the area under the curve faxb œ
1
Èx 1 ,
and the sum s50 can be considered an
50
approximation of either integral using rectangles with ?x œ 1. The sum s50 œ !
n œ1
integral
1
Èn 1
is an overestimate of the
'151 Èx1 1 dx. The sum s50 represents a left-hand sum (that is, the we are choosing the left-hand endpoint of
each subinterval for ci ) and because f is a decreasing function, the value of f is a maximum at the left-hand endpoint of
each sub interval. The area of each rectangle overestimates the true area, thus '1
51
manner, s50 underestimates the integral '0
50
1
Èx 1 dx.
1
Èx 1 dx
50
!
n œ1
1
Èn 1 .
In a similar
In this case, the sum s50 represents a right-hand sum and because
f is a decreasing function, the value of f is aminimum at the right-hand endpoint of each subinterval. The area of each
50
rectangle underestimates the true area, thus !
n œ1
1
Èn 1
œ ’2Èx 1“ œ 2È52 2È2 ¸ 11.6 and '0
51
50
1
50
11.6 !
n œ1
1
Èn 1
Ên
1
Èx 1 dx
50
1
Èx 1 dx.
Evaluating the integrals we find '1
51
1
Èx 1 dx
50
œ ’2Èx 1“ œ 2È51 2È1 ¸ 12.3. Thus,
0
12.3.
nb1
(b) sn 1000 Ê '1
'0
1
Èx 1 dx
nb1
œ ’2Èx 1“
1
2
œ 2Èn 1 2È2 1000 Ê n Š500 2È2‹ ¸ 251414.2
251415.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
588
Chapter 10 Infinite Sequences and Series
30
48. (a) Since we are using s30 œ !
n œ1
1
n4
_
to estimate !
n œ1
of the area under the curve faxb œ
1
x4
_
the error is given by !
1
n4 ,
nœ31
1
n4 .
We can consider this sum as an estimate
30 using rectangles with ?x œ 1 and ci is the right-hand endpoint of
when x
each subinterval. Since f is a decreasing function, the value of f is a minimum at the right-hand endpoint of each
_
subinterval, thus !
nœ31
'30
_
1
n4
1
x4 dx
œ lim '30
b
bÄ_
1
x4 dx
b
œ lim ’ 3x1 3 “ œ lim Š 3b1 3
bÄ_
bÄ_
30
1
‹
3a30b3
¸ 1.23 ‚ 105 .
Thus the error 1.23 ‚ 105 Þ
(b) We want S sn 0.000001 Ê 'n
_
œ
lim ˆ 3b1 3
bÄ_
1 ‰
3n3
œ
1
3n3
49. We want S sn 0.01 Ê 'n
_
œ
1
2n2
_
1
x3 dx
1
x2 4 dx
bÄ_
n œ1
1
n2 4
_
10
n0.1
b
bÄ_
1
x2 4 dx
1
x4 dx
b
œ lim ’ 3x1 3 “
bÄ_
n
70.
b
œ lim ’ 2x1 2 “ œ lim ˆ 2b1 2
bÄ_
bÄ_
n
1 ‰
2n2
¸ 1.195
1
n3
n œ1
bÄ_
1
1 ˆ n ‰
2 tan
2
1
x3 dx
bÄ_
8
b
1
4
œ lim 'n
8 Ê S ¸ s8 œ !
0.1 Ê lim 'n
b
¸ 69.336 Ê n
É 1000000
3
3
1
x3 dx
œ lim 'n
b
œ lim ’ 21 tan1 ˆ 2x ‰“
bÄ_
n
0.1 Ê n 2tanˆ 12 0.2‰ ¸ 9.867 Ê n
10 Ê S ¸ s10
1
x1.1 dx
0.00001 Ê 'n
_
1
x1.1 dx
œ lim 'n
b
bÄ_
1
x1.1 dx
b
œ lim ’ x10
lim ˆ b10
0.1 “ œ
0.1
bÄ_
bÄ_
n
10 ‰
n0.1
0.00001 Ê n 100000010 Ê n 1060
52. S sn 0.01 Ê 'n
_
œ
_
1
x4 dx
¸ 0.57
51. S sn 0.00001 Ê 'n
œ
_
0.01 Ê 'n
œ lim ˆ 12 tan1 ˆ b2 ‰ 12 tan1 ˆ n2 ‰‰ œ
10
0.000001 Ê 'n
0.000001 Ê n
0.01 Ê n È50 ¸ 7.071 Ê n
50. We want S sn 0.1 Ê 'n
œ!
1
x4 dx
lim Š 2aln1bb2
bÄ_
1
dx
xaln xb3
1
‹
2aln nb2
n
n
k œ1
k œ1
0.01 Ê 'n
_
œ
1
2aln nb2
œ lim 'n
b
1
dx
xaln xb3
bÄ_
È50
0.01 Ê n e
1
dx
xaln xb3
b
œ lim ’ 2aln1xb2 “
bÄ_
¸ 1177.405 Ê n
n
1178
53. Let An œ ! ak and Bn œ ! 2k aa2k b , where {ak } is a nonincreasing sequence of positive terms converging to
0. Note that {An } and {Bn } are nondecreasing sequences of positive terms. Now,
Bn œ 2a# 4a% 8a) á 2n aa2n b œ 2a# a2a% 2a% b a2a) 2a) 2a) 2a) b á
ˆ2aa2n b 2aa2n b á 2aa2n b ‰ Ÿ 2a" 2a# a2a$ 2a% b a2a& 2a' 2a( 2a) b á
ðóóóóóóóóóóóóóóñóóóóóóóóóóóóóóò
2n1 terms
_
ˆ2aa2nc1 b 2aa2nc1 1b á 2aa2n b ‰ œ 2Aa2n b Ÿ 2 ! ak . Therefore if ! ak converges,
k œ1
then {Bn } is bounded above Ê ! 2k aa2k b converges. Conversely,
_
An œ a" aa# a$ b aa% a& a' a( b á an a" 2a# 4a% á 2n aa2n b œ a" Bn a" ! 2k aa2k b .
k œ1
_
Therefore, if ! 2 aa2k b converges, then {An } is bounded above and hence converges.
k
k œ1
54. (a) aa2n b œ
_
Ê !
n œ2
"
2n ln a2n b
"
n ln n
œ
"
2n †n(ln 2)
_
_
n œ2
n œ2
Ê ! 2 n a a2 n b œ ! 2 n
"
#n †n(ln 2)
œ
"
ln #
_
!
n œ2
"
n
, which diverges
diverges.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.3 The Integral Test
"
#np
(b) aa2n b œ
55. (a)
_
_
n œ1
nœ1
Ê ! 2 n a a2 n b œ ! 2 n †
"
#pc1
'2_ x(lndxx)
u œ ln x
• Ä
du œ dx
x
œœ
;”
_
n
œ ! ˆ #p"c1 ‰ , a geometric series that
nœ1
'2_ x dxln x œ
p œ 1.
cpb1
lim ’ up 1 “
bÄ_
b
ln 2
œ lim Š 1 " p ‹ cbp1 (ln 2)p1 d
bÄ_
Ê the improper integral converges if p 1 and diverges if p 1.
_, p "
For p œ 1:
"
a2n bpc1
nœ1
'ln_2 ucp du œ
(ln 2)cpb1 , p 1
"
p1
_
œ!
1 or p 1, but diverges if p Ÿ 1.
converges if
p
"
#np
589
lim cln (ln x)d b2 œ lim cln (ln b) ln (ln 2)d œ _, so the improper integral diverges if
bÄ_
bÄ_
_
"
n(ln n)p
(b) Since the series and the integral converge or diverge together, !
n œ2
converges if and only if p 1.
56. (a) p œ 1 Ê the series diverges
(b) p œ 1.01 Ê the series converges
_
(c) !
n œ2
"
n aln n$ b
"
3
œ
_
"
n(ln n)
!
n œ2
; p œ 1 Ê the series diverges
(d) p œ 3 Ê the series converges
57. (a) From Fig. 10.11(a) in the text with f(x) œ
Ÿ 1 '1 f(x) dx Ê ln (n 1) Ÿ 1
n
Ÿ ˆ1
"
#
"
3
á
"‰
n
"
#
"
x
and ak œ
"
3
á
(b) From the graph in Fig. 10.11(b) with f(x) œ
Ê 0
cln (n 1) ln nd œ ˆ1
If we define an œ 1
"
#
œ
nb1
, we have '1
"
n
"
3
"
n
"
x
"
n1
"
"
# 3
,
"
x
dx Ÿ 1
"
#
"
3
á
"
n
Ÿ 1 ln n Ê 0 Ÿ ln (n 1) ln n
ln n Ÿ 1. Therefore the sequence ˜ˆ1
1 and below by 0.
"
n1
"
k
nb1
'n
"
x
á
"
n 1
"
#
"
3
á n" ‰ ln n™ is bounded above by
dx œ ln (n 1) ln n
ln (n 1)‰ ˆ1
"
#
"
3
á
"
n
ln n‰ .
ln n, then 0 an1 an Ê an1 an Ê {an } is a decreasing sequence of
nonnegative terms.
_
_
#
b
1, and '1 ecx dx œ lim cex d " œ lim ˆeb e1 ‰ œ ec1 Ê '1 ecx dx converges by
#
58. ex Ÿ ex for x
bÄ_
bÄ_
_
n#
the Comparison Test for improper integrals Ê ! e
n œ0
10
59. (a) s10 œ !
'10_ x1
n œ1
"
n3
dx œ lim
3
bÄ_
Ê 1.97531986
_
(b) s œ !
n œ1
"
n3
10
60. (a) s10 œ !
'10_ x1
nœ1
4
¸
"
n4
(b) s œ !
n œ1
"
n4
¸
c2 b
lim ’ x2 “
bÄ_
'11_ x1
'10b x4 dx œ
1
3993
10
4
dx œ lim
c3 b
bÄ_
c2 b
lim ’ x2 “
bÄ_
œ lim ˆ 2b1 2
bÄ_
bÄ_
10
s 1.082036583
1.08229 1.08237
2
'11b x3 dx œ
1 ‰
200
œ
11
œ lim ˆ 2b1 2
bÄ_
1 ‰
242
œ
1
242
and
1
200
Ê 1.20166 s 1.20253
1
200
lim ’ x3 “
#
nœ1
œ 1.202095; error Ÿ
œ 1.082036583;
Ê 1.082036583
bÄ_
s 1.97531986
1.20166 1.20253
2
bÄ_
dx œ lim
3
'10b x3 dx œ
1
242
dx œ lim
_
'11_ x1
œ 1.97531986;
_
œ 1 ! en converges by the Integral Test.
1.20253 1.20166
2
'11b x4 dx œ
c3 b
lim ’ x3 “
bÄ_
œ lim ˆ 3b1 3
bÄ_
1
3000
œ 1.08233; error Ÿ
œ 0.000435
1 ‰
3000
œ
11
œ lim ˆ 3b1 3
bÄ_
1
3000
Ê 1.08229 s 1.08237
1.08237 1.08229
2
œ 0.00004
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1 ‰
3993
œ
1
3993
and
590
Chapter 10 Infinite Sequences and Series
10.4 COMPARISON TESTS
_
1. Compare with !
n œ1
n
which is a convergent p-series, since p œ 2 1. Both series have nonnegative terms for n
1, we have n2 Ÿ n2 30 Ê
_
2. Compare with !
n œ1
n
"
n2 ,
"
n3 ,
1
n2
1
n2 30 .
_
Then by Comparison Test, !
1
n2 30
n œ1
1. For
converges.
which is a convergent p-series, since p œ 3 1. Both series have nonnegative terms for n
1, we have n4 Ÿ n4 2 Ê
1
n4
Ê
1
n 4 2
n
n4
Ê
n
n 4 2
1
n3
n
n 4 2
_
n1
n 4 2 .
Then by Comparison Test, !
n œ1
1. For
n1
n 4 2
converges.
_
3. Compare with !
n œ2
n
_
n œ2
"
n,
_
n œ1
For n
1
Èn 1
"
,
n3Î2
n2
1
n
_
n œ1
1. For n
œ È5 !
n œ1
1
n3Î2
"
3n ,
Ê
cos2 n
n3Î2
n œ2
1
Èn 1
n2
n
n
n
n2
œ
1
n
Ê
Ÿ
1
.
n3Î2
n2
n2 n
3
2
n2
n
n
1
n.
diverges.
_
Thus !
n œ2
n2
n2 n
n 3 an 4 b
n4 4
_
_
È5
.
n3Î2
_
cos2 n
n3Î2
Then by Comparison Test, !
n œ1
_
8. Compare with !
n œ1
n
The series !
nœ1
n4
n4 4
"
Èn ,
1, we have Èn
Ê n2 2 nÈn n
Ê
Èn 1
È n2 3
1
Èn .
diverges.
1.
converges.
1
n †3 n
1
n3Î2
Ÿ
1
3n .
_
Then by Comparison Test, !
n œ1
is a convergent p-series, since p œ
3
2
1
n †3 n
converges.
_
Ÿ
5
n3
Ê É nn444 Ÿ É n53 œ
È5
n3Î2
nœ1
1. For n
_
n2 3 Ê
2 Ê 2È n 1
n ˆn 2 È n 1 ‰
n2 3
1Ê
"
#
Ÿ 5.
Ÿ 1. Both series have nonnegative terms for n
3 Ê nˆ2Èn 1‰
n 2È n 1
n2 3
Èn 1
È n2 3
n œ1
n4 4n3
n4 4
n œ1
1
n
Ê
3 Ê 2 nÈ n n
3n
ˆÈ n 1 ‰
n2 3
2
1
n
ÊÊ
ˆÈ n 1 ‰
n2 3
_
Then by Comparison Test, !
1, we have
Then by Comparison Test, ! É nn444 converges.
which is a divergent p-series, since p œ
1 Ê 2È n
È5
n3Î2
1, and the series !
converges by Theorem 8 part 3. Both series have nonnegative terms for n
Ÿ5Ê
2. For
1. Both series have nonnegative terms for n
n3 Ÿ n4 Ê 4n3 Ÿ 4n4 Ê n4 4n3 Ÿ n4 4n4 œ 5n4 Ê n4 4n3 Ÿ 5n4 20 œ 5an4 4b Ê
Ê
2. For
which is a convergent geometric series, since lrl œ ¹ 13 ¹ 1. Both series have nonnegative terms for
3n Ê
nœ1
_
_
which is a convergent p-series, since p œ
1, we have n † 3n
7. Compare with !
Ÿ 1. Both series have nonnegative terms for n
Then by Comparison Test, !
1
Èn .
1
n2
1, we have 0 Ÿ cos2 n Ÿ 1 Ê
6. Compare with !
"
#
which is a divergent p-series, since p œ 1 Ÿ 1. Both series have nonnegative terms for n
2, we have n2 n Ÿ n2 Ê
5. Compare with !
n
which is a divergent p-series, since p œ
2, we have Èn 1 Ÿ Èn Ê
4. Compare with !
n
"
Èn ,
diverges.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
3
2
É 1n
1. For
Section 10.4 Comparison Tests
_
"
n2 ,
9. Compare with !
nc2
n3 c n2 b 3
1 În 2
œ lim
_
!
n œ1
n œ1
nÄ_
n2
n3 n2 3
n3 2n2
3
2
nÄ_ n n 3
œ lim
3n2 4n
2
nÄ_ 3n 2n
œ lim
6n 4
nÄ_ 6n 2
œ lim
œ lim
6
nÄ_ 6
an
nÄ_ bn
1. lim
œ 1 0. Then by Limit Comparison Test,
converges.
_
"
Èn ,
10. Compare with !
n œ1
É nn2bb12
œ lim
which is a convergent p-series, since p œ 2 1. Both series have positive terms for n
591
which is a divergent p-series, since p œ
n
œ lim É nn2
2 œ É lim
n2 n
2
nÄ_ n 2
2
nÄ_ 1ÎÈn
nÄ_
œ É lim
nÄ_
"
#
2n 1
2n
Ÿ 1. Both series have positive terms for n
œ É lim
2
nÄ_ 2
an
nÄ_ bn
1. lim
œ È1 œ 1 0. Then by Limit Comparison
_
Test, ! É nn212 diverges.
n œ1
_
"
n,
11. Compare with !
nan b 1b
Šn2
œ lim
n œ2
b 1‹an c 1b
n3 + n2
3
2
nÄ_ n n n 1
œ lim
1 În
nÄ_
_
Test, !
n œ2
n an 1 b
an2 1ban 1b
_
n œ1
lim an
nÄ_ bn
1.
nÄ_
_
n œ1
5n
È n 4n
œ lim
†
nÄ_ 1ÎÈn
6n 2
nÄ_ 6n 2
œ lim
œ lim
6
nÄ_ 6
œ 1 0. Then by Limit Comparison
which is a convergent geometric series, since lrl œ ¹ 12 ¹ 1. Both series have positive terms for
œ lim
13. Compare with !
3n2 2n
2
nÄ_ 3n 2n 1
œ lim
an
nÄ_ bn
2. lim
diverges.
"
2n ,
12. Compare with !
n
which is a divergent p-series, since p œ 1 Ÿ 1. Both series have positive terms for n
"
Èn ,
2n
3 b 4n
1Î2 n
4n
3
4n
nÄ_
œ lim
4n ln 4
n
nÄ_ 4 ln 4
œ lim
_
œ 1 0. Then by Limit Comparison Test, !
which is a divergent p-series, since p œ
n œ1
1
2
_
nÄ_
converges.
Ÿ 1. Both series have positive terms for n
n
œ lim ˆ 54 ‰ œ _. Then by Limit Comparison Test, !
5n
n
nÄ_ 4
œ lim
2n
3 4n
n œ1
5n
Èn†4n
an
nÄ_ bn
1. lim
diverges.
_
n
14. Compare with ! ˆ 25 ‰ , which is a convergent geometric series, since lrl œ ¹ 25 ¹ 1. Both series have positive terms for
n œ1
n
1.
œ exp
b 3 ‰n
ˆ 2n
5n b 4
n
15 ‰n
15 ‰
lim ˆ 10n 15 ‰ œ exp lim lnˆ 10n
œ exp lim n lnˆ 10n
n œ
10n 8
10n 8
nÄ_ a2Î5b
nÄ_ 10n 8
nÄ_
nÄ_
b 15 ‰
10
lnˆ 10n
10b 8
70n2
70n2
10n b 8
lim
œ exp lim 10n b151În10n
œ exp lim a10n 15
2
2
1 În
ba10n 8b œ exp nlim
nÄ_
nÄ_
nÄ_
Ä_ 100n 230n 120
lim an
nÄ_ bn
œ lim
œ exp lim
œ exp lim
140n
nÄ_ 200n 230
_
15. Compare with !
n œ2
œ lim
"
ln n
nÄ_ 1În
n œ1
lnŠ1 n"2 ‹
1În2
_
3 ‰n
œ e7Î10 0. Then by Limit Comparison Test, ! ˆ 2n
converges.
5n 4
n œ1
which is a divergent p-series, since p œ 1 Ÿ 1. Both series have positive terms for n
n
nÄ_ ln n
_
nÄ_
"
n,
œ lim
16. Compare with !
œ lim
140
nÄ_ 200
"
n2 ,
œ lim
1
nÄ_ 1În
_
œ lim n œ _. Then by Limit Comparison Test, !
nÄ_
n œ2
"
ln n
an
nÄ_ bn
2. lim
diverges.
which is a convergent p-series, since p œ 2 1. Both series have positive terms for n
1
œ lim
nÄ_
1
2
" Š n3 ‹
n2
Š n23 ‹
œ lim
1
"
nÄ_ 1 n2
_
œ 1 0. Then by Limit Comparison Test, ! lnˆ1
n œ1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"‰
n2
an
nÄ_ bn
1. lim
converges.
592
Chapter 10 Infinite Sequences and Series
_
"
Èn
17. diverges by the Limit Comparison Test (part 1) when compared with !
n œ1
, a divergent p-series:
"
lim
Œ #Èn È
$ n
Š È"n ‹
nÄ_
Èn
$ n
2È n È
œ n lim
Ä_
ˆ "
œ n lim
Ä _ #n
1Î6
‰œ
"
#
18. diverges by the Direct Comparison Test since n n n n Èn 0 Ê
_
"
n
term of the divergent series !
nœ1
3
n Èn
"
n
, which is the nth
"
n
or use Limit Comparison Test with bn œ
19. converges by the Direct Comparison Test;
sin# n
2n
Ÿ
"
#n
, which is the nth term of a convergent geometric series
20. converges by the Direct Comparison Test;
1 cos n
n#
Ÿ
2
n#
2n
3n 1
21. diverges since n lim
Ä_
œ
2
3
Š nn# È"n ‹
"
n#
converges
Á0
22. converges by the Limit Comparison Test (part 1) with
lim
nÄ_
and the p-series !
"
n$Î#
, the nth term of a convergent p-series:
ˆ n n " ‰ œ 1
œ n lim
Ä_
" ‹
Š $Î#
n
23. converges by the Limit Comparison Test (part 1) with
lim
Š n(n 10n1)(n" 2) ‹
Š n"# ‹
nÄ_
10n# n
n# 3n 2
œ n lim
Ä_
œ n lim
Ä_
20n 1
2n 3
24. converges by the Limit Comparison Test (part 1) with
lim
n# (n
"
n#
, the nth term of a convergent p-series:
œ n lim
Ä_
"
n#
œ 10
20
2
, the nth term of a convergent p-series:
5n$
3n
2) Šn# 5‹
Š n"# ‹
nÄ_
œ n lim
Ä_
5n$ 3n
n$ 2n# 5n 10
15n# 3
3n# 4n 5
œ n lim
Ä_
n
œ n lim
Ä_
30n
6n 4
œ5
n
n
n ‰
25. converges by the Direct Comparison Test; ˆ 3n n 1 ‰ ˆ 3n
œ ˆ "3 ‰ , the nth term of a convergent geometric series
26. converges by the Limit Comparison Test (part 1) with
"
Š $Î# ‹
n
lim
nÄ_ Š "
È$
n
2
$
‹
É n n$ 2 œ lim É1
œ n lim
Ä_
nÄ_
"
n$Î#
, the nth term of a convergent p-series:
œ1
2
n$
27. diverges by the Direct Comparison Test; n ln n Ê ln n ln ln n Ê
"
n
_
28. converges by the Limit Comparison Test (part 2) when compared with !
n œ1
#
lim
nÄ_
’ (lnn$n) “
Š n"# ‹
œ n lim
Ä_
(ln n)#
n
œ n lim
Ä_
2(ln n) Š n" ‹
1
œ 2 n lim
Ä_
29. diverges by the Limit Comparison Test (part 3) with
lim
nÄ_
’È
1
“
n ln n
ˆ n" ‰
œ n lim
Ä_
Èn
ln n
"
n
Š 2È n ‹
ˆ n" ‰
"
n#
"
ln n
"
ln (ln n)
_
and !
n œ3
"
n
, a convergent p-series:
œ0
, the nth term of the divergent harmonic series:
"
œ n lim
Ä_
ln n
n
œ n lim
Ä_
Èn
2
œ_
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
diverges
Section 10.4 Comparison Tests
"
n&Î%
30. converges by the Limit Comparison Test (part 2) with
lim
n)#
’ (ln$Î#
“
n
nÄ_ Š
"
‹
n&Î%
(ln n)#
n"Î%
œ n lim
Ä_
œ n lim
Ä_
ˆ 2 lnn n ‰
"
n
31. diverges by the Limit Comparison Test (part 3) with
lim
nÄ_
ˆ 1 "ln n ‰
ˆ "n ‰
œ n lim
Ä_
n
1 ln n
32. diverges by the Integral Test:
"
Š "n ‹
œ n lim
Ä_
, the nth term of a convergent p-series:
œ 8 n lim
Ä_
"
Š $Î%
‹
4n
ln n
n"Î%
œ 8 n lim
Ä_
ˆ n" ‰
Š
"
‹
4n$Î%
œ 32 n lÄ
im_
"
n"Î%
œ 32 † 0 œ 0
, the nth term of the divergent harmonic series:
œ n lim
nœ_
Ä_
'2_ lnx(x11) dx œ 'ln_3 u du œ
" u# ‘ b œ lim
ln 3
lim
bÄ_ 2
"
bÄ_ #
ab# ln# 3b œ _
"
33. converges by the Direct Comparison Test with n$Î#
, the nth term of a convergent p-series: n# 1 n for
"
"
n 2 Ê n# an# 1b n$ Ê nÈn# 1 n$Î# Ê $Î#
or use Limit Comparison Test with
nÈ n# 1
n
"
n$Î#
Èn
n# 1
34. converges by the Direct Comparison Test with
n# 1
Èn
Ê n# 1 Ènn$Î# Ê
_
35. converges because !
n œ1
_
!
nœ1
"
n2n
"n
n2n
n$Î# Ê
_
œ!
n œ1
"
n2n
_
"
#n
!
n œ1
593
, the nth term of a convergent p-series: n# 1 n#
"
n$Î#
_
n œ1
or use Limit Comparison Test with
"
.
n$Î#
which is the sum of two convergent series:
converges by the Direct Comparison Test since
36. converges by the Direct Comparison Test: !
1
n# .
"
n #n
"
#n
_
n 2n
n# 2n
_
, and !
œ ! ˆ n2" n
nœ1
nœ1
"‰
n#
"
2n
and
is a convergent geometric series
"
n2n
"
n#
Ÿ
"
#n
"
n#
, the sum of
the nth terms of a convergent geometric series and a convergent p-series
37. converges by the Direct Comparison Test:
38. diverges; n lim
Š3
Ä_
nc1
"
3n ‹
ˆ"
œ n lim
Ä_ 3
"
3nc1 1
"‰
3n
"
3
œ
"
3nc1
, which is the nth term of a convergent geometric series
Á0
_
n
39. converges by Limit Comparison Test: compare with ! ˆ 15 ‰ , which is a convergent geometric series with lrl œ
n œ1
lim
nÄ_
1
1
Š n2n b
b 3n † 5n ‹
a 1 Î5 b n
œ n lim
Ä_
n1
n2 3n
œ n lim
Ä_
1
2n 3
_
n œ1
3
Š 23n b
b 4n ‹
n
n
a 3 Î4 b n
œ n lim
Ä_
8n 12n
9n 12n
œ n lim
Ä_
8 ‰n
ˆ 12
1
9 ‰n
ˆ 12
1
œ
1
1
_
n œ1
œ
œ
2
n
lim 2 aln 2b
n Ä _ 2n aln 2b2
1
5
1,
œ 1 0.
41. diverges by Limit Comparison Test: compare with !
n
lim 2 ln 2 1
n Ä _ 2n ln 2
1,
œ 0.
n
40. converges by Limit Comparison Test: compare with ! ˆ 34 ‰ , which is a convergent geometric series with lrl œ
lim
nÄ_
1
5
Š 2n 2cnn ‹
n
1
n,
which is a divergent p-series, n lim
Ä_
†
1 În
2 n
œ n lim
Ä _ 2n
n
œ 1 0.
_
_
n œ1
n œ1
42. diverges by the definition of an infinite series: ! lnˆ n n 1 ‰ œ ! ln n ln an 1b‘, sk œ aln 1 ln 2b aln 2 ln 3b
Þ Þ Þ alnak 1b ln kb aln k ln ak 1bb œ ln ak 1b Ê lim sk œ _
kÄ_
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
594
Chapter 10 Infinite Sequences and Series
_
43. converges by Comparison Test with !
n œ2
sk œ ˆ1 12 ‰ ˆ 12 13 ‰ Þ Þ Þ ˆ k 1 2
Ê nan 1ban 2b!
_
which converges since !
1
nan 1b
nan 1b Ê n!
n œ2
1 ‰
k1
ˆ k 1 1 k1 ‰ œ 1
nan 1b Ê
1
n!
_
1
n3 ,
n œ1
œ
œ
n2
lim
n Ä _ n2 3n 2
œ n lim
Ä_
45. diverges by the Limit Comparison Test (part 1) with
"‰
n
ˆsin
lim
n Ä _ ˆ "n ‰
œ lim
xÄ0
sin x
x
nœ2
Ê lim sk œ 1; for n
1
k
2, an 2b!
kÄ_
1
nan1b
an
which is a convergent p-series, n lim
Ä_
2n
2n 3
œ n lim
Ä_
c 1bx
2bx
1 În 3
œ10
2
2
"
n
, the nth term of the divergent harmonic series:
"
n
, the nth term of the divergent harmonic series:
œ1
46. diverges by the Limit Comparison Test (part 1) with
ˆtan "n ‰
lim
n Ä _ ˆ "n ‰
Ÿ
_
œ ! ’ n 1 1 n1 “, and
an
44. converges by Limit Comparison Test: compare with !
n 3 a n 1 bx
lim
n Ä _ an 2ban 1bnan 1bx
1
nan 1b
œ n lim
Š " ‹
Ä _ cos "
n
ˆsin n" ‰
ˆ "n ‰
œ lim ˆ cos" x ‰ ˆ sinx x ‰ œ 1 † 1 œ 1
xÄ0
tanc" n
n1.1
47. converges by the Direct Comparison Test:
_
1
#
n1.1
1
and !
n œ1
1
#
œ
#
n1.1
_
!
nœ1
"
n1.1
is the product of a
convergent p-series and a nonzero constant
48. converges by the Direct Comparison Test: sec" n
1
#
Ê
secc" n
n1 3
Þ
ˆ 1# ‰
n1 3
Þ
_
and !
n œ1
ˆ 1# ‰
n1 3
Þ
œ
1
#
_
!
n œ1
"
n1 3
Þ
is the
product of a convergent p-series and a nonzero constant
49. converges by the Limit Comparison Test (part 1) with
œ n lim
Ä_
" ec2n
1 ec2n
" ec2n
1 ec2n
: n lim
Ä_
"
n#
: n lim
Ä_
52. converges by the Limit Comparison Test (part 1) with
"
123án
lim
nÄ_
54.
œ
Š nan 2b 1b ‹
Š n"# ‹
"
1 2# 3# á n#
Š n"# ‹
œ n lim
coth n œ n lim
Ä_
Ä_
en ecn
en ecn
n
Š tanh
‹
n#
Š n"# ‹
œ n lim
tanh n œ n lim
Ä_
Ä_
en e
en e
n
n
œ1
51. diverges by the Limit Comparison Test (part 1) with 1n : n lim
Ä_
53.
n
Š coth
‹
n#
œ1
50. converges by the Limit Comparison Test (part 1) with
œ n lim
Ä_
"
n#
"
ˆ n(n #
1) ‰
œ
œ n lim
Ä_
œ
"
2
n(n 1) .
2n#
n# n
n(n b 1)(2n b 1)
6
œ
1
Š nÈ
n n‹
ˆ 1n ‰
"
n# : n lim
Ä_
Š
œ n lim
Ä_
Èn n ‹
n#
Š n"# ‹
1
n n
È
œ 1.
n
È
œ n lim
nœ1
Ä_
The series converges by the Limit Comparison Test (part 1) with
œ n lim
Ä_
4n
2n 1
6
n(n 1)(2n 1)
Ÿ
œ n lim
Ä_
6
n$
4
2
"
n# :
œ 2.
Ê the series converges by the Direct Comparison Test
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1
Section 10.4 Comparison Tests
595
an
55. (a) If n lim
œ 0, then there exists an integer N such that for all n N, ¹ bann 0¹ 1 Ê 1 bann 1
Ä _ bn
Ê an bn . Thus, if ! bn converges, then ! an converges by the Direct Comparison Test.
an
(b) If n lim
œ _, then there exists an integer N such that for all n N, bann 1 Ê an bn . Thus, if
Ä _ bn
! bn diverges, then ! an diverges by the Direct Comparison Test.
_
56. Yes, !
n œ1
an
n
converges by the Direct Comparison Test because
an
n
an
an
57. n lim
œ _ Ê there exists an integer N such that for all n N,
Ä _ bn
!
then
bn converges by the Direct Comparison Test
an
bn
1 Ê an bn . If ! an converges,
58. ! an converges Ê n lim
a œ 0 Ê there exists an integer N such that for all n N, 0 Ÿ an 1 Ê an# an
Ä_ n
Ê ! a#n converges by the Direct Comparison Test
59. Since an 0 and n lim
a œ _ Á 0, by nth term test for divergence, ! an diverges.
Ä_ n
60. Since an 0 and n lim
a n2 † an b œ 0, compare !an with ! n"# , which is a convergent p-series; n lim
Ä_
Ä_
an
1 În 2
œ n lim
a n2 † an b œ 0 Ê !an converges by Limit Comparison Test
Ä_
_
61. Let _ q _ and p 1. If q œ 0, then !
n œ2
_
!
n œ2
1
nr
where 1 r p, then n lim
Ä_
œ n lim
Ä_
1
aln nbcq npcr
qc1
lim qaln nb
n Ä _ ap rbnpcr
aln nbq
np
1 În r
œ 0. If q 0, n lim
Ä_
œ n lim
Ä_
qc2
q
ap rbnpcr aln nb1cq
aln nbq
np
œ n lim
Ä_
aln nbq
npcr
_
œ!
nœ2
aln nbq
npcr ,
œ n lim
Ä_
1
np ,
which is a convergent p-series. If q Á 0, compare with
and p r 0. If q 0 Ê q 0 and n lim
Ä_
qaln nbqc1 ˆ 1n ‰
ap rbnpcrc1
œ n lim
Ä_
qaln nbqc1
ap rbnpcr .
aln nbq
npcr
If q 1 Ÿ 0 Ê 1 q
œ 0, otherwise, we apply L'Hopital's Rule again. n lim
Ä_
qc2
0 and
qaq 1baln nbqc2 ˆ 1n ‰
ap rb2 npcrc1
qaq 1baln nb
qaq 1baln nb
q aq 1 b
œ n lim
. If q 2 Ÿ 0 Ê 2 q 0 and n lim
œ n lim
œ 0; otherwise, we
Ä _ ap rb2 npcr
Ä _ ap rb2 npcr
Ä _ ap rb2 npcr aln nb2cq
apply L'Hopital's Rule again. Since q is finite, there is a positive integer k such that q k Ÿ 0 Ê k q 0. Thus, after k
qaq 1bâaq k 1baln nbqck
q a q 1 bâa q k 1 b
œ n lim
Ä _ ap rbk npcr aln nbkcq
ap rbk npcr
_
q
series ! alnnnpb converges.
n œ1
applications of L'Hopital's Rule we obtain n lim
Ä_
0 in every case, by Limit Comparison Test, the
_
62. Let _ q _ and p Ÿ 1. If q œ 0, then !
n œ2
_
!
nœ2
1
np ,
aln nbq
np
which is a divergent p-series. Then n lim
Ä_
where 0 p r Ÿ 1. n lim
Ä_
lim
aln nbq
np
ar pbn
rcpc1
n Ä _ aqbaln nbcqc1 ˆ 1n ‰
œ n lim
Ä_
aln nbq
np
1 În r
œ
q
lim aln nb
n Ä _ npcr
rcp
ar pbn
.
aqbaln nbcqc1
1 În p
œ
_
œ!
nœ2
1
np ,
which is a divergent p-series. If q 0, compare with
_
œ n lim
aln nbq œ _. If q 0 Ê q 0, compare with !
Ä_
nœ2
nrcp
lim
cq
n Ä _ aln nb
otherwise, we apply L'Hopital's Rule again to obtain n lim
Ä_
0 and n lim
Ä_
2 rcpc1
a r pb n
aqbaq 1baln nbcqc2 ˆ 1n ‰
2 rcp
ar pbnrcp aln nbqb1
aqb
œ n lim
Ä_
qb2
œ _,
a r pb2 nrcp
.
aqbaq 1baln nbcqc2
a r pb n aln nb
œ n lim
œ _, otherwise, we
aqbaq 1b
Ä_
apply L'Hopital's Rule again. Since q is finite, there is a positive integer k such that q k Ÿ 0 Ê q k
q 2 Ÿ 0 Ê q 2
0 and n lim
Ä_
k applications of L'Hopital's Rule we obtain n lim
Ä_
1
nr ,
since r p 0. Apply L'Hopital's to obtain
If q 1 Ÿ 0 Ê q 1
a r pb2 nrcp
aqbaq 1baln nbcqc2
œ 0. Since the limit is
k rcp
a r pb n
aqbaq 1bâaq k 1baln nbcqck
œ n lim
Ä_
k rcp
If
0. Thus, after
qbk
a r pb n aln nb
aqbaq 1bâaq k 1b
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
œ _.
596
Chapter 10 Infinite Sequences and Series
_
q
Since the limit is _ if q 0 or if q 0 and p 1, by Limit comparison test, the series ! alnnpncbr diverges. Finally if q 0
_
and p œ 1 then !
nœ2
Ê aln nbq
_
1Ê
aln nb
np
q
_
œ !
aln nbq
n
nœ2
1
n.
aln nb
n
q
_
. Compare with !
nœ2
_
Thus !
n œ2
aln nbq
n
nœ1
1
n,
which is a divergent p-series. For n
q
nœ1
63. Converges by Exercise 61 with q œ 3 and p œ 4.
1
2
and p œ 12 .
65. Converges by Exercise 61 with q œ 1000 and p œ 1.001.
66. Diverges by Exercise 62 with q œ
1
5
1
diverges by Comparison Test. Thus, if _ q _ and p Ÿ 1,
the series ! alnnpncbr diverges.
64. Diverges by Exercise 62 with q œ
3, ln n
and p œ 0.99.
67. Converges by Exercise 61 with q œ 3 and p œ 1.1.
68. Diverges by Exercise 62 with q œ 12 and p œ 12 .
69. Example CAS commands:
Maple:
a := n -> 1./n^3/sin(n)^2;
s := k -> sum( a(n), n=1..k );
# (a)]
limit( s(k), k=infinity );
pts := [seq( [k,s(k)], k=1..100 )]:
# (b)
plot( pts, style=point, title="#69(b) (Section 10.4)" );
pts := [seq( [k,s(k)], k=1..200 )]:
# (c)
plot( pts, style=point, title="#69(c) (Section 10.4)" );
pts := [seq( [k,s(k)], k=1..400 )]:
# (d)
plot( pts, style=point, title="#69(d) (Section 10.4)" );
evalf( 355/113 );
Mathematica:
Clear[a, n, s, k, p]
a[n_]:= 1 / ( n3 Sin[n]2 )
s[k_]= Sum[ a[n], {n, 1, k}]
points[p_]:= Table[{k, N[s[k]]}, {k, 1, p}]
points[100]
ListPlot[points[100]]
points[200]
ListPlot[points[200]
points[400]
ListPlot[points[400], PlotRange Ä All]
To investigate what is happening around k = 355, you could do the following.
N[355/113]
N[1 355/113]
Sin[355]//N
a[355]//N
N[s[354]]
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.5 The Ratio and Root Tests
597
N[s[355]]
N[s[356]]
_
_
70. (a) Let S œ ! n12 , which is a convergent p-series. By Example 5 in Section 10.2, ! nan 1 1b converges to 1. By Theorem 8,
n œ1
_
Sœ!
n œ1
_
1
n2
n œ1
_
œ!
_
!
1
nan 1b
n œ1
_
!
1
n2
n œ1
_
œ!
1
nan 1b
nœ1
nœ1
1
n an 1 b
_
!
nœ1
Š n12
1
nan 1b ‹
also converges.
_
(b) Since ! nan 1 1b converges to 1 (from Example 5 in Section 10.2), S œ 1 ! Š n12
n œ1
n œ1
_
(c) The new series is comparible to
! 13 ,
n
n œ1
1
nan 1b ‹
_
œ 1 ! n2 an1 1b
n œ1
_
so it will converge faster because its terms Ä 0 faster than the terms of ! n12 .
n œ1
1000
1000
(d) The series 1 ! n2 an1 1b gives a better approximation. Using Mathematica, 1 ! n2 an1 1b œ 1.644933568, while
nœ1
1000000
!
n œ1
1
n2
nœ1
œ 1.644933067. Note that
1
6
œ 1.644934067. The error is 4.99 ‚ 107 compared with 1 ‚ 106 .
2
10.5 THE RATIO AND ROOT TESTS
1.
2.
3.
2n
n!
2nb"
0 for all n
n2
3n
1; lim Œ
nÄ_
"b !
2n
n!
an
0 for all n
an 1 b!
an 1b2
an
1; lim Œ
nÄ_
2 †2
lim Š an"
b†n! †
n
œ
nÄ_
b1b b 2
3nb1
nb2
n
3
3
lim ˆ n3
n †3 †
œ
nÄ_
b1bc1b!
b1bb1b2
anc1b!
anb1b2
aan
0 for all n
1; lim Œ
nÄ_
aan
_
n!
2n ‹
œ
_
n
œ lim ˆ n 2 " ‰ œ 0 1 Ê ! 2n! converges
nÄ_
3n ‰
n2
n3 ‰
ˆ1‰
œ lim ˆ 3n
6 œ lim 3 œ
nÄ_
lim Š na†nan21b2b! †
nÄ_
nœ1
nÄ_
an "b2
an 1b! ‹
_
1 Ê ! n 3n 2 converges
1
3
nœ1
n
3n 4n 1
œ lim Š nn22n
4n 4 ‹ œ lim Š 2n 4 ‹
3
2
2
nÄ_
nÄ_
1b!
œ lim ˆ 6n 2 4 ‰ œ _ 1 Ê ! aann
diverges
1 b2
nÄ_
4.
2nb1
n†3n 1
n œ1
0 for all n
_
1; lim
nÄ_
an
2an1b1
1b†3an1b
2n1
n†3n 1
1
nb1
lim Š an21b†3†n2 1 †3 †
œ
nÄ_
n †3 n 1
2n1 ‹
œ lim ˆ 3n2n 3 ‰ œ lim ˆ 23 ‰ œ
nÄ_
nÄ_
2
3
1
nb1
Ê ! n2†3nc1 converges
nœ1
5.
n4
4n
œ lim ˆ 14
nÄ_
6.
3nb2
ln n
1
n
b1b4
4nb1
n4
4n
an
0 for all n
1; lim Œ
nÄ_
0 for all n
3
2n2
1
n3
1 ‰
4n4
2; lim Œ
nÄ_
_
œ
œ
3anb1bb2
ln anb1b
3nb2
ln n
1
4
4
lim Š an4n †14b †
4n
n4 ‹
nÄ_
_
œ lim Š n
4
nÄ_
4n3 6n2 4n 1
‹
4n4
4
1 Ê ! n4n converges
œ
n œ1
nb2
lim Š ln3an †31b †
nÄ_
ln n
3nb2 ‹
œ lim Š ln 3anlnn1b ‹ œ lim Š
nÄ_
nÄ_
3
n
1
nb1
‹ œ lim ˆ 3n n 3 ‰
nÄ_
nb2
œ lim ˆ 31 ‰ œ 3 1 Ê ! 3ln n diverges
nÄ_
7.
n 2 a n 2 b!
nx32n
n œ2
an
0 for all n
1; lim
nÄ_
b 1b2aan b 1b b 2b!
b 1bx32an 1b
an
n2 an 2b!
nx32n
œ
7
ˆ 6n 15 ‰
ˆ6‰
œ lim Š 3n27n2 15n
18n ‹ œ lim 54n 18 œ lim 54 œ
2
nÄ_
nÄ_
nÄ_
2
3ban 2b!
lim Š an an1ba1nb
†
†nx32n †32
nÄ_
1
9
_
nx32n
n 2 an 2 b! ‹
œ lim Š n
nÄ_
1 Ê ! n annx32n2b! converges
2
n œ1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
3
5n2 7n 3
‹
9n3 9n2
598
8.
Chapter 10 Infinite Sequences and Series
n †5 n
a2n 3b lnan 1b
0 for all n
1b†a2n 3b
lim Š 5an
†
na2n 5b
œ
1;
lim Œ
nÄ_
lnan 1b
lnan 2b ‹
nÄ_
a2an
an b 1b†5n 1
1b 3b lnaan 1b
n†5n
a2n 3b lnan 1b
n 1b†5 †5
lim Š a2na
5b lnan 2b †
n
œ
1b
nÄ_
a2n 3b lnan 1b
‹
n †5 n
lnan 1b
15
25 ‰
lim Š 10n 2n2 25n
† lim Š
‹ † lim Š lnan 2b ‹ œ lim ˆ 20n
5n
4n 5
2
œ
nÄ_
nÄ_
_
nÄ_
nÄ_
1
nb1
1
nb2
‹
n †5
!
‰
ˆ n2‰
ˆ 1‰
œ lim ˆ 20
4 † lim n 1 œ 5 † lim 1 œ 5 † 1 œ 5 1 Ê
a2n 3b lnan 1b diverges
nÄ_
9.
10.
nÄ_
7
a2n 5bn
nÄ_
4n
a3n bn
nÄ_
n 1
12. ’lnˆe2 1n ‰“
_
nÄ_
nÄ_
n
n œ1
n
n
3‰
ˆ 4n
2; lim É
œ
3n 5
3‰
lim ˆ 4n
3n 5 œ
nÄ_
n 1
nÄ_
Ê ! ’lnˆe2 1n ‰“
diverges
8
ˆ3 1n ‰2n
8
n
1; lim É
œ
ˆ3 1 ‰2n
nÄ_
lim ˆ 43 ‰ œ
nÄ_
n
1; lim Ê’lnˆe2 1n ‰“
0 for all n
n 1
n œ1
4 ‰
lim ˆ 3n
œ 0 1 Ê ! a3n4 bn converges
n
0 for all n
_
n
È
lim Š 2n 7 5 ‹ œ 0 1 Ê ! a2n 7 5bn converges
nÄ_
_
4
n
1; lim É
œ
a3n bn
0 for all n
3 ‰n
11. ˆ 4n
3n 5
n œ2
7
n
1; lim É
œ
a2n 5bn
0 for all n
n
œ
4
3
nÄ_
_
n
3‰
1 Ê ! ˆ 4n
diverges
3n 5
n œ1
1 1 În
lim ’lnˆe2 1n ‰“
nÄ_
œ lnae2 b œ 2 1
n œ1
13.
0 for all n
n
16.
"
n1bn
nÄ_
n
14. ’sinŠ È1n ‹“
n
15. ˆ1 n1 ‰
lim Œ ˆ
n
È
8
3 1n ‰
0 for all n
0 for all n
1;
17. converges by the Ratio Test:
converges
nÄ_
n
n œ1
nÄ_
2
nœ1
n
È
nÄ_
”
œ n lim
Ä_
nÄ_
È
(n b 1) 2
2 n b1 •
”
_
n
È
1
! 1"bn converges
lim Š n È
n n‹ œ 0 1 Ê
n
lim Š n1În 1 1 ‹ œ
lim anb1
n Ä _ an
8
1 ‰2n
n œ1 3 n
n
n
lim ˆ1 n1 ‰ œ e1 1 Ê ! ˆ1 n1 ‰ converges
nÄ_
nÄ_
_
1Ê !ˆ
_
2
n
n
lim Ɉ1 n1 ‰ œ
"
n
2; lim É
n1bn œ
1
9
lim sinŠ È1n ‹ œ sina0b œ 0 1 Ê ! ’sinŠ È1n ‹“ converges
nÄ_
2
œ
_
n
n
1; lim Ê
’sinŠ È1n ‹“ œ
0 for all n
2
È
n 2
#n
•
Š (nenbb1)1 ‹
n œ2
È
È
n
(n 1) 2
ˆ1 n" ‰ 2 ˆ "# ‰ œ
œ n lim
† 2È2 œ n lim
Ä _ #nb1
Ä_
n
"
#
1
2
18. converges by the Ratio Test:
lim anb1
n Ä _ an
œ n lim
Ä_
19. diverges by the Ratio Test: n lim
Ä_
anb1
an
œ n lim
Ä_
20. diverges by the Ratio Test: n lim
Ä_
anb1
an
œ n lim
Ä_
#
Š nen ‹
Š (nenbb1)!
1 ‹
ˆ en!n ‰
b 1)! ‹
Š (n
10nb1
ˆ 10n!n ‰
œ n lim
Ä_
21. converges by the Ratio Test:
œ n lim
Ä_
(n ")!
enb1
†
en
n!
œ n lim
Ä_
(n ")!
10nb1
†
10n
n!
Š (n10bn1)1 ‹
"!
Š n10n ‹
†
œ n lim
Ä_
"!
lim anb1
n Ä _ an
(n 1)2
enb1
œ n lim
Ä_
(n ")"!
10n 1
†
en
lim
n2 œ n Ä
_
œ n lim
Ä_
œ n lim
Ä_
10n
n"!
ˆ1 n" ‰# ˆ "e ‰ œ
n"
e
n
10
"
e
1
œ_
œ_
ˆ1 n" ‰"! ˆ 1"0 ‰ œ
œ n lim
Ä_
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
10
1
Section 10.5 The Ratio and Root Tests
ˆ nn 2 ‰n œ lim ˆ1
22. diverges; n lim
a œ n lim
Ä_ n
Ä_
nÄ_
23. converges by the Direct Comparison Test:
2(1)n
(1.25)n
2 ‰ n
n
599
œ e# Á 0
n
n
œ ˆ 45 ‰ c2 (1)n d Ÿ ˆ 45 ‰ (3) which is the nth term of a convergent
geometric series
24. converges; a geometric series with krk œ ¸ 23 ¸ 1
ˆ1 3n ‰n œ lim ˆ1
25. diverges; n lim
a œ n lim
Ä_ n
Ä_
nÄ_
ˆ1
26. diverges; n lim
a œ n lim
Ä_ n
Ä_
" ‰n
3n
3 ‰ n
n
œ n lim
1
Ä_
27. converges by the Direct Comparison Test:
ln n
n$
n
n$
œ
œ e$ ¸ 0.05 Á 0
Š "3 ‹
n
"
n#
n
"Î$
¸ 0.72 Á 0
œe
2, the nth term of a convergent p-series.
for n
n
n (ln n)
n
È
É
28. converges by the nth-Root Test: n lim
an œ n lim
nn œ n lim
Ä_
Ä_
Ä_
29. diverges by the Direct Comparison Test:
with "n .
"
n
"
n#
œ
n1
n#
ln n
n
"
n
for n
" ‰n
n#
ˆˆ n"
œ n lim
Ä_
anb1
an
œ n lim
Ä_
(n 1) ln (n 1)
#nb1
†
33. converges by the Ratio Test: n lim
Ä_
anb1
an
œ n lim
Ä_
(n 2)(n 3)
(n 1)!
†
n!
(n 1)(n 2)
34. converges by the Ratio Test: n lim
Ä_
anb1
an
œ n lim
Ä_
(n 1)$
en 1
œ
35. converges by the Ratio Test: n lim
Ä_
anb1
an
œ n lim
Ä_
(n 4)!
3! (n 1)! 3nb1
anb1
an
œ n lim
Ä_
†
anb1
an
œ n lim
Ä_
(n 1)!
(2n 3)!
38. converges by the Ratio Test: n lim
Ä_
anb1
an
œ n lim
Ä_
(n 1)!
(n 1)nb1
"
ˆ1 "n ‰n
œ
"
e
en
n$
†
(n 1)2nb1 (n 2)!
3nb1 (n 1)!
37. converges by the Ratio Test: n lim
Ä_
œ n lim
Ä_
ln n
n
œ n lim
Ä_
Š "n ‹
1
œ01
" ‰n ‰1În
n#
ˆ"
œ n lim
Ä_ n
"‰
n#
3
32. converges by the Ratio Test: n lim
Ä_
36. converges by the Ratio Test: n lim
Ä_
œ n lim
Ä_
"# ˆ "n ‰ for n 2 or by the Limit Comparison Test (part 1)
n
n
ˆ n"
È
É
30. converges by the nth-Root Test: n lim
an œ n lim
Ä_
Ä_
31. diverges by the Direct Comparison Test:
a(ln n)n b1În
ann b1În
2n
n ln (n)
"
e
†
nn
n!
1
œ01
œ n lim
Ä_
3n n!
n2n (n 1)!
(2n 1)!
n!
†
"
#
1
3! n! 3n
(n 3)!
†
œ
n4
3(n 1)
"
3
œ
1
2‰
ˆ n n 1 ‰ ˆ 32 ‰ ˆ nn
œ n lim
1 œ
Ä_
œ n lim
Ä_
n"
(2n 3)(2n 2)
œ01
ˆ n ‰n œ lim
œ n lim
Ä _ n1
nÄ_
"
ˆ n bn " ‰n
1
n
n
n
È
39. converges by the Root Test: n lim
an œ n lim
œ n lim
Ä_
Ä _ É (ln n)n
Ä_
n n
È
ln n
œ n lim
Ä_
"
ln n
œ01
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
2
3
1
œ01
600
Chapter 10 Infinite Sequences and Series
n n
È
Èln n
n
n
n
È
40. converges by the Root Test: n lim
an œ n lim
œ n lim
Ä_
Ä _ É (ln n)nÎ2
Ä_
Ä_
Ä_
n n
lim È
n
Èln n
lim
n
œ
œ01
n
È
n œ 1‹
Šn lim
Ä_
41. converges by the Direct Comparison Test:
œ
n! ln n
n(n 2)!
ln n
n(n 1)(n 2)
"
(n 1)(n #)
œ
n
n(n 1)(n 2)
"
n#
which is the nth-term of a convergent p-series
an 1
an
42. diverges by the Ratio Test: n lim
Ä_
œ n lim
Ä_
3n 1
(n 1)$ 2n
1
†
n$ 2n
3n
†
a2nbx
nx‘2
43. converges by the Ratio Test: n lim
Ä_
anb1
an
œ n lim
Ä_
an1bx‘2
2(n 1)‘x
44. converges by the Ratio Test: n lim
Ä_
anb1
an
œ n lim
Ä_
a2n 5bˆ2nb1 3‰
3nb1 2
œ n lim
’ 2n 5 “ † n lim
’ 2 † 6 4 † 2 3† 3 6 “ œ 1 †
Ä _ 2n 3
Ä _ 3 † 6 n 9 † 3 n 2† 2 n 6
n
n
n
45. converges by the Ratio Test: n lim
Ä_
anb1
an
œ n lim
Ä_
2
3
œ
ˆ 1 b nsin n ‰ an
an
"n
Š 1 b tan
n
47. diverges by the Ratio Test: n lim
Ä_
anb1
an
œ n lim
Ä_
œ n lim
Ä_
†
ˆ #3 ‰ œ
an1b2
(2n 2)(2n 1)
3n 2
a2n 3ba2n 3b
3
#
1
œ n lim
Ä_
œ n lim
’ 2n 5 †
Ä _ 2n 3
n2 2n 1
4n2 6n 2
œ
œ n lim
Ä_
3n 1
2n 5
œ n lim
Ä_
" tan " n
n
œ
3
#
œ 0 since the numerator
1
2
‰
48. diverges; an1 œ n n 1 an Ê an1 œ ˆ n n 1 ‰ ˆ n n 1 an1 ‰ Ê an1 œ ˆ n n 1 ‰ ˆ n n 1 ‰ ˆ nn
1 an2
a
"
n
n
1
n
2
3
"
Ê an1 œ ˆ n 1 ‰ ˆ n ‰ ˆ n 1 ‰ â ˆ # ‰ a" Ê an1 œ n 1 Ê an1 œ n 1 , which is a constant times the
general term of the diverging harmonic series
49. converges by the Ratio Test: n lim
Ä_
50. converges by the Ratio Test:
n ln n
n 10
0 and a" œ
Ê an1 œ
n ln n
n 10
"
#
œ n lim
Ä_
lim anb1
n Ä _ an
œ n lim
Ä_
anb1
an
œ n lim
Ä_
51. converges by the Ratio Test: n lim
Ä_
52.
anb1
an
Š 2n ‹ an
an
Œ
Èn n
#
œ n lim
Ä_
an
an
œ n lim
Ä_
Š 1 bnln n ‹ an
an
2
n
œ01
n n
È
œ n lim
Ä_
n
œ
"ln n
n
"
#
1
œ n lim
Ä_
Ê an 0; ln n 10 for n e"! Ê n ln n n 10 Ê
an an ; thus an1 an
53. diverges by the nth-Term Test: a" œ
"
3
"
#
"
n
œ01
n ln n
n 10
1
Ê n lim
a Á 0, so the series diverges by the nth-Term Test
Ä_ n
3
3
6 "
%! "
2 "
2 "
2 "
É
É
É
É
, a# œ É
3 , a$ œ Ê
3 œ
3 , a% œ ËÊ
3 œ
3 ,á ,
%
n! "
n! "
n "
É
an œ É
a œ 1 because šÉ
3 Ê n lim
3 › is a subsequence of š
3 › whose limit is 1 by Table 8.1
Ä_ n
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1
4
1
2†6n 4†2n 3†3n 6
3 †6 n 9 † 3 n 2 † 2 n 6 “
œ01
‹ an
an
c1‰
ˆ 3n
2n b 5 an
an
n3
(n 1)3
1
2
3
anb1
46. converges by the Ratio Test: n lim
œ n lim
Ä _ an
Ä_
1
approaches 1 # while the denominator tends to _
œ n lim
Ä_
Section 10.5 The Ratio and Root Tests
54. converges by the Direct Comparison Test: a" œ
n!
"
#
# $
#
' %
'
#%
, a# œ ˆ "# ‰ , a$ œ Šˆ "# ‰ ‹ œ ˆ "# ‰ , a% œ Šˆ "# ‰ ‹ œ ˆ "# ‰ , á
n
Ê an œ ˆ "# ‰ ˆ "# ‰ which is the nth-term of a convergent geometric series
anb1
an
55. converges by the Ratio Test: n lim
Ä_
n"
"
œ n lim
œ
1
#
Ä _ 2n 1
2nb1 (n 1)! (n 1)!
(2n 2)!
œ n lim
Ä_
†
(2n)!
2n n! n!
2(n 1)(n 1)
(2n #)(2n 1)
œ n lim
Ä_
(3n 3)!
1)! (n 2)!
anb1
56. diverges by the Ratio Test: n lim
œ n lim
† n! (n (3n)!
Ä _ an
Ä _ (n 1)! (n 2)! (n 3)!
(3n 3)(3 2)(3n 1)
2 ‰ ˆ 3n 1 ‰
œ n lim
œ n lim
3 ˆ 3n
n#
n 3 œ 3 † 3 † 3 œ 27 1
Ä _ (n 1)(n 2)(n 3)
Ä_
n
n (n!)
n
È
57. diverges by the Root Test: n lim
an ´ n lim
œ n lim
Ä_
Ä _ É an n b#
Ä_
n
œ_1
n!
n#
n
n (n!)
n (n!)
É
58. converges by the Root Test: n lim
œ n lim
œ n lim
É
ann bn
Ä_
Ä_
Ä_
nn#
"
Ÿ n lim
œ01
Ä_ n
n!
nn
ˆ " ‰ ˆ 2n ‰ ˆ 3n ‰ â ˆ n n 1 ‰ ˆ nn ‰
œ n lim
Ä_ n
"
#n ln 2
n n
n
È
59. converges by the Root Test: n lim
an œ n lim
œ n lim
Ä_
Ä _ É 2 n#
Ä_
n
#n
œ n lim
Ä_
n
n
n
È
60. diverges by the Root Test: n lim
an œ n lim
œ n lim
Ä_
Ä _ É a#n b#
Ä_
n
4
œ_1
n
n
anb1
an
61. converges by the Ratio Test: n lim
Ä_
1†3 â (2n 1)
(2†4 â #n) a3n 1b
62. converges by the Ratio Test: an œ
Ê n lim
Ä_
(2n 2)!
c2nb1 (n 1)!d# a3nb1 1b
#
œ n lim
Š 4n 6n 2 ‹
Ä _ 4n# 8n 4
63. Ratio: n lim
Ä_
anb1
an
†
a1 3cn b
a3 3cn b
œ n lim
Ä_
œ n lim
Ä_
a2n n!b# a3n 1b
(2n)!
œ1†
"
(n 1)p
†
"
3
np
1
œ
"
3
anb1
an
œ n lim
Ä_
"
(ln (n 1))p
†
†
1†2†3†4 â (2n 1)(2n)
(2†4 â 2n)# a3n 1b
œ
œ n lim
Ä_
4n 2n n!
1†3† â †(2n 1)
œ
œ n lim
Ä_
2n "
(4†#)(n 1)
(2n)!
a2n n!b# a3n 1b
(2n ")(2n 2) a3n 1b
2# (n 1)# a3n 1 1b
1
(ln n)p
1
"
n n ‰p
ˆÈ
"
(1)p
œ
œ ’n lim
Ä_
œ 1 Ê no conclusion
ln n
ln (n 1) “
p
œ ”n lim
Ä_
ˆ "n ‰
p
ˆ n b 1 ‰ • œ Šn lim
Ä_
"
n"
n ‹
œ (1)p œ 1 Ê no conclusion
"
n
n
È
Root: n lim
an œ n lim
É
(ln n)p œ
Ä_
Ä_
"
p
lim (ln n)1În ‹
ŠnÄ_
ˆ
"
‰
; let f(n) œ (ln n)1În , then ln f(n) œ
ln (ln n)
n ln n
Ê n lim
ln f(n) œ n lim
œ n lim
œ n lim
n
1
Ä_
Ä_
Ä_
Ä_
"
ln fÐnÑ
!
n
È an œ
œ n lim
e
œ
e
œ
1;
therefore
lim
Ä_
nÄ_
"
n ln n
p
lim (ln n)1În ‹
ŠnÄ_
65. an Ÿ
n
2n
_
_
for every n and the series !
nœ1
n
#n
œ
ˆ n ‰p œ 1p œ 1 Ê no conclusion
œ n lim
Ä_ n1
n "
n
È
É
Root: n lim
an œ n lim
np œ n lim
Ä_
Ä_
Ä_
64. Ratio: n lim
Ä_
1†3† â †(2n 1)(2n 1)
4nb1 2nb1 (n 1)!
œ01
ln (ln n)
n
œ 0 Ê n lim
(ln n)1În
Ä_
œ (1)" p œ 1 Ê no conclusion
converges by the Ratio Test since n lim
Ä_
(n ")
2nb1
†
2n
n
œ
"
#
Ê ! an converges by the Direct Comparison Test
nœ1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1
p
"
4
1
601
602
Chapter 10 Infinite Sequences and Series
2
66.
2n
n!
0 for all n
1; lim
nÄ_
_
2
2anb1b
anb1b!
2
2n
n!
n2 b2nb1
œ
lim Š a2n1b†n! †
nÄ_
n!
‹
2n2
2nb1
†4 ‰
ˆ 2†4 1ln 4 ‰
œ lim Š 2n1 ‹ œ lim ˆ n2
1 œ lim
n
nÄ_
n
nÄ_
nÄ_
n2
œ _ 1 Ê ! 2n! diverges
n œ1
10.6 ALTERNATING SERIES, ABSOLUTE AND CONDITIONAL CONVERGENCE
1.
converges by the Alternating Convergence Test since: un œ
Ê
1
Èn1
Ÿ
1
Èn
Ê un1 Ÿ un ;
lim un œ
lim 1
nÄ_ Èn
nÄ_
1
Èn
0 for all n
1; n
1Ê n1
_
_
n œ1
n œ1
n Ê Èn 1
Èn
œ 0.
2. converges absolutely Ê converges by the Alternating Convergence Test since ! kan k œ !
"
n$Î#
which is a
convergent p-series
3. converges Ê converges by Alternating Series Test since: un œ
Ê an 1b3n1
n 3n Ê
1
an1b3nb1
Ÿ
1
n 3n
Ê un1 Ÿ un ;
1
n3n
0 for all n
lim un œ
4. converges Ê converges by Alternating Series Test since: un œ
n Ê 3n 1
4
aln nb2
0 for all n
2; n
2Ên1
Ÿ
1
aln nb2
Ê
4
aln nb2
Ê un 1 Ÿ u n ;
5. converges Ê converges by Alternating Series Test since: un œ
n
n2 1
0 for all n
Ê ln an 1b
lim un œ
nÄ_
ln n Ê aln an 1bb2
lim 4 2
nÄ_ aln nb
Ê n3 2n2 2n
Ê
n
n 2 1
aln nb2 Ê
1
aln an1bb2
Ÿ
4
aln an1bb2
Ê un1 Ÿ un ;
lim un œ
nÄ_
lim 2 n
nÄ_ n 1
1 Ê 2n2 2n
1; n
n3 n2 n 1 Ê nŠan 1b2 1‹
2
lim n2 5
nÄ_ n 4
7. diverges Ê diverges by nth Term Test for Divergence since:
2n
2
nÄ_ n
lim
œ1Ê
œ
lim 10
nÄ_ n 2
œ_Ê
5
lim a1bn1 nn2
4 œ does not exist
2
lim a1bn1 2n2 œ does not exist
n
nÄ_
_
_
n œ1
n œ1
10n
a n 1 bx ,
n
10
_
n ‰n
ˆ n ‰n Á 0 Ê ! (1)n1 ˆ 10
1 Ê n lim
diverges
Ä _ 10
n œ1
10. converges by the Alternating Series Test because f(x) œ ln x is an increasing function of x Ê
un1 for n
1; also un
0 for n
1 and
"
lim
n Ä _ ln n
11. converges by the Alternating Series Test since f(x) œ
Ê un
un1 ; also un
0 for n
which converges by the
œ01
9. diverges by the nth-Term Test since for n 10 Ê
Ê un
an2 1ban 1b
nÄ_
8. converges absolutely Ê converges by the Absolute Convergence Test since ! kan k œ !
anb1
nÄ_ an
n2 n 1
œ 0.
6. diverges Ê diverges by nth Term Test for Divergence since:
Ratio Test, since lim
n
œ 0.
n3 n2 n 1 Ê nan2 2n 2b
n 1
an1b2 1
3n
œ 0.
lim 1 n
nÄ_ n 3
nÄ_
1Ên1
1; n
ln x
x
is decreasing
œ0
Ê f w (x) œ
1 and n lim
u œ n lim
Ä_ n
Ä_
"
ln x
ln n
n
1 ln x
x#
œ n lim
Ä_
0 when x e Ê f(x) is decreasing
Š "n ‹
1
œ0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.6 Alternating Series, Absolute and Conditional Convergence
12. converges by the Alternating Series Test since f(x) œ ln a1 x" b Ê f w (x) œ
Ê un
un1 ; also un
unb1 ; also un
0 for x 0 Ê f(x) is decreasing
ˆ1 n" ‰‹ œ ln 1 œ 0
1 and n lim
u œ n lim
ln ˆ1 "n ‰ œ ln Šn lim
Ä_ n
Ä_
Ä_
0 for n
13. converges by the Alternating Series Test since f(x) œ
Ê un
"
x(x 1)
1 and n lim
u œ
Ä_ n
0 for n
3È n 1
Èn 1
14. diverges by the nth-Term Test since n lim
Ä_
_
_
nœ1
nœ1
Èx "
x1
1 x 2È x
2Èx (x 1)#
Ê f w (x) œ
Èn "
lim
n Ä _ n1
0 Ê f(x) is decreasing
œ0
3É 1
œ n lim
Ä_
"
n
"
1 Š Èn ‹
œ3Á0
" ‰n
15. converges absolutely since ! kan k œ ! ˆ 10
a convergent geometric series
16. converges absolutely by the Direct Comparison Test since ¹ (1)
nb1
(0.1)n
n
¹œ
"
(10)n n
n
" ‰
ˆ 10
which is the nth term
of a convergent geometric series
17. converges conditionally since
"
Èn
"
Èn 1
"
Èn
0 and n lim
Ä_
_
_
n œ1
n œ1
œ 0 Ê convergence; but ! kan k œ !
"
n"Î#
is a divergent p-series
18. converges conditionally since
_
_
! kan k œ !
nœ1
nœ1
"
1 Èn
"
1 Èn
"
1 Èn 1
is a divergent series since
_
_
n œ1
nœ1
19. converges absolutely since ! kan k œ !
n
n $ 1
n!
#n
20. diverges by the nth-Term Test since n lim
Ä_
21. converges conditionally since
_
œ!
n œ1
"
n3
diverges because
"
n3
"
n3
"
(n 1) 3
"
4n
"
1 Èn
0 and n lim
Ä_
"
1 È n
and
n
n $ 1
"
#È n
_
and !
"
n#
nœ1
"
n"Î#
œ 0 Ê convergence; but
is a divergent p-series
which is the nth-term of a converging p-series
œ_
0 and n lim
Ä_
_
and !
n œ1
"
n
"
n 3
_
œ 0 Ê convergence; but ! kan k
n œ1
is a divergent series
_
22. converges absolutely because the series ! ¸ sinn# n ¸ converges by the Direct Comparison Test since ¸ sinn# n ¸ Ÿ
n œ1
3n
5n
23. diverges by the nth-Term Test since n lim
Ä_
œ1Á0
nb1
24. converges absolutely by the Direct Comparison Test since ¹ (n2)5n ¹ œ
2nb1
n 5 n
n
2 ˆ 25 ‰ which is the nth term
of a convergent geometric series
25. converges conditionally since f(x) œ
un unb1 0 for n
_
œ!
n œ1
"
n#
_
!
nœ1
"
n
"
x#
"
x
Ê f w (x) œ ˆ x2$
"‰
x#
0 Ê f(x) is decreasing and hence
_
_
n œ1
n œ1
ˆ " "n ‰ œ 0 Ê convergence; but ! kan k œ !
1 and n lim
Ä _ n#
603
1 n
n#
is the sum of a convergent and divergent series, and hence diverges
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
n#
604
Chapter 10 Infinite Sequences and Series
26. diverges by the nth-Term Test since n lim
a œ n lim
101În œ 1 Á 0
Ä_ n
Ä_
27. converges absolutely by the Ratio Test: n lim
Š uunbn 1 ‹ œ n lim
Ä_
Ä_ ”
28. converges conditionally since f(x) œ
Ê un unb1 0 for n
'2_ x dxln x œ
lim
Š "x ‹
bÄ_
_
_
n œ1
n œ1
Ê ! kan k œ !
'2b ln x dx œ
"
n ln n
n 1
•œ
2
3
1
1d
Ê f w (x) œ cln(x(x)
ln x)# 0 Ê f(x) is decreasing
"
x ln x
"
n ln n
2 and n lim
Ä_
(n")# ˆ 23 ‰
n
n# ˆ 23 ‰
œ 0 Ê convergence; but by the Integral Test,
lim cln (ln x)d b2 œ lim cln (ln b) ln (ln 2)d œ _
bÄ_
bÄ_
diverges
_
" xb#
29. converges absolutely by the Integral Test since '1 atan" xb ˆ 1 " x# ‰ dx œ lim ’ atan #
bÄ_
œ lim ’atan
bÄ_
"
#
"
bb atan
#
1b “ œ
30. converges conditionally since f(x) œ
œ
1 Š lnxx ‹ ln
x Š lnxx ‹
(x ln x)#
œ n lim
Ä_
Š "n ‹
1 Š n" ‹
_
_
n œ1
nœ1
! kan k œ !
œ
"
#
1 ln x
(x ln x)#
#
#
’ˆ 1# ‰ ˆ 14 ‰ “ œ
ln x
x ln x
Ê f w (x) œ
0 Ê un
1
31 #
32
Š "x ‹ (x ln x) (ln x) Š1 x" ‹
(x ln x)#
un1 0 when n e and n lim
Ä_
œ 0 Ê convergence; but n ln n n Ê
ln n
n ln n
b
“
"
nln n
"
n
Ê
ln n
n ln n
ln n
nln n
"
n
so that
diverges by the Direct Comparison Test
31. diverges by the nth-Term Test since n lim
Ä_
_
_
n œ1
nœ1
n
n1
œ1Á0
n
32. converges absolutely since ! kan k œ ! ˆ "5 ‰ is a convergent geometric series
33. converges absolutely by the Ratio Test: n lim
Š uunbn 1 ‹ œ n lim
Ä_
Ä_
("00)nb1
(n1)!
_
_
n œ1
n œ1
34. converges absolutely by the Direct Comparison Test since ! kan k œ !
†
n!
(100)n
œ n lim
Ä_
"
n# 2n 1
and
"00
n1
œ01
"
n# 2n 1
"
n#
nth-term of a convergent p-series
_
_
n œ1
n œ1
_
35. converges absolutely since ! kan k œ ! ¹ (nÈ1)n ¹ œ !
_
36. converges conditionally since !
n œ1
_
_
n œ1
n œ1
! kan k œ !
"
n
cos n1
n
n
n œ1
_
œ!
nœ1
(1)n
n
"
n$Î#
is a convergent p-series
is the convergent alternating harmonic series, but
diverges
1)
n
È
kan k œ n lim
37. converges absolutely by the Root Test: n lim
Š (n(2n)
n ‹
Ä_
Ä_
n
1 În
œ n lim
Ä_
n"
#n
œ
"
#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1
which is the
Section 10.6 Alternating Series, Absolute and Conditional Convergence
38. converges absolutely by the Ratio Test: n lim
¹ anb1 ¹ œ n lim
Ä _ an
Ä_
a(n 1)!b#
((2n 2)!)
39. diverges by the nth-Term Test since n lim
kan k œ n lim
Ä_
Ä_
œ n lim
Ä_
ˆ n # 1 ‰n1 œ _ Á 0
n lim
Ä_
(n 1)(n 2)â(n (n 1))
#nc1
œ n lim
Ä_
(2n)!
2n n! n
(n 1)# 3
(2n 2)(2n 3)
œ
3
4
Èn 1 Èn
1
†
Èn 1 Èn
Èn 1 Èn
œ
"
Èn 1 Èn
_
decreasing sequence of positive terms which converges to 0 Ê !
n œ1
_
_
n œ1
nœ1
"
Èn 1 Èn
Èn
lim
nÄ_
"
1
Èn
"
Èn
(n 1)#
(2n 2)(2n 1)
œ n lim
Ä_
œ
"
4
1
(n ")(n 2)â(2n)
2n n
†
(2n 1)!
n! n! 3n
1
41. converges conditionally since
! kan k œ !
(2n)!
(n!)#
(n 1)! (n 1)! 3nb1
(2n 3)!
40. converges absolutely by the Ratio Test: n lim
¹ anabn 1 ¹ œ n lim
Ä_
Ä_
œ n lim
Ä_
†
605
and š Èn 1" Èn › is a
(")n
Èn 1 Èn
diverges by the Limit Comparison Test (part 1) with
œ n lim
Ä_
Èn
Èn 1 Èn
œ n lim
Ä_
1
É1 1n 1
œ
converges; but
"
Èn ;
a divergent p-series:
"
#
È
#
n n
42. diverges by the nth-Term Test since n lim
ŠÈn# n n‹ œ n lim
ŠÈn# n n‹ † Š Ènn#
‹
Ä_
Ä_
n n
œ n lim
Ä_
n
Èn# nn
œ n lim
Ä_
"
É1 "n 1
"
#
œ
Á0
É n Èn Èn
43. diverges by the nth-Term Test since n lim
ŠÉn Èn Èn‹ œ n lim
ŠÉ n È n È n ‹
Ä_
Ä_ –
Én Èn Èn —
Èn
œ n lim
Ä_
É n Èn Èn
œ n lim
Ä_
"
É1
"
Èn 1
"
#
œ
Á0
44. converges conditionally since š Èn "Èn 1 › is a decreasing sequence of positive terms converging to 0
_
(")n
Èn Èn 1
Ê !
n œ1
_
so that !
nœ1
converges; but n lim
Ä_
"
Èn Èn 1
"
Èn
Š È"n ‹
Š Èn
1
‹
Èn
È n È n 1
œ n lim
Ä_
_
diverges by the Limit Comparison Test with !
nœ1
45. converges absolutely by the Direct Comparison Test since sech (n) œ
"
Èn
œ n lim
Ä_
"
1É1 "n
"
#
œ
which is a divergent p-series
2
en ecn
œ
2en
e2n 1
2en
e2n
œ
2
en
which is the
nth term of a convergent geometric series
_
_
n œ1
nœ1
46. converges absolutely by the Limit Comparison Test (part 1): ! kan k œ !
Apply the Limit Comparison Test with
lim
nÄ_
47.
1
4
1
6
n2
Œ
2
en c ecn
1
en
1
8
1
10
œ n lim
Ä_
1
12
1
14
n 1 Ê 2 an 2 b
2en
en ecn
1
en ,
the n-th term of a convergent geometric series:
œ n lim
Ä_
_
ÞÞÞ œ !
n œ1
2
1 ec2n
(")nb1
2 an 1 b ;
2an 1b Ê
2
en ecn
œ2
converges by Alternating Series Test since: un œ
1
2aan 1b 1b
Ÿ
1
2 an 1 b
Ê u n 1 Ÿ un ;
lim un œ
nÄ_
1
2 an 1 b
lim 1
nÄ_ 2an1b
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
0 for all n
œ 0.
1;
606
Chapter 10 Infinite Sequences and Series
48. 1
1
4
1
9
1
16
1
25
1
36
1
49
1
64
_
_
_
n œ1
n œ1
n œ1
Þ Þ Þ œ ! an ; converges by the Absolute Convergence Test since ! kan k œ !
"
n#
which is a convergent p-series
49. kerrork ¸(1)' ˆ "5 ‰¸ œ 0.2
51. kerrork ¹(1)'
(0.01)&
5 ¹
50. kerrork ¸(1)' ˆ 10" & ‰¸ œ 0.00001
œ 2 ‚ 10""
52. kerrork k(1)% t% k œ t% 1
53. kerrork 0.001 Ê un1 0.001 Ê
1
an 1b2 3
0.001 Ê an 1b2 3 1000 Ê n 1 È997 ¸ 30.5753 Ê n
54. kerrork 0.001 Ê un1 0.001 Ê
n1
an 1b2 1
0.001 Ê an 1b2 1 1000an 1b Ê n
¸ 998.9999 Ê n
31
998È9982 4a998b
2
999
55. kerrork 0.001 Ê un1 0.001 Ê
1
3
ˆan 1b 3Èn 1‰
3
0.001 Ê Šan 1b 3Èn 1‹ 1000
2
È
Ê ŠÈn 1‹ 3Èn 1 10 0 Ê Èn 1 œ 3 29 40 œ 2 Ê n œ 3 Ê n
56. kerrork 0.001 Ê un1 0.001 Ê
1
lnalnan 3bb
4
0.001 Ê lnalnan 3bb 1000 Ê n 3 ee
1000
¸ 5.297 ‚ 10323228467
which is the maximum arbitrary-precision number represented by Mathematica on the particular computer solving this
problem..
57.
"
(2n)!
58.
"
n!
Ê (2n)!
5
10'
10'
5
Ê
5
10'
59. (a) an
10'
5
œ 200,000 Ê n
n! Ê n
an1 fails since
_
_
nœ1
nœ1
"
3
9 Ê 11
"
#
"
#!
5 Ê 1
"
#!
"
3!
"
4!
_
_
nœ1
nœ1
"
4!
"
5!
"
6!
"
6!
"
7!
"
8!
¸ 0.54030
"
8!
¸ 0.367881944
n
n
n
n
(b) Since ! kan k œ ! ˆ 3" ‰ ˆ "# ‰ ‘ œ ! ˆ "3 ‰ ! ˆ "# ‰ is the sum of two absolutely convergent
series, we can rearrange the terms of the original series to find its sum:
ˆ "3
"
9
"
27
60. s#! œ 1
"
#
"
3
á ‰ ˆ "#
"
4
á
"
19
"
4
"
20
"
8
በœ
ˆ "3 ‰
1 ˆ 3" ‰
ˆ "# ‰
1 ˆ "# ‰
œ
"
#
1 œ #"
"
#
†
"
#1
¸ 0.6687714032 Ê s#!
¸ 0.692580927
_
61. The unused terms are ! (1)j 1 aj œ (1)n 1 aan 1 an 2 b (1)n 3 aan 3 an 4 b á
jœn 1
œ (1)n 1 caan 1 an 2 b aan 3 an 4 b á d . Each grouped term is positive, so the remainder
has the same sign as (1)n 1 , which is the sign of the first unused term.
62. sn œ
"
1 †2
"
#†3
"
3 †4
á
"
n(n 1)
n
œ!
k œ1
"
k(k 1)
n
œ ! ˆ k"
k œ1
œ ˆ1 "# ‰ ˆ "# 3" ‰ ˆ 3" 4" ‰ ˆ 4" 5" ‰ á ˆ n"
" ‰
k1
" ‰
n1
which are the first 2n terms
of the first series, hence the two series are the same. Yes, for
n
sn œ ! ˆ k"
k œ1
" ‰
k1
œ ˆ1 "# ‰ ˆ "# 3" ‰ ˆ 3" 4" ‰ ˆ 4" 5" ‰ á ˆ n " 1 n" ‰ ˆ n"
" ‰
n1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
œ 1
"
n1
Section 10.6 Alternating Series, Absolute and Conditional Convergence
607
ˆ1 n " 1 ‰ œ 1 Ê both series converge to 1. The sum of the first 2n 1 terms of the first
Ê n lim
s œ n lim
Ä_ n
Ä_
ˆ1 n " 1 ‰ œ 1.
series is ˆ1 n " 1 ‰ n " 1 œ 1. Their sum is n lim
s œ n lim
Ä_ n
Ä_
_
_
_
_
n œ1
n œ1
n œ1
n œ1
63. Theorem 16 states that ! kan k converges Ê ! an converges. But this is equivalent to ! an diverges Ê ! kan k diverges
_
_
n œ1
n œ1
64. ka" a# á an k Ÿ ka" k ka# k á kan k for all n; then ! kan k converges Ê ! an converges and these imply that
_
_
n œ1
n œ1
º ! an º Ÿ ! kan k
_
65. (a) ! kan bn k converges by the Direct Comparison Test since kan bn k Ÿ kan k kbn k and hence
n œ1
_
! aan bn b converges absolutely
n œ1
_
_
_
(b) ! kbn k converges Ê ! bn converges absolutely; since ! an converges absolutely and
nœ1
_
nœ1
nœ1
_
_
! bn converges absolutely, we have ! can (bn )d œ ! aan bn b converges absolutely by part (a)
nœ1
_
_
_
nœ1
n œ1
nœ1
nœ1
nœ1
_
(c) ! kan k converges Ê kkk ! kan k œ ! kkan k converges Ê ! kan converges absolutely
66. If an œ bn œ (1)n
"
Èn
_
, then ! (1)n
nœ1
67. s" œ "# , s# œ "# 1 œ
"
#
s$ œ 1
s% œ s$
s& œ s%
s' œ s&
s( œ s'
"
4
"
6
"
8
"
3 ¸ 0.1766,
"
"
"
#4 #6 #8
"
5 ¸ 0.312,
"
"
"
46 48 50
"
#
"
Èn
nœ1
_
_
nœ1
nœ1
converges, but ! an bn œ !
"
n
diverges
,
"
10
"
1#
"
14
"
16
"
18
"
#0
"
2#
¸ 0.5099,
"
30
"
3#
"
34
"
36
"
38
"
40
"
42
"
44
¸ 0.512,
"
52
"
54
"
56
"
58
"
60
"
62
"
64
"
66
¸ 0.51106
N" 1
68. (a) Since ! kan k converges, say to M, for % 0 there is an integer N" such that º ! kan k Mº
nœ1
N" 1
N" 1
_
nœ1
nœ1
nœN"
Í » ! kan k ! kan k ! kan k »
%
#
_
Í » ! k an k »
nœN"
%
#
_
Í ! kan k
nœN"
%
#
%
#
. Also, ! an
converges to L Í for % 0 there is an integer N# (which we can choose greater than or equal to N" ) such
that ksN# Lk
%
#
_
. Therefore, ! kan k
nœN"
%
#
and ksN# Lk
%
#
.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
608
Chapter 10 Infinite Sequences and Series
_
k
nœ1
nœ1
(b) The series ! kan k converges absolutely, say to M. Thus, there exists N" such that º ! kan k Mº %
whenever k N" . Now all of the terms in the sequence ekbn kf appear in ekan kf. Sum together all of the
N
terms in ekbn kf, in order, until you include all of the terms ekan kf nœ" 1 , and let N# be the largest index in the
N#
N#
_
n œ1
nœ1
n œ1
sum ! kbn k so obtained. Then º ! kbn k Mº % as well Ê ! kbn k converges to M.
10.7 POWER SERIES
_
nb1
1. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ xxn ¹ 1 Ê kxk 1 Ê 1 x 1; when x œ 1 we have ! (1)n , a divergent
Ä_
Ä_
n œ1
_
series; when x œ 1 we have ! 1, a divergent series
n œ1
(a) the radius is 1; the interval of convergence is 1 x 1
(b) the interval of absolute convergence is 1 x 1
(c) there are no values for which the series converges conditionally
nb1
2. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (x(x5)5)n ¹ 1 Ê kx 5k 1 Ê 6 x 4; when x œ 6 we have
Ä_
Ä_
_
_
n œ1
nœ1
! (1)n , a divergent series; when x œ 4 we have ! 1, a divergent series
(a) the radius is 1; the interval of convergence is 6 x 4
(b) the interval of absolute convergence is 6 x 4
(c) there are no values for which the series converges conditionally
nb1
1)
"
"
3. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (4x
(4x 1)n ¹ 1 Ê k4x 1k 1 Ê 1 4x 1 1 Ê # x 0; when x œ # we
Ä_
Ä_
_
_
_
_
_
n œ1
n œ1
n œ1
n œ1
n œ1
have ! (1)n (1)n œ ! (1)2n œ ! 1n , a divergent series; when x œ 0 we have ! (1)n (1)n œ ! (1)n ,
a divergent series
(a) the radius is "4 ; the interval of convergence is "# x 0
(b) the interval of absolute convergence is "# x 0
(c) there are no values for which the series converges conditionally
nb1
4. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (3xn2)1
Ä_
Ä_
Ê 1 3x 2 1 Ê
"
3
†
n
(3x 2)n ¹
ˆ n ‰ 1 Ê k3x 2k 1
1 Ê k3x 2k n lim
Ä _ n1
x 1; when x œ
"
3
_
n œ1
(b) the interval of absolute convergence is
"
3
(c) the series converges conditionally at x œ
nb1
2)
5. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹ (x 10
nb1
Ä_
Ä_
n œ1
"
n
conditionally convergent; when x œ 1 we have !
(a) the radius is "3 ; the interval of convergence is
_
we have !
"
3
(")n
n
which is the alternating harmonic series and is
, the divergent harmonic series
Ÿx1
x1
"
3
10n
(x 2)n ¹
1 Ê
kx 2 k
10
1 Ê kx 2k 10 Ê 10 x 2 10
_
_
nœ1
nœ1
Ê 8 x 12; when x œ 8 we have ! (")n , a divergent series; when x œ 12 we have ! 1, a divergent series
(a) the radius is "0; the interval of convergence is 8 x 12
(b) the interval of absolute convergence is 8 x 12
(c) there are no values for which the series converges conditionally
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.7 Power Series
nb1
6. n lim
k2xk 1 Ê k2xk 1 Ê "# x
¹ uunbn 1 ¹ 1 Ê n lim
¹ (2x)
(2x)n ¹ 1 Ê n lim
Ä_
Ä_
Ä_
_
! (")n , a divergent series; when x œ
n œ1
"
#
"
#
; when x œ "# we have
_
we have ! 1, a divergent series
n œ1
(a) the radius is "# ; the interval of convergence is "# x
(b) the interval of absolute convergence is "# x
"
#
"
#
(c) there are no values for which the series converges conditionally
nb1
1)x
7. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (n (n
3) †
Ä_
Ä_
(n 2)
nxn ¹
1 Ê kxk n lim
Ä_
_
Ê 1 x 1; when x œ 1 we have ! (")n
n œ1
_
have !
n œ1
n
n#,
n
n#
(n 1)(n 2)
(n 3)(n)
1 Ê kxk 1
, a divergent series by the nth-term Test; when x œ " we
a divergent series
(a) the radius is "; the interval of convergence is " x "
(b) the interval of absolute convergence is " x "
(c) there are no values for which the series converges conditionally
nb1
8. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (x n2)1
Ä_
Ä_
†
n
(x 2)n ¹
ˆ
1 Ê kx 2k n lim
Ä_
_
Ê 1 x 2 1 Ê 3 x 1; when x œ 3 we have !
n œ1
_
!
n œ1
(1)n
n ,
"
n,
n ‰
n1
1 Ê kx 2k 1
a divergent series; when x œ " we have
a convergent series
(a) the radius is "; the interval of convergence is 3 x Ÿ "
(b) the interval of absolute convergence is 3 x "
(c) the series converges conditionally at x œ 1
nb1
x
9. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹
Ä_
Ä _ (n 1)Èn 1 3nb1
Ê
kx k
3
nÈ n 3n
xn ¹
1 Ê
kxk
3
n
n
‹
n 1 ‹ ŠÉ n lim
Ä _ n1
Šn lim
Ä_
_
(1)(1) 1 Ê kxk 3 Ê 3 x 3; when x œ 3 we have !
n œ1
_
when x œ 3 we have !
n œ1
1
,
n$Î#
(")n
,
n$Î#
1
an absolutely convergent series;
a convergent p-series
(a) the radius is 3; the interval of convergence is 3 Ÿ x Ÿ 3
(b) the interval of absolute convergence is 3 Ÿ x Ÿ 3
(c) there are no values for which the series converges conditionally
nb1
10. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (xÈn1) 1 †
Ä_
Ä_
Èn
(x 1)n ¹
1 Ê kx 1k Én lim
Ä_
_
Ê 1 x 1 1 Ê 0 x 2; when x œ 0 we have !
n œ1
_
we have !
n œ1
1
,
n"Î#
(")n
,
n"Î#
n
n1
609
1 Ê kx 1k 1
a conditionally convergent series; when x œ 2
a divergent series
(a) the radius is 1; the interval of convergence is 0 Ÿ x 2
(b) the interval of absolute convergence is 0 x 2
(c) the series converges conditionally at x œ 0
nb1
ˆ " ‰ 1 for all x
11. n lim
† n! ¹ 1 Ê kxk n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ x
Ä_
Ä _ (n 1)! xn
Ä _ n1
(a) the radius is _; the series converges for all x
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
610
Chapter 10 Infinite Sequences and Series
(b) the series converges absolutely for all x
(c) there are no values for which the series converges conditionally
nb1
nb1
ˆ " ‰ 1 for all x
12. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ 3 x † 3nn!xn ¹ 1 Ê 3 kxk n lim
Ä_
Ä _ (n 1)!
Ä _ n1
(a) the radius is _; the series converges for all x
(b) the series converges absolutely for all x
(c) there are no values for which the series converges conditionally
13. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹4
Ä_
Ä_
nb1 2nb2
x
n1
†
n
4n x2n ¹
ˆ 4n ‰ œ 4x# 1 Ê x#
1 Ê x# n lim
Ä _ n1
_
_
n œ1
nœ1
n
2n
Ê 12 x 12 ; when x œ 12 we have ! 4n ˆ 12 ‰ œ !
_
!
n œ1
4n ˆ 1 ‰2n
n 2
_
œ!
n œ1
1
n,
1
n
1
4
, a divergent p-series; when x œ
1
2
we have
a divergent p-series
(a) the radius is 12 ; the interval of convergence is 12 x
(b) the interval of absolute convergence is 12 x
1
2
1
2
(c) there are no values for which the series converges conditionally
nb1
14. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹ (x 1)
Ä_
Ä _ an 1b2 3nb1
n2 3n
(x 1)n ¹
_
2
1 Ê lx 1l n lim
Š n ‹ œ 13 lx 1l 1
Ä _ 3an 1b2
_
Ê 2 x 4; when x œ 2 we have ! (n2 3)3n œ ! (n1)
, an absolutely convergent series; when x œ 4 we have
2
n
n œ1
_
n
nœ1
_
n
! (3)
! 12 , an absolutely convergent series.
n2 3n œ
n
n œ1
n œ1
(a) the radius is 3; the interval of convergence is 2 Ÿ x Ÿ 4
(b) the interval of absolute convergence is 2 Ÿ x Ÿ 4
(c) there are no values for which the series converges conditionally
nb1
x
15. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹
Ä_
Ä _ È(n 1)# 3
È n# 3
¹
xn
_
Ê 1 x 1; when x œ 1 we have !
n œ1
_
!
n œ1
"
È n# 3
1 Ê kxk Én lim
Ä_
(")n
È n# 3
n# 3
n# 2n 4
" Ê kxk 1
, a conditionally convergent series; when x œ 1 we have
, a divergent series
(a) the radius is 1; the interval of convergence is 1 Ÿ x 1
(b) the interval of absolute convergence is 1 x 1
(c) the series converges conditionally at x œ 1
n 1
x
16. n lim
†
¹ uun n 1 ¹ 1 Ê n lim
¹
Ä_
Ä _ È(n 1)# 3
È n# 3
¹
xn
_
Ê 1 x 1; when x œ 1 we have !
nœ1
1 Ê kxk Én lim
Ä_
"
È n# 3
n# 3
n# 2n 4
" Ê kxk 1
_
, a divergent series; when x œ 1 we have !
nœ1
a conditionally convergent series
(a) the radius is 1; the interval of convergence is 1 x Ÿ 1
(b) the interval of absolute convergence is 1 x 1
(c) the series converges conditionally at x œ 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
(")n
È n# 3
,
Section 10.7 Power Series
3)
17. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (n 1)(x
5nb1
Ä_
Ä_
nb1
†
5n
n(x 3)n ¹
1 Ê
kx 3 k
lim
5
nÄ_
ˆ n n " ‰ 1 Ê
_
Ê kx 3k 5 Ê 5 x 3 5 Ê 8 x 2; when x œ 8 we have !
n œ1
_
series; when x œ 2 we have !
n œ1
n5n
5n
n(5)n
5n
kx 3 k
5
611
1
_
œ ! (1)n n, a divergent
n œ1
_
œ ! n, a divergent series
n œ1
(a) the radius is 5; the interval of convergence is 8 x 2
(b) the interval of absolute convergence is 8 x 2
(c) there are no values for which the series converges conditionally
nb1
18. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹ (n 1)x
Ä_
Ä _ 4nb1 an# 2n 2b
_
Ê 4 x 4; when x œ 4 we have !
n œ1
4 n an # 1 b
¹
nxn
n(1)n
n# 1
1 Ê
kx k
4 n lim
Ä_
#
(n 1) n
1
¹ n an# a2n 2bb ¹ 1 Ê kxk 4
_
, a conditionally convergent series; when x œ 4 we have !
n œ1
n
n# 1
,
a divergent series
(a) the radius is 4; the interval of convergence is 4 Ÿ x 4
(b) the interval of absolute convergence is 4 x 4
(c) the series converges conditionally at x œ 4
19. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹
Ä_
Ä_
Èn 1 xnb1
3nb1
†
3n
È n xn ¹
1 Ê
kx k
3
ˆ n n 1 ‰ 1 Ê
Én lim
Ä_
kx k
3
1 Ê kxk 3
_
_
n œ1
n œ1
Ê 3 x 3; when x œ 3 we have ! (1)n Èn , a divergent series; when x œ 3 we have ! Èn, a divergent series
(a) the radius is 3; the interval of convergence is 3 x 3
(b) the interval of absolute convergence is 3 x 3
(c) there are no values for which the series converges conditionally
20. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹
Ä_
Ä_
nbÈ
1
n 1 (2x5)nb1
¹
n n (2x5)n
È
1 Ê k2x 5k n lim
Š
Ä_
nbÈ
1
n1
‹
n n
È
1
t
È
lim
t
Ä_
Ê k2x 5k Œ tlim
n n 1 Ê k2x 5k 1 Ê 1 2x 5 1 Ê 3 x 2; when x œ 3 we have
È
n
_
Ä_
_
n
n
n
! (1) È
È
n, a divergent series since n lim
n œ 1; when x œ 2 we have ! È
n, a divergent series
Ä_
n œ1
n œ1
(a) the radius is "# ; the interval of convergence is 3 x 2
(b) the interval of absolute convergence is 3 x 2
(c) there are no values for which the series converges conditionally
_
_
_
n œ1
n œ1
21. First, rewrite the series as ! a2 (1)n bax 1bn1 œ ! 2ax 1bn1 ! (1)n ax 1bn1 . For the series
n œ1
_
n
! 2ax 1bn1 : lim ¹ unb1 ¹ 1 Ê lim ¹ 2ax1nbc1 ¹ 1 Ê lx 1l lim 1 œ lx 1l 1 Ê 2 x 0; For the
un
nÄ_
n Ä _ 2 ax 1 b
nÄ_
n œ1
_
nb1
n
(1) ax1b
series ! (1)n ax 1bn1 : n lim
1 œ lx 1l 1
¹ uunbn 1 ¹ 1 Ê n lim
¹
¹ 1 Ê lx 1ln lim
Ä_
Ä _ (1)n ax1bnc1
Ä_
n œ1
_
Ê 2 x 0; when x œ 2 we have ! a2 (1)n ba1bn1 , a divergent series; when x œ 0 we have
n œ1
_
! a2 (1)n b, a divergent series
n œ1
(a) the radius is 1; the interval of convergence is 2 x 0
(b) the interval of absolute convergence is 2 x 0
(c) there are no values for which the series converges conditionally
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
612
Chapter 10 Infinite Sequences and Series
( 1)
22. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹
Ä_
Ä_
Ê
_
!
n œ1
17
9
x
19
9 ;
(1)n 32n ˆ 1 ‰n
3n
9
when x œ
_
œ!
n œ1
(1)n
3n ,
17
9
3
ax 2bnb1
3an 1b
nb1 2nb2
_
we have !
n œ1
†
(1)n 32n ˆ 1 ‰n
9
3n
(b) the interval of absolute convergence is
17
9
(c) the series converges conditionally at x œ
23.
_
œ!
nœ1
1
3n ,
9n
n1
œ 9lx 2l 1
a divergent series; when x œ
19
9
we have
a conditionally convergent series.
(a) the radius is 19 ; the interval of convergence is
lim ¹ uunbn 1 ¹
nÄ_
1 Ê lx 2ln lim
Ä_
3n
¹
(1)n 32n ax 2bn
1 Ê n lim
Ä_ »
Š1
n
"
nb1
1‹
xnb1
Š1 "n ‹ xn
n
17
9
x
xŸ
19
9
19
9
19
9
"
t
lim Š1 t ‹
e
Ä_
» 1 Ê kxk lim Š1 " ‹n 1 Ê kxk ˆ e ‰ 1 Ê kxk 1
n
nÄ_
t
_
n
Ê 1 x 1; when x œ 1 we have ! (1)n ˆ1 "n ‰ , a divergent series by the nth-Term Test since
n œ1
lim ˆ1
nÄ_
" ‰n
n
_
n
œ e Á 0; when x œ 1 we have ! ˆ1 n" ‰ , a divergent series
n œ1
(a) the radius is "; the interval of convergence is 1 x 1
(b) the interval of absolute convergence is 1 x 1
(c) there are no values for which the series converges conditionally
24. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ ln (nxnln1)xn
Ä_
Ä_
nb1
¹ 1 Ê kxk n lim
Ä_ º
ˆn " 1‰
ˆ n" ‰ º
ˆ n ‰ 1 Ê kxk 1
1 Ê kxk n lim
Ä _ n1
_
Ê 1 x 1; when x œ 1 we have ! (1)n ln n, a divergent series by the nth-Term Test since n lim
ln n Á 0;
Ä_
n œ1
_
when x œ 1 we have ! ln n, a divergent series
n œ1
(a) the radius is 1; the interval of convergence is 1 x 1
(b) the interval of absolute convergence is 1 x 1
(c) there are no values for which the series converges conditionally
nb1 nb1
x
ˆ1 n" ‰n ‹ Š lim (n 1)‹ 1
25. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (n 1)
¹ 1 Ê kxk Šn lim
nn xn
Ä_
Ä_
Ä_
nÄ_
Ê e kxk n lim
(n 1) 1 Ê only x œ 0 satisfies this inequality
Ä_
(a) the radius is 0; the series converges only for x œ 0
(b) the series converges absolutely only for x œ 0
(c) there are no values for which the series converges conditionally
nb1
26. n lim
(n 1) 1 Ê only x œ 4 satisfies this inequality
¹ uunbn 1 ¹ 1 Ê n lim
¹ (n n!1)!(x(x4)4)n ¹ 1 Ê kx 4k n lim
Ä_
Ä_
Ä_
(a) the radius is 0; the series converges only for x œ 4
(b) the series converges absolutely only for x œ 4
(c) there are no values for which the series converges conditionally
nb1
27. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹ (x 2)
Ä_
Ä _ (n 1) 2nb1
n2n
(x 2)n ¹
1 Ê
kx 2 k
lim
#
nÄ_
ˆ n n 1 ‰ 1 Ê
kx 2 k
#
1 Ê kx 2k 2
_
_
n œ1
n œ1
! (1)
Ê 2 x 2 2 Ê 4 x 0; when x œ 4 we have ! "
n , a divergent series; when x œ 0 we have
n
the alternating harmonic series which converges conditionally
(a) the radius is 2; the interval of convergence is 4 x Ÿ 0
(b) the interval of absolute convergence is 4 x 0
(c) the series converges conditionally at x œ 0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
nb1
,
Section 10.7 Power Series
nb1
613
nb1
(n 2)(x 1)
ˆ n 2 ‰ 1 Ê 2 kx 1k 1
28. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ ((2)2)n (n
1)(x 1)n ¹ 1 Ê 2 kx 1k n lim
Ä_
Ä_
Ä _ n1
Ê kx 1k
_
"
#
Ê "# x 1
"
#
"
#
Ê
x 3# ; when x œ
"
#
_
we have ! (n 1) , a divergent series; when x œ
n œ1
we have ! (1) (n 1), a divergent series
n
n œ1
(a) the radius is "# ; the interval of convergence is
(b) the interval of absolute convergence is
"
#
"
#
x
x
3
#
3
#
(c) there are no values for which the series converges conditionally
nb1
x
29. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹
Ä_
Ä _ (n 1) aln (n 1)b#
Ê kxk (1) Œn lim
Ä_
_
!
nœ1
(1)n
n(ln n)#
ˆ "n ‰
ˆ nb" 1 ‰
#
n(ln n)#
xn ¹
1 Ê kxk Šn lim
Ä_
n1
n ‹
1 Ê kxk Šn lim
Ä_
#
n
ln n
‹
n 1 ‹ Šn lim
Ä _ ln (n 1)
#
1
1 Ê kxk 1 Ê 1 x 1; when x œ 1 we have
_
which converges absolutely; when x œ 1 we have !
nœ1
"
n(ln n)#
which converges
(a) the radius is "; the interval of convergence is 1 Ÿ x Ÿ 1
(b) the interval of absolute convergence is 1 Ÿ x Ÿ 1
(c) there are no values for which the series converges conditionally
nb1
x
30. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹
Ä_
Ä _ (n 1) ln (n 1)
n ln (n)
xn ¹
1 Ê kxk Šn lim
Ä_
ln (n)
n
‹
n 1 ‹ Šn lim
Ä _ ln (n 1)
_
(1)n
n ln n
Ê kxk (1)(1) 1 Ê kxk 1 Ê 1 x 1; when x œ 1 we have !
n œ2
_
when x œ 1 we have !
n œ2
"
n ln n
1
, a convergent alternating series;
which diverges by Exercise 38, Section 9.3
(a) the radius is "; the interval of convergence is 1 Ÿ x 1
(b) the interval of absolute convergence is 1 x 1
(c) the series converges conditionally at x œ 1
2nb3
5)
31. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹ (4x
(n 1)$Î#
Ä_
Ä_
n$Î#
(4x 5)2n
1
¹ 1 Ê (4x 5)# Šn lim
Ä_
Ê k4x 5k 1 Ê 1 4x 5 1 Ê 1 x
absolutely convergent; when x œ
3
#
_
we have !
n œ1
(")2nb1
n$Î#
3
#
_
; when x œ 1 we have !
n œ1
$Î#
1 Ê (4x 5)# 1
(1)2nb1
n$Î#
_
œ!
n œ1
"
n$Î#
which is
, a convergent p-series
(a) the radius is "4 ; the interval of convergence is 1 Ÿ x Ÿ
(b) the interval of absolute convergence is 1 Ÿ x Ÿ
n
n1‹
3
#
3
#
(c) there are no values for which the series converges conditionally
nb2
32. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (3x2n1)4
Ä_
Ä_
†
2n 2
(3x 1)nb1 ¹
ˆ 2n 2 ‰ 1 Ê k3x 1k 1
1 Ê k3x 1k n lim
Ä _ 2n 4
_
Ê 1 3x 1 1 Ê 23 x 0; when x œ 23 we have !
n œ1
_
when x œ 0 we have !
n œ1
(")nb1
2n 1
_
œ!
nœ1
"
#n 1
(1)nb1
2n 1
, a conditionally convergent series;
, a divergent series
(a) the radius is "3 ; the interval of convergence is 32 Ÿ x 0
(b) the interval of absolute convergence is 23 x 0
(c) the series converges conditionally at x œ 23
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
3
#
614
Chapter 10 Infinite Sequences and Series
nb1
x
ˆ 1 ‰ 1 for all x
33. n lim
† 2†4†6xân a2nb ¹ 1 Ê kxk n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹
Ä_
Ä _ 2†4†6âa2nba2an 1bb
Ä _ 2n 2
(a) the radius is _; the series converges for all x
(b) the series converges absolutely for all x
(c) there are no values for which the series converges conditionally
nb2
3 5 7 2n 1 2 n 1
1x
2n 3 n
34. n lim
† 3†5†7âan2n2 1bxnb1 ¹ 1 Ê kxk n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ † † âa an ba1ba2 2nb1 b b
Š a2an 1bb2 ‹ 1 Ê only
Ä_
Ä_
Ä_
x œ 0 satisfies this inequality
(a) the radius is 0; the series converges only for x œ 0
(b) the series converges absolutely only for x œ 0
(c) there are no values for which the series converges conditionally
_
35. For the series !
n œ1
12ân n
12 22 â n2 x ,
recall 1 2 â n œ
nan b 1b
_
2
2 n
nan 1b
2
and 12 22 â n2 œ
nan 1ba2n 1b
6
_
nb1
rewrite the series as ! Œ n n b 1 2 2n b 1 xn œ ! ˆ 2n 3 1 ‰xn ; then n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ a2an3x
1 b 1 b †
Ä
_
Ä
_
6
n œ1
nœ1
a
ba
b
so that we can
a2n 1b
3xn ¹
1
_
Ê kxk n lim
¹ a2n 1b ¹ 1 Ê kxk 1 Ê 1 x 1; when x œ 1 we have ! ˆ 2n 3 1 ‰a1bn , a conditionally
Ä _ a2n 3b
n œ1
_
convergent series; when x œ 1 we have ! ˆ 2n 3 1 ‰, a divergent series.
n œ1
(a) the radius is 1; the interval of convergence is 1 Ÿ x 1
(b) the interval of absolute convergence is 1 x 1
(c) the series converges conditionally at x œ 1
_
36. For the series ! ŠÈn 1 Èn‹ax 3bn , note that Èn 1 Èn œ
n œ1
_
can rewrite the series as !
n œ1
Ê lx 3ln lim
Ä_
a x 3 bn
Èn 1 Èn ;
Èn 1 Èn
Èn 2 Èn 1
Èn 1 Èn
1
†
nb1
Èn 1 Èn
Èn 1 Èn
then n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹ ax 3 b
Ä_
Ä _ Èn 2 Èn 1
ax 3bn
_
n œ1
1
Èn 1 Èn ,
so that we
¹1
a 1 b n
Èn 1 Èn ,
n œ1
_
1
Èn 1 Èn
Èn 1 Èn
1 Ê lx 3l 1 Ê 2 x 4; when x œ 2 we have !
convergent series; when x œ 4 we have !
œ
a conditionally
a divergent series;
(a) the radius is 1; the interval of convergence is 2 Ÿ x 4
(b) the interval of absolute convergence is 2 x 4
(c) the series converges conditionally at x œ 2
nb1
an 1bxx
37. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹
Ä_
Ä _ 3†6†9âa3nba3an 1bb
3†6†9âa3nb
¹
nx xn
2 nb1
2 4 6 2n 2 n 1
x
38. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹ a † † âa ba a bbb
Ä_
Ä _ a2†5†8âa3n 1ba3an 1b 1bb2
9
9
Ê lxl 4 Ê R œ 4
2 nb1
n 1 x
39. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹ aa bxb
Ä_
Ä _ 2nb1 a2an 1bbx
2n a2nbx
¹
an x b 2 x n
an 1 b
1 Ê lxln lim
¹
¹1Ê
Ä _ 3 an 1 b
a2†5†8âa3n 1bb2
¹
a2†4†6âa2nbb2 xn
lx l
3
1 Ê lxl 3 Ê R œ 3
2
1 Ê lxln lim
¹ a2n 2b ¹ 1 Ê
Ä _ a3n 2b2
2
an 1 b
1 Ê lxln lim
¹
¹1Ê
Ä _ 2a2n 2ba2n 1b
lx l
8
4 lx l
9
1
1 Ê lxl 8 Ê R œ 8
2
n
n ‰n n
n
Ɉ
ˆ n ‰n 1 Ê lxle1 1 Ê lxl e Ê R œ e
È
40. n lim
un 1 Ê n lim
x 1 Ê lxl n lim
n1
Ä_
Ä_
Ä _ n1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.7 Power Series
nb1
nb1
41. n lim
3 1 Ê lxl
¹ uunbn 1 ¹ 1 Ê n lim
¹ 3 3n xxn ¹ 1 Ê lxl n lim
Ä_
Ä_
Ä_
_
_
! 3n ˆ 1 ‰n œ ! a1bn , which diverges; at x œ
3
n œ0
n œ0
1
3
1
3
_
_
n œ0
nœ0
615
Ê 31 x 31 ; at x œ 31 we have
_
n
we have ! 3n ˆ 13 ‰ œ ! 1 , which diverges. The series ! 3n xn
_
œ ! a3xbn is a convergent geometric series when 13 x
n œ0
1
3
and the sum is
nœ0
1
1 3x .
nb1
e
4
42. n lim
1 1 Ê lex 4l 1 Ê 3 ex 5 Ê ln 3 x ln 5;
¹ uunbn 1 ¹ 1 Ê n lim
¹ a aex 4b bn ¹ 1 Ê lex 4l n lim
Ä_
Ä_
Ä_
x
_
_
_
_
nœ0
_
nœ0
nœ0
nœ0
n
n
at x œ ln 3 we have ! ˆeln 3 4‰ œ ! a1bn , which diverges; at x œ ln 5 we have ! ˆeln 5 4‰ œ ! 1, which
diverges. The series ! aex 4bn is a convergent geometric series when ln 3 x ln 5 and the sum is
n œ0
2nb2
43. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (x 4n1)1
Ä_
Ä_
4n
(x 1)2n ¹
†
1 Ê
(x 1)#
lim
4
nÄ_
_
Ê 2 x 1 2 Ê 1 x 3; at x œ 1 we have !
n œ0
_
we have !
n œ0
_
!
n œ0
(x ")2n
4n
"
#
4
4n
nœ0
œ!
nœ0
4
(x
4
9n
(x 1)2n ¹
†
1 Ê
(x 1)#
lim
9
nÄ_
n œ0
!
n œ0
nœ0
n œ0
k1k 1 Ê (x 1)# 9 Ê kx 1k 3
(3)2n
9n
_
œ ! 1 which diverges; at x œ 2 we have
n œ0
_
œ ! " which also diverges; the interval of convergence is 4 x 2; the series
(x 1)
9n
"
_
4
4
")# “ œ 4 x# 2x 1 œ 3 2x x#
_
!
_
n
œ ! 44n œ ! 1, which diverges; at x œ 3
is a convergent geometric series when 1 x 3 and the sum is
Ê 3 x 1 3 Ê 4 x 2; when x œ 4 we have !
n œ0
_
k1k 1 Ê (x 1)# 4 Ê kx 1k 2
nœ0
2nb2
32n
9n
1
5 ex .
œ ! 1, a divergent series; the interval of convergence is 1 x 3; the series
44. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (x 9n1)1
Ä_
Ä_
_
œ
_
# n
Šˆ x # 1 ‰ ‹
"
’
n
œ!
2
4n
_
œ
"
1 Šxc
# ‹
_
2n
(2)2n
4n
1
1 ae x 4 b
nœ0
2n
_
n
#
œ ! Šˆ x3 1 ‰ ‹ is a convergent geometric series when 4 x 2 and the sum is
n œ0
1
1 Šxb
3 ‹
#
œ
"
’
9
(x 1)#
“
9
œ
9
9 x# 2x 1
45. n lim
¹ uunbn 1 ¹ 1 Ê n lim
Ä_
Ä_ º
œ
9
8 2x x#
ˆÈx 2‰nb1
2nb1
†
2n
ˆÈ x 2 ‰ n º
1 Ê ¸Èx 2¸ 2 Ê 2 Èx 2 2 Ê 0 Èx 4
_
_
Ê 0 x 16; when x œ 0 we have ! (1)n , a divergent series; when x œ 16 we have ! (1)n , a divergent
nœ0
nœ0
_
series; the interval of convergence is 0 x 16; the series !
n œ0
0 x 16 and its sum is
1Œ
"
Èx c 2 œ
#
Œ
2c
"
Èx
#
2
œ
Èx 2 n
Š # ‹
is a convergent geometric series when
2
4 Èx
nb1
46. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (ln(lnx)x)n ¹ 1 Ê kln xk 1 Ê 1 ln x 1 Ê e" x e; when x œ e" or e we
Ä_
Ä_
_
_
_
nœ0
nœ0
nœ0
obtain the series ! 1n and ! (1)n which both diverge; the interval of convergence is e" x e; ! (ln x)n œ
when e" x e
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
1 ln x
616
Chapter 10 Infinite Sequences and Series
47. n lim
¹ uunbn 1 ¹ 1 Ê n lim
Šx
Ä_
Ä_ º
#
1
3 ‹
n1
n
† ˆ x# 3 1 ‰ º 1 Ê
ax# 1b
lim
3
nÄ_
x# "
3
k1k 1 Ê
1 Ê x# 2
_
Ê kxk È2 Ê È2 x È2 ; at x œ „ È2 we have ! (1)n which diverges; the interval of convergence is
n œ0
_
È2 x È2 ; the series !
n œ0
"
#
1 Š x 3b 1 ‹
œ
"
#
Š 3 c x3 c 1 ‹
œ
#
Š x 3 1 ‹
n
is a convergent geometric series when È2 x È2 and its sum is
3
# x#
ax
48. n lim
¹ uun n 1 ¹ 1 Ê n lim
¹
Ä_
Ä_
# 1 bn
2n
1
2n
¹
a x # 1 bn
†
1
1 Ê kx# 1k 2 Ê È3 x È3 ; when x œ „ È3 we
_
_
n œ0
n œ0
have ! 1n , a divergent series; the interval of convergence is È3 x È3 ; the series ! Š x
convergent geometric series when È3 x È3 and its sum is
nb1
49. n lim
¹ (x #n3)
b1
Ä_
†
2n
(x 3)n ¹
"
#
1 Šx 2 1‹
"
œ
2
œ
Šx# 1 ‹
#
#
n
1
‹
2
is a
2
3 x#
_
1 Ê kx 3k 2 Ê 1 x 5; when x œ 1 we have ! (1)n which diverges;
nœ1
_
when x œ 5 we have ! (1) which also diverges; the interval of convergence is 1 x 5; the sum of this
n
n œ1
convergent geometric series is
œ
2
x1
"
3
1 Šxc
# ‹
œ
2
x1
n
. If f(x) œ 1 #" (x 3) 4" (x 3)# á ˆ #" ‰ (x 3)n á
n
then f w (x) œ #" "# (x 3) á ˆ #" ‰ n(x 3)n1 á is convergent when 1 x 5, and diverges
2
(x 1)#
when x œ 1 or 5. The sum for f w (x) is
, the derivative of
2
x1
.
n
50. If f(x) œ 1 "# (x 3) 4" (x 3)# á ˆ "# ‰ (x 3)n á œ
œx
(x 3)#
4
(x 3)$
12
_
n (x 3)n
n 1
1
á ˆ "# ‰
2
x1
then ' f(x) dx
_
á . At x œ 1 the series ! n21 diverges; at x œ 5
n œ1
2
the series ! (n1)
1 converges. Therefore the interval of convergence is 1 x Ÿ 5 and the sum is
n
n œ1
2 ln kx 1k (3 ln 4), since '
dx œ 2 ln kx 1k C, where C œ 3 ln 4 when x œ 3.
2
x1
51. (a) Differentiate the series for sin x to get cos x œ 1
œ
x#
#!
x%
4!
x'
6!
)
x"!
1 x8! 10!
á .
2nb2
a
b
#
n
!
lim ¹ x
† x#8 ¹ œ x2 n lim
n Ä _ (2n 2)!
Ä_
(b) sin 2x œ 2x
2$ x$
3!
2& x&
5!
2( x(
7!
"
6!
œ 2x
52. (a)
(b)
d
x
5x%
5!
7x'
7!
9x)
9!
11x"!
11!
á
The series converges for all values of x since
Š a2n 1ba" 2n 2b ‹ œ 0 1 for all x.
1†00†
"
4!
$ $
( (
2 x
3!
aex b œ 1
& &
2 x
5!
2x
2!
3x#
3!
2 x
7!
0†
4x$
4!
' ex dx œ ex C œ x x#
#
#
(c) ex œ 1 x x#!
ˆ1 † 3!" 1 † #"!
ˆ1 † 5!" 1 † 4!"
x$
3!
"
#!
"
#!
"
3!
* *
2 x
9!
2* x*
9!
2"" x""
11!
"
#
0†
"" ""
á
0†
5x%
5!
2 x
11!
á œ 2x
8x$
3!
&
(
*
""
128x
512x
2048x
32x
5! 7! 9! 11! á
"
"‰ $
‰ # ˆ
(c) 2 sin x cos x œ 2 (0 † 1) (0 † 0 1 † 1)x ˆ0 † "
# 1 † 0 0 † 1 x 0 † 0 1 † # 0 † 0 1 † 3! x
ˆ0 † 4!" 1 † 0 0 † #" 0 † 3!" 0 † 1‰ x% ˆ0 † 0 1 † 4!" 0 † 0 #" † 3!" 0 † 0 1 † 5!" ‰ x&
ˆ0 †
3x#
3!
"
5!
0 † 1‰ x' á ‘ œ 2 ’x
á œ1x
x#
#!
x$
3!
x%
4!
4x$
3!
16x&
5!
á“
á œ ex ; thus the derivative of ex is ex itself
x$
x%
x&
x
3! 4! 5! á C, which is the general antiderivative of e
%
&
x4! x5! á ; ecx † ex œ 1 † 1 (1 † 1 1 † 1)x ˆ1 † #"! 1 † 1 #"!
† 1 3!" † 1‰ x$ ˆ1 † 4!" 1 † 3!" #"! † #"! 3!" † 1 4!" † 1‰ x%
† 3!" 3!" † #"! 4!" † 1 5!" † 1‰ x& á œ 1 0 0 0 0 0 á
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
† 1‰ x#
Section 10.8 Taylor and Maclaurin Series
53. (a) ln ksec xk C œ ' tan x dx œ ' Šx
#
œ
x
#
%
x
1#
'
)
x
45
17x
2520
1
#
converges when
#
(b) sec x œ
d(tan x)
dx
œ
when 1# x
d
dx
x
x$
3
œ1x
x$
6
œx
$
x
6
x&
24
&
x
24
(b) sec x tan x œ
when
1
#
17x'
45
61x(
5040
(
"
4
62x)
315
x#
#
17x(
315
5x%
24
5 ‰ %
24 x
62x*
2835
61x'
720
62x*
2835
á ‹ dx
á ‹ œ 1 x#
d(sec x)
dx
*
œ
277x
72,576
d
dx
á ‹ Š1
61
ˆ 720
á ,
277x*
72,576
61x
5040
2x%
3
x#
#
17x'
45
x%
12
62x)
315
x'
45
17x)
2520
31x"!
14,175
á ,
á , converges
1
#
x
x#
2
5
48
1
#
5
48
5x%
24
x#
#
5x%
24
61 ‰ '
720 x
61x'
720
61x'
720
á‹
á
á ‹ dx
á C; x œ 0 Ê C œ 0 Ê ln ksec x tan xk
á , converges when 1# x
Š1
x#
#
5x%
24
61x'
720
5x$
6
á‹ œ x
1
#
61x&
120
277x(
1008
á , converges
1
#
(c) (sec x)(tan x) œ Š1
x#
#
2
œ x ˆ "3 #" ‰ x$ ˆ 15
1# x
1
#
x
17x(
315
2x&
15
54. (a) ln ksec x tan xk C œ ' sec x dx œ ' Š1
œx
5
œ 1 ˆ "# "# ‰ x# ˆ 24
2x%
3
2x&
15
á C; x œ 0 Ê C œ 0 Ê ln ksec xk œ
(c) sec# x œ (sec x)(sec x) œ Š1
#
"!
31x
14,175
1#
Šx
x$
3
617
1
#
5x%
24
"
6
_
61x'
720
á ‹ Šx
5 ‰ &
24 x
17
ˆ 315
"
15
x$
3
5
72
2x&
15
17x(
315
61 ‰ (
720 x
á‹
á œ x
5x$
6
61x&
120
277x(
1008
á ,
_
55. (a) If f(x) œ ! an xn , then f ÐkÑ (x) œ ! n(n 1)(n 2)â(n (k 1)) an xnk and f ÐkÑ (0) œ k!ak
n œ0
Ê ak œ
f ÐkÑ (0)
k!
n œk
_
; likewise if f(x) œ ! bn xn , then bk œ
n œ0
f ÐkÑ (0)
k!
Ê ak œ bk for every nonnegative integer k
_
(b) If f(x) œ ! an xn œ 0 for all x, then f ÐkÑ (x) œ 0 for all x Ê from part (a) that ak œ 0 for every nonnegative integer k
n œ0
10.8 TAYLOR AND MACLAURIN SERIES
1. f(x) œ e2x , f w (x) œ 2e2x , f ww (x) œ 4e2x , f www (x) œ 8e2x ; f(0) œ e2a0b œ ", f w (0) œ 2, f ww (0) œ 4, f www (0) œ 8 Ê P! (x) œ 1,
P" (x) œ 1 2x, P# (x) œ 1 x 2x# , P$ (x) œ 1 x 2x# 43 x3
2. f(x) œ sin x, f w (x) œ cos x , f ww (x) œ sin x , f www (x) œ cos x; f(0) œ sin 0 œ 0, f w (0) œ 1, f ww (0) œ 0, f www (0) œ 1
Ê P! (x) œ 0, P" (x) œ x, P# (x) œ x, P$ (x) œ x 16 x3
3. f(x) œ ln x, f w (x) œ
"
x
, f ww (x) œ x"# , f www (x) œ
2
x$ ;
f(1) œ ln 1 œ 0, f w (1) œ 1, f ww (1) œ 1, f www (1) œ 2 Ê P! (x) œ 0,
P" (x) œ (x 1), P# (x) œ (x 1) "# (x 1)# , P$ (x) œ (x 1) "# (x 1)# 3" (x 1)$
4. f(x) œ ln (1 x), f w (x) œ
f w (0) œ
5. f(x) œ
œ 1, f ww (0) œ (1)
1
1
"
x
(1 x)" , f ww (x) œ (1 x)# , f www (x) œ 2(1 x)$ ; f(0) œ ln 1 œ 0,
œ 1, f www (0) œ 2(1)$ œ 2 Ê P! (x) œ 0, P" (x) œ x, P# (x) œ x
œ x" , f w (x) œ x# , f ww (x) œ 2x$ , f www (x) œ 6x% ; f(2) œ
Ê P! (x) œ
P$ (x) œ
"
1x œ
#
"
#
"
"
"
"
# , P" (x) œ # 4 (x 2), P# (x) œ #
"
"
"
#
$
4 (x 2) 8 (x 2) 16 (x 2)
"
#
x#
#,
P$ (x) œ x
, f w (2) œ 4" , f ww (2) œ 4" , f www (x) œ 83
"4 (x 2) "8 (x 2)# ,
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
x#
#
x$
3
618
Chapter 10 Infinite Sequences and Series
6. f(x) œ (x 2)" , f w (x) œ (x 2)# , f ww (x) œ 2(x 2)$ , f www (x) œ 6(x 2)% ; f(0) œ (2)" œ
œ 4" , f ww (0) œ 2(2)$ œ
P$ (x) œ
"
#
x
4
x#
8
"
4
, f www (0) œ 6(2)% œ 38 Ê P! (x) œ
x$
16
"
#
, P" (x) œ
f ww ˆ 14 ‰ œ sin
P# (x) œ
È2
#
1
4 œ
È2
ˆx
#
È
, f www ˆ 14 ‰ œ cos 14 œ #2 Ê
È2
È
1‰
ˆx 14 ‰# , P$ (x) œ #2
4 4
x4 , P# (x) œ
"
#
, f w (0) œ (2)#
x
4
È2
È2
1
w ˆ1‰
# ,f
4 œ cos 4 œ #
È
È
È
P! œ #2 , P" (x) œ #2 #2 ˆx 14 ‰ ,
È2
È
È
ˆx 14 ‰ 42 ˆx 14 ‰# 1#2 ˆx 14 ‰$
#
7. f(x) œ sin x, f w (x) œ cos x, f ww (x) œ sin x, f www (x) œ cos x; f ˆ 14 ‰ œ sin
È2
#
"
#
"
#
1
4
œ
x#
8
,
,
8. f(x) œ tan x, f w (x) œ sec2 x, f ww (x) œ 2sec2 x tan x, f www (x) œ 2sec4 x 4sec2 x tan2 x; f ˆ 14 ‰ œ tan 14 œ 1 ,
f w ˆ 14 ‰ œ sec2 ˆ 14 ‰ œ 2 , f ww ˆ 14 ‰ œ 2sec2 ˆ 14 ‰ tan ˆ 14 ‰ œ 4 , f www ˆ 14 ‰ œ 2sec4 ˆ 14 ‰ 4sec2 ˆ 14 ‰ tan2 ˆ 14 ‰ œ 16 Ê P! (x) œ 1 ,
2
2
3
P" (x) œ 1 2 ˆx 14 ‰ , P# (x) œ 1 2 ˆx 14 ‰ 2 ˆx 14 ‰ , P$ (x) œ 1 2 ˆx 14 ‰ 2 ˆx 14 ‰ 83 ˆx 14 ‰
9. f(x) œ Èx œ x"Î# , f w (x) œ ˆ "# ‰ x"Î# , f ww (x) œ ˆ 4" ‰ x$Î# , f www (x) œ ˆ 83 ‰ x&Î# ; f(4) œ È4 œ 2,
"
3
f w (4) œ ˆ "# ‰ 4"Î# œ "4 , f ww (4) œ ˆ "4 ‰ 4$Î# œ 32
,f www (4) œ ˆ 38 ‰ 4&Î# œ 256
Ê P! (x) œ 2, P" (x) œ 2 "4 (x 4),
P# (x) œ 2 4" (x 4)
"
64
(x 4)# , P$ (x) œ 2 "4 (x 4)
"
64
(x 4)#
"
51#
(x 4)$
10. f(x) œ (1 x)"Î# , f w (x) œ "# (1 x)"Î# , f ww (x) œ "4 (1 x)$Î# , f www (x) œ 38 (1 x)&Î# ; f(0) œ (1)"Î# œ 1,
f w (0) œ "# (1)"Î# œ "# , f ww (0) œ "4 (1)$Î# œ "4 , f www (0) œ 83 (1)&Î# œ 83 Ê P! (x) œ 1,
P" (x) œ 1 2" x, P# (x) œ 1 2" x 8" x# , P$ (x) œ 1 2" x 8" x#
1
16
x$
11. f(x) œ ex , f w (x) œ ex , f ww (x) œ ex , f www (x) œ ex Ê á f ÐkÑ (x) œ a1bk ex ; f(0) œ ea0b œ ", f w (0) œ 1,
_
f ww (0) œ 1, f www (0) œ 1, á ß f ÐkÑ (0) œ (1)k Ê ex œ 1 x 12 x# 16 x3 á œ !
n œ0
(1)n n
n! x
12. f(x) œ x ex , f w (x) œ x ex ex , f ww (x) œ x ex 2ex , f www (x) œ x ex 3ex Ê á f ÐkÑ (x) œ x ex k ex ; f(0) œ a0bea0b œ 0,
_
f w (0) œ 1, f ww (0) œ 2, f www (0) œ 3, á ß f ÐkÑ (0) œ k Ê x x# 12 x3 á œ !
n œ0
1
n
a n 1 b! x
13. f(x) œ (1 x)" Ê f w (x) œ (1 x)# , f ww (x) œ 2(1 x)$ , f www (x) œ 3!(1 x)% Ê á f ÐkÑ (x)
œ (1)k k!(1 x)k1 ; f(0) œ 1, f w (0) œ 1, f ww (0) œ 2, f www (0) œ 3!, á ß f ÐkÑ (0) œ (1)k k!
_
_
n œ0
nœ0
Ê 1 x x# x$ á œ ! (x)n œ ! (1)n xn
14. f(x) œ
2x
1x
Ê f w (x) œ
œ 6(1 x)$ , f www (x) œ 18(1 x)% Ê á f ÐkÑ (x) œ 3ak!b(1 x)
3
ww
(1 x)# , f (x)
_
f w (0) œ 3, f ww (0) œ 6, f www (0) œ 18, á ß f ÐkÑ (0) œ 3ak!b Ê 2 3x 3x# 3x$ á œ 2 ! 3xn
n œ1
_
15. sin x œ !
n œ0
_
16. sin x œ !
nœ0
_
(")n x2nb1
(#n1)!
Ê sin 3x œ !
(")n x2nb1
(#n1)!
Ê sin
n œ0
_
17. 7 cos (x) œ 7 cos x œ 7 !
n œ0
x
#
_
œ!
nœ0
(")n x2n
(2n)!
(")n (3x)2nb1
(#n1)!
2n 1
(")n ˆ x# ‰
(#n1)!
œ7
7x#
#!
_
(")n 32nb1 x2nb1
(#n1)!
œ 3x
(")n x2nb1
#2n 1 (2n1)!
x
#
œ!
n œ0
_
œ!
nœ0
7x%
4!
7x'
6!
œ
3$ x$
3!
x$
2$ †3!
3& x&
5!
x&
2& †5!
á
á
á , since the cosine is an even function
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
k 1
; f(0) œ 2,
Section 10.8 Taylor and Maclaurin Series
_
18. cos x œ !
n œ0
19. cosh x œ
_
œ!
n œ0
œ!
n œ0
ex ecx
#
Ê 5 cos 1x œ 5 !
nœ0
œ
"
#
’Š1 x#
œ
"
#
’Š1 x
x#
#!
x$
3!
(1)n (1x)2n
(#n)!
x%
4!
œ5
51 # x#
2!
51 % x%
4!
á ‹ Š1 x
x#
#!
5 1 ' x'
6!
x$
3!
á
x%
4!
á ‹“ œ 1
x#
#!
x%
4!
x'
6!
á
x2n
(2n)!
20. sinh x œ
_
_
(1)n x2n
(2n)!
619
ex ecx
#
x#
#!
x$
3!
x%
4!
á ‹ Š1 x
x#
#!
x$
3!
x%
4!
á ‹“ œ x
x$
3!
x&
5!
x'
6!
á
x2n 1
(2n 1)!
21. f(x) œ x% 2x$ 5x 4 Ê f w (x) œ 4x$ 6x# 5, f ww (x) œ 12x# 12x, f www (x) œ 24x 12, f Ð4Ñ (x) œ 24
Ê f ÐnÑ (x) œ 0 if n 5; f(0) œ 4, f w (0) œ 5, f ww (0) œ 0, f www (0) œ 12, f Ð4Ñ (0) œ 24, f ÐnÑ (0) œ 0 if n 5
24 %
$
%
$
Ê x% 2x$ 5x 4 œ 4 5x 12
3! x 4! x œ x 2x 5x 4
22. f(x) œ
x#
x1
Ê f w (x) œ
2x x#
; f ww (x)
ax 1b2
f www (0) œ 6, f ÐnÑ (0) œ a1bn nx if n
œ
2
;
ax 1 b 3
f www (x) œ
6
ax 1 b 4
Ê f ÐnÑ (x) œ
a1bn nx
;
ax 1bnb1
f(0) œ 0, f w (0) œ 0, f ww (0) œ 2,
_
2 Ê x# x3 x4 x5 Þ Þ Þ œ ! a1bn xn
n œ2
23. f(x) œ x$ 2x 4 Ê f w (x) œ 3x# 2, f ww (x) œ 6x, f www (x) œ 6 Ê f ÐnÑ (x) œ 0 if n 4; f(2) œ 8, f w (2) œ 10,
6
#
$
f ww (2) œ 12, f www (2) œ 6, f ÐnÑ (2) œ 0 if n 4 Ê x$ 2x 4 œ 8 10(x 2) 12
2! (x 2) 3! (x 2)
œ 8 10(x 2) 6(x 2)# (x 2)$
24. f(x) œ 2x$ x# 3x 8 Ê f w (x) œ 6x# 2x 3, f ww (x) œ 12x 2, f www (x) œ 12 Ê f ÐnÑ (x) œ 0 if n
f w (1) œ 11, f ww (1) œ 14, f www (1) œ 12, f ÐnÑ (1) œ 0 if n 4 Ê 2x$ x# 3x 8
12
#
$
#
$
œ 2 11(x 1) 14
2! (x 1) 3! (x 1) œ 2 11(x 1) 7(x 1) 2(x 1)
4; f(1) œ 2,
25. f(x) œ x% x# 1 Ê f w (x) œ 4x$ 2x, f ww (x) œ 12x# 2, f www (x) œ 24x, f Ð4Ñ (x) œ 24, f ÐnÑ (x) œ 0 if n 5;
f(2) œ 21, f w (2) œ 36, f ww (2) œ 50, f www (2) œ 48, f Ð4Ñ (2) œ 24, f ÐnÑ (2) œ 0 if n 5 Ê x% x# 1
48
24
#
$
%
#
$
%
œ 21 36(x 2) 50
2! (x 2) 3! (x 2) 4! (x 2) œ 21 36(x 2) 25(x 2) 8(x 2) (x 2)
26. f(x) œ 3x& x% 2x$ x# 2 Ê f w (x) œ 15x% 4x$ 6x# 2x, f ww (x) œ 60x$ 12x# 12x 2,
f www (x) œ 180x# 24x 12, f Ð4Ñ (x) œ 360x 24, f Ð5Ñ (x) œ 360, f ÐnÑ (x) œ 0 if n 6; f(1) œ 7,
f w (1) œ 23, f ww (1) œ 82, f www (1) œ 216, f Ð4Ñ (1) œ 384, f Ð5Ñ (1) œ 360, f ÐnÑ (1) œ 0 if n 6
216
384
360
#
$
%
&
Ê 3x& x% 2x$ x# 2 œ 7 23(x 1) 82
2! (x 1) 3! (x 1) 4! (x 1) 5! (x 1)
œ 7 23(x 1) 41(x 1)# 36(x 1)$ 16(x 1)% 3(x 1)&
27. f(x) œ x# Ê f w (x) œ 2x$ , f ww (x) œ 3! x% , f www (x) œ 4! x& Ê f ÐnÑ (x) œ (1)n (n 1)! xn2 ;
f(1) œ 1, f w (1) œ 2, f ww (1) œ 3!, f www (1) œ 4!, f ÐnÑ (1) œ (1)n (n 1)! Ê x"#
_
œ 1 2(x 1) 3(x 1)# 4(x 1)$ á œ ! (1)n (n 1)(x 1)n
n œ0
28. f(x) œ
1
a1 xb3
Ê f w (x) œ 3(1 x)4 , f ww (x) œ 12(1 x)5 , f www (x) œ 60 (1 x)6 Ê f ÐnÑ (x) œ
fa0b œ 1, f w a0b œ 3, f ww a0b œ 12, f www a0b œ 60, á , f ÐnÑ a0b œ
_
œ!
n œ0
an 2b!
2
Ê
1
a1 xb3
an 2b!
2
(1 x)n3 ;
œ 1 3x 6x# 10x3 á
an 2ban 1b n
x
2
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
620
Chapter 10 Infinite Sequences and Series
29. f(x) œ ex Ê f w (x) œ ex , f ww (x) œ ex Ê f ÐnÑ (x) œ ex ; f(2) œ e# , f w (2) œ e# , á f ÐnÑ (2) œ e#
Ê ex œ e# e# (x 2)
e#
#
(x 2)#
e$
3!
_
(x 2)$ á œ !
n œ0
e#
n!
(x 2)n
30. f(x) œ 2x Ê f w (x) œ 2x ln 2, f ww (x) œ 2x (ln 2)# , f www (x) œ 2x (ln 2)3 Ê f ÐnÑ (x) œ 2x (ln 2)n ; f(1) œ 2, f w (1) œ 2 ln 2,
f ww (1) œ 2(ln 2)# , f www (1) œ 2(ln 2)$ , á , f ÐnÑ (1) œ 2(ln 2)n
2(ln 2)#
#
Ê 2x œ 2 (2 ln 2)(x 1)
(x 1)#
2(ln 2)3
3!
_
(x 1)3 á œ !
n œ0
2(ln 2)n (x1)n
n!
31. f(x) œ cosˆ2x 12 ‰, f w (x) œ 2 sinˆ2x 12 ‰, f ww (x) œ 4 cosˆ2x 12 ‰, f www (x) œ 8 sinˆ2x 12 ‰,
f a4b axb œ 24 cosˆ2x 12 ‰ß f a5b axb œ 25 sinˆ2x 12 ‰ß . . ; fˆ 14 ‰ œ 1, f w ˆ 14 ‰ œ 0, f ww ˆ 14 ‰ œ 4, f www ˆ 14 ‰ œ 0, f a4b ˆ 14 ‰ œ 24 ,
2
4
f a5b ˆ 14 ‰ œ 0, . . ., f Ð2nÑ ˆ 14 ‰ œ a1bn 22n Ê cosˆ2x 12 ‰ œ 1 2ˆx 14 ‰ 32 ˆx 14 ‰ . . .
_
œ!
n œ0
a1bn 22n ˆ
x
a2nbx
2n
14 ‰
7 Î2
32. f(x) œ Èx 1, f w (x) œ 12 ax 1b1Î2 , f ww (x) œ 14 ax 1b3Î2 , f www (x) œ 38 ax 1b5Î2 , f a4b (x) œ 15
, . . .;
16 ax 1b
1
1
3
15
1
1
1
5
f(0) œ 1, f w (0) œ , f ww (0) œ , f www (0) œ , f a4b (0) œ , . . . Ê Èx 1 œ 1 x x2 x3
x4 Þ Þ Þ
2
4
8
16
_
a1bn 2n
a2nbx x
33. The Maclaurin series generated by cos x is !
n œ0
by
_
!
n œ0
2
1x
2
8
16
which converges on a_, _b and the Maclaurin series generated
_
is 2 ! xn which converges on a1, 1b. Thus the Maclaurin series generated by faxb œ cos x
n œ0
a1bn 2n
a2nbx x
128
2
1x
is given by
_
2 ! xn œ 1 2x 25 x2 Þ Þ Þ Þ which converges on the intersection of a_, _b and a1, 1b, so the
nœ0
interval of convergence is a1, 1b.
_
34. The Maclaurin series generated by ex is !
n œ0
xn
nx
which converges on a_, _b. The Maclaurin series generated by
_
faxb œ a1 x x2 bex is given by a1 x x2 b !
n œ0
_
35. The Maclaurin series generated by sin x is !
n œ0
_
generated by lna1 xb is !
n œ1
a1bnc1 n
x
n
xn
nx
œ 1 12 x2 23 x3 Þ Þ Þ Þ which converges on a_, _bÞ
a1bn
2n1
a2n 1bx x
which converges on a_, _b and the Maclaurin series
which converges on a1, 1b. Thus the Maclaurin series genereated by
_
faxb œ sin x † lna1 xb is given by Œ !
n œ0
_
a1bn
a1bnc1 n
2n1
Π! n x
a2n 1bx x
n œ1
œ x2 12 x3 61 x4 Þ Þ Þ Þ which converges on
the intersection of a_, _b and a1, 1b, so the interval of convergence is a1, 1b.
_
36. The Maclaurin series generated by sin x is !
n œ0
a1bn
2n1
a2n 1bx x
_
genereated by faxb œ x sin2 x is given by xŒ !
n œ0
œ x3 13 x5
_
37. If ex œ !
n œ0
f ÐnÑ (a)
n!
2 7
45 x
which converges on a_, _b. The Maclaurin series
2
a 1 b n
2n1
a2n 1bx x
_
œ xŒ !
nœ0
_
a 1 b n
a 1 b n
2n1
2n1
Π! a2n 1bx x
a2n 1bx x
n œ0
. . . which converges on a_, _bÞ
(x a)n and f(x) œ ex , we have f ÐnÑ (a) œ ea f or all n œ 0, 1, 2, 3, á
!
Ê ex œ ea ’ (x 0!a)
(x a)"
1!
(x a)#
2!
á “ œ ea ’1 (x a)
(x a)#
2!
á “ at x œ a
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.9 Convergence of Taylor Series
621
38. f(x) œ ex Ê f ÐnÑ (x) œ ex for all n Ê f ÐnÑ (1) œ e for all n œ 0, 1, 2, á
Ê ex œ e e(x 1)
e
#!
(x 1)#
e
3!
(x 1)$ á œ e ’1 (x 1)
f ww (a)
f www (a)
#
$
w
# (x a) 3! (x a) á Ê f (x)
www
œ f w (a) f ww (a)(x a) f 3!(a) 3(x a)# á Ê f ww (x) œ f ww (a) f www (a)(x
Ðn 2Ñ
Ê f ÐnÑ (x) œ f ÐnÑ (a) f Ðn1Ñ (a)(x a) f # (a) (x a)# á
w
w
Ðn Ñ
Ðn Ñ
(x 1)#
2!
39. f(x) œ f(a) f w (a)(x a)
Ê f(a) œ f(a) 0, f (a) œ f (a) 0, á , f
(a) œ f
a)
(x 1)$
3!
f Ð4Ñ (a)
4!
á“
4 † 3(x a)# á
(a) 0
40. E(x) œ f(x) b! b" (x a) b# (x a)# b$ (x a)$ á bn (x a)n
Ê 0 œ E(a) œ f(a) b! Ê b! œ f(a); from condition (b),
lim
xÄa
Ê
Ê
f(x) f(a) b" (x a) b# (x a)# b$ (x a)$ á bn (x a)n
(x a)n
œ0
w
a)# á nbn (x a)n 1
lim f (x) b" 2b# (x a) n(x3b$ (xa)
œ0
n 1
xÄa
f ww (x) 2b# 3! b$ (x a) á n(n ")bn (x a)n
w
b" œ f (a) Ê xlim
n(n 1)(x a)n 2
Äa
"
#
f ww (a) Ê xlim
Äa
" www
œ b$ œ 3! f (a) Ê xlim
Äa
Ê b# œ
g(x) œ f(a) f w (a)(x a)
f www (x) 3! b$ á n(n 1)(n 2)bn (x a)n
n(n 1)(n #)(x a)n
f
ÐnÑ
(x) n! bn
n!
f ww (a)
2!
œ 0 Ê bn œ
(x a)# á
3
3
"
n!
f ÐnÑ (a)
n!
2
œ0
œ0
f ÐnÑ (a); therefore,
(x a)n œ Pn (x)
#
41. f(x) œ ln (cos x) Ê f w (x) œ tan x and f ww (x) œ sec# x; f(0) œ 0, f w (0) œ 0, f ww (0) œ 1 Ê L(x) œ 0 and Q(x) œ x2
42. f(x) œ esin x Ê f w (x) œ (cos x)esin x and f ww (x) œ ( sin x)esin x (cos x)# esin x ; f(0) œ 1, f w (0) œ 1, f ww (0) œ 1
Ê L(x) œ 1 x and Q(x) œ 1 x
"Î#
43. f(x) œ a1 x# b
x#
#
Ê f w (x) œ x a1 x# b
f ww (0) œ 1 Ê L(x) œ 1 and Q(x) œ 1
$Î#
and f ww (x) œ a1 x# b
$Î#
3x# a1 x# b
&Î#
; f(0) œ 1, f w (0) œ 0,
x#
#
44. f(x) œ cosh x Ê f w (x) œ sinh x and f ww (x) œ cosh x; f(0) œ 1, f w (0) œ 0, f ww (0) œ 1 Ê L(x) œ 1 and Q(x) œ 1
45. f(x) œ sin x Ê f w (x) œ cos x and f ww (x) œ sin x; f(0) œ 0, f w (0) œ 1, f ww (0) œ 0 Ê L(x) œ x and Q(x) œ x
46. f(x) œ tan x Ê f w (x) œ sec# x and f ww (x) œ 2 sec# x tan x; f(0) œ 0, f w (0) œ 1, f ww œ 0 Ê L(x) œ x and Q(x) œ x
10.9 CONVERGENCE OF TAYLOR SERIES
_
1. ex œ 1 x
x#
#!
á œ !
2. ex œ 1 x
x#
#!
á œ !
nœ0
_
nœ0
xn
n!
Ê e5x œ 1 (5x)
(5x)#
#!
á œ 1 5x
xn
n!
Ê exÎ2 œ 1 ˆ #x ‰
ˆ x# ‰#
#!
á œ1
_
3. sin x œ x
x$
3!
x&
5!
á œ!
4. sin x œ x
x$
3!
x&
5!
á œ!
n œ0
_
nœ0
(1)n x2n 1
(#n1)!
Ê 5 sin (x) œ 5 ’(x)
(1)n x2n 1
(#n1)!
Ê sin
1x
#
œ
1x
#
ˆ 1#x ‰$
3!
(x)$
3!
ˆ 1#x ‰&
5!
x
#
x#
2# #!
(x)&
5!
ˆ 1#x ‰(
7!
5# x#
#!
_
5$ x$
3!
x$
2$ 3!
á œ!
nœ0
_
á œ !
nœ0
_
(1)n xn
2n n!
x
á “ œ ! 5((1)
#n1)!
n 1 2n 1
n œ0
_
1
x
á œ ! (21)
2n 1 (#n1)!
nœ0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
(1)n 5n xn
n!
n 2n 1 2n 1
x#
#
622
Chapter 10 Infinite Sequences and Series
_
5. cos x œ !
n œ0
_
6. cos x œ !
n œ0
x$
2†2!
œ1
Ê cos 5x2 œ !
a1bn x2n
(2n)!
$Î#
cos Š xÈ ‹
2
x'
2# †4!
_
7. lna1 xb œ !
n œ1
_
nœ0
10.
Ê
x*
2$ †6!
1
1x
œ ! xn Ê
_
n œ0
n œ0
_
nœ0
_
(1)n x2n
(2n)!
13. cos x œ !
n œ0
œ
x%
4!
x'
6!
_
(1)n x2nb1
(2n1)!
14. sin x œ !
n œ0
œ Šx
x$
3!
_
n œ0
_
16. cos x œ !
n œ0
œ1
"
#
œ!
n œ1
nœ0
nœ0
_
xnb1
n!
œ!
n œ0
nœ0
x"!
10!
x#
#
1 cos x œ
x(
7!
_
x#
#
_
a1bn 32nb1 x8nb4
n
œ!
nœ0
x4
2
x$
#!
œ x x#
_
(1)n x2nb1
(#n1)!
nœ0
_
(1)n x2n
(#n)!
n œ0
x&
4!
(1)n x2nb3
(2n1)!
œ!
1!
14 x 18 x2
x%
3!
x#
#
œ
x8
4
œ 3x4 9x12
n
"
#
x6
3
9 6
16 x
...
243 20
5 x
27 9
64 x
1 3
16 x
...
x(
5!
x*
7!
2187 28
7 x
...
á
œ x$
x&
3!
11
x#
2
x%
4!
á
x'
6!
x)
8!
x"!
10!
á
n œ2
x*
9!
x""
11!
x$
3!
_
œ Œ!
n œ0
á‹ x
_
(1)n x2n
(2n)!
Ê x# cos ax# b œ x# !
nœ0
n œ0
_
"
#
(2x)%
2†4!
(2x)'
2†6!
"
#
"
#
_
œ!
nœ1
"
#
! (1) (2x) œ
(2n)!
œ
n
2n
n œ0
(2x))
2†8!
(1)n x2nb1
(#n1)!
x$
3!
(1)n (1x)2n
(#n)!
_
cos 2x
#
(1)nb1 (2x)2n
#†(2n)!
nœ1
œ x2
n 2n
Ê x cos 1x œ x !
á
x
á œ ! ((1)
#n)!
Ê sin x x
2x ‰
18. sin# x œ ˆ 1cos
œ
#
_
_
(1)n x2n
(2n)!
(2x)#
2†2!
x&
5!
15. cos x œ !
17. cos# x œ
x)
8!
_
Ê x# sin x œ x# Œ !
Ê
15625x12
6!
(1)n x3n
2n (2n)!
nœ0
_
(1)n x2nb1
(2n1)!
_
œ!
nœ0
a1bnc1 x2n
n
œ!
n
n 1
œ #" ! ˆ #" x‰ œ ! ˆ #" ‰ xn œ
xn
n!
Ê xex œ x Œ !
12. sin x œ !
_
n
_
n
nœ0
_
xn
n!
11. ex œ !
(#n)!
nœ0
2nb1
a1bn ˆ3x4 ‰
2n 1
_
" 1
# 1 "# x
œ
1
2x
œ ! a1bn ˆ 34 x3 ‰ œ ! a1bn ˆ 34 ‰ x3n œ 1 34 x3
1
1 34 x3
n œ0
n œ0
nœ1
nœ0
_
œ!
625x8
4!
2n
"Î#
$
a1bn ŒŠ x# ‹
_
a1bnc1 ˆx2 ‰
n
Ê lna1 x2 b œ !
_
œ ! a1bn xn Ê
n œ0
25x4
#!
œ1
á
Ê tan1 a3x4 b œ !
1
1x
_
œ
(1)n 52n x4n
(2n)!
œ!
"Î#
$
cos ŒŠ x# ‹
_
a1bnc1 xn
n
_
2n
(1)n 5x2 ‘
(2n)!
n œ0
a1bn x2nb1
2n 1
8. tan1 x œ !
9.
_
(1)n x2n
(2n)!
cos 2x œ
_
n œ1
"
#
_
nœ0
2n
"# Š1
x(
7!
_
n œ0
(2x)#
2!
(1)n (2x)2n
2†(2n)!
(2x)#
#!
x*
9!
x$
3!
(")n 12n x2nb1
(#n)!
œ!
"# ’1
á œ1!
œ!
(1)n ax# b
(#n)!
"
#
x&
5!
œ
x
(")n x4n
(#n)!
(2x)%
4!
x""
11!
œx
2
(2x)'
6!
_
n œ1
(2x)'
6!
n œ2
1 # x$
2!
œ x#
œ1!
(2x)%
4!
_
á œ!
x'
2!
(2x))
8!
(1)n x2n 1
(2n1)!
1 % x&
4!
1 ' x(
6!
x"!
4!
x"%
6!
á
á
á“
(1)n 22n 1 x2n
(2n)!
á‹ œ
(2x)#
2†2!
(2x)%
2†4!
(2x)'
2†6!
(1)n 22n 1 x2n
(2n)!
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
á
...
Section 10.9 Convergence of Taylor Series
19.
x#
12x
_
n œ1
22.
"
1x
_
n œ0
n œ0
(1)nc1 (2x)n
n
20. x ln (1 2x) œ x !
21.
_
œ x# ˆ 1"2x ‰ œ x# ! (2x)n œ ! 2n xn2 œ x# 2x$ 2# x% 2$ x& á
_
(1)nc1 2n xn
n
œ!
n œ1
_
œ ! xn œ 1 x x# x$ á Ê
2
a 1 x b$
d
dx
nœ0
œ
d#
dx#
ˆ 1" x ‰ œ
d
dx
Š (1"x)# ‹ œ
d
dx
1
œ 2x#
ˆ 1" x ‰ œ
2# x$
#
"
(1x)#
2$ x%
4
2% x&
5
á
_
_
nœ1
nœ0
œ 1 2x 3x# á œ ! nxn1 œ ! (n 1)xn
_
a1 2x 3x# á b œ 2 6x 12x# á œ ! n(n 1)xn2
n œ2
_
œ ! (n 2)(n 1)xn
n œ0
3
5
7
23. tan1 x œ x 13 x3 15 x5 17 x7 Þ Þ Þ Ê x tan1 x2 œ xŠx2 13 ax2 b 15 ax2 b 17 ax2 b Þ Þ Þ ‹
_
œ x3 13 x7 15 x11 17 x15 Þ Þ Þ œ !
n œ1
x3
3!
24. sin x œ x
œx
4 x3
3!
16 x5
5!
x2
2!
25. ex œ 1 x
œ Š1 x
26. sin x œ x
œ Š1
_
2
x
2!
x5
5!
x2
2!
x3
3!
4
1) x
œ ! Š ((2n)!
n œ0
x3
3!
x
4!
n 2n
x3
3!
x5
5!
x7
7!
64 x7
7!
a1bn x4nc1
2n 1
á Ê sin x † cos x œ "# sin 2x œ "# Š2x
á œx
á and
1
1x
2 x3
3
2x5
15
4 x7
315
_
á œ!
n œ0
a2xb3
3!
x7
7!
œ 1 x x2 x3 á Ê ex
6
x
6!
á and cos x œ 1
á ‹ Šx
3
x
3!
x2
2!
x4
4!
x6
6!
a2xb7
7!
á‹
1
1x
25 4
24 x
_
á œ ! ˆ n!1 a1bn ‰xn
n œ0
á Ê cos x sin x
5
x
3
lna1 x2 b œ x3 Šx2 12 ax2 b 13 ax2 b 14 ax2 b á ‹
x
5!
7
(1)n 22n x2nb1
(#n1)!
á ‹ a1 x x2 x3 á b œ 2 32 x2 56 x3
a2xb5
5!
x
7!
á‹ œ 1 x
x2
2!
x3
3!
x4
4!
x5
5!
x6
6!
x7
7!
á
(1)n x2nb1
(#n1)! ‹
27. lna1 xb œ x 12 x2 13 x3 14 x4 á Ê
œ 13 x3 16 x5 19 x7
1 9
12 x
_
2
3
4
nc1
á œ ! a13nb x2n1
n œ1
28. lna1 xb œ x 12 x2 13 x3 14 x4 á and lna1 xb œ x 12 x2 13 x3 14 x4 á Ê lna1 xb lna1 xb
_
œ ˆx 12 x2 13 x3 14 x4 á ‰ ˆx 12 x2 13 x3 14 x4 á ‰ œ 2x 23 x3 25 x5 á œ ! 2n 2 1 x2n1
n œ0
29. ex œ 1 x
œ Š1 x
x2
2!
x2
2!
x3
3!
x3
3!
á and sin x œ x
á ‹Šx
x3
3!
x5
5!
x3
3!
x5
5!
x7
7!
á ‹ œ x x2 13 x3
x7
7!
á Ê ex † sin x
1 5
30 x
ÞÞÞÞ
30. lna1 xb œ x 12 x2 31 x3 41 x4 á and 1 " x œ 1 x x# x$ á Ê ln1a1xxb œ lna1 xb †
7 4
œ ˆx 12 x2 13 x3 14 x4 á ‰a1 x x# x$ á b œ x 12 x2 56 x3 12
x ÞÞÞÞ
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
1x
623
624
Chapter 10 Infinite Sequences and Series
2
31. tan1 x œ x 13 x3 15 x5 17 x7 Þ Þ Þ Ê atan1 xb œ atan1 xbatan1 xb
44 8
6
œ ˆx 13 x3 15 x5 17 x7 Þ Þ Þ‰ˆx 13 x3 15 x5 17 x7 Þ Þ Þ‰ œ x2 23 x4 23
45 x 105 x Þ Þ Þ Þ
32. sin x œ x
x3
3!
x5
5!
x7
7!
á and cos x œ 1
œ cos x † "# sin 2x œ "# Š1
33. sin x œ x
x3
3!
x5
5!
x7
7!
3
Ê esin x œ 1 Šx
x
3!
x2
2!
x4
4!
x6
6!
x2
2!
x
5!
7
x
7!
x2
2!
á ‹ 12 Šx
x6
6!
a2xb
3!
á ‹Š2x
á and ex œ 1 x
5
x4
4!
3
á Ê cos2 x † sin x œ cos x † cos x † sin x
x3
3!
á
3
x5
5!
x
3!
a2xb5
5!
x7
7!
a2xb7
7!
á ‹ œ x 76 x3
2
á ‹ 16 Šx
x3
3!
x5
5!
x7
7!
61 5
120 x
1247 7
5040 x
ÞÞÞ
3
á‹ á
œ 1 x 12 x2 18 x4 Þ Þ Þ Þ
x3
x5
x7
1 3
1 5
1 7
1 3
1 5
1
1
ˆ
3! 5! 7! á and tan x œ x 3 x 5 x 7 x Þ Þ Þ Ê sinatan xb œ x 3 x 5 x
3
5
1 ˆ
1 ˆ
16 ˆx 31 x3 51 x5 71 x7 Þ Þ Þ‰ 120
x 13 x3 15 x5 17 x7 Þ Þ Þ‰ 5040
x 13 x3 15 x5 17 x7
5 7
x 12 x3 38 x5 16
x ÞÞÞ
34. sin x œ x
œ
71 x7 Þ Þ Þ‰
7
Þ Þ Þ‰ á
35. Since n œ 3, then f a4b axb œ sin x, lf a4b axbl Ÿ M on Ò0, 0.1Ó Ê lsin xl Ÿ 1 on Ò0, 0.1Ó Ê M œ 1. Then lR3 a0.1bl Ÿ 1 l0.14x 0l
4
œ 4.2 ‚ 106 Ê error Ÿ 4.2 ‚ 106
36. Since n œ 4, then f a5b axb œ ex , lf a5b axbl Ÿ M on Ò0, 0.5Ó Ê lex l Ÿ Èe on Ò0, 0.5Ó Ê M œ 2.7. Then
lR4 a0.5bl Ÿ 2.7 l0.55x 0l œ 7.03 ‚ 104 Ê error Ÿ 7.03 ‚ 104
5
kxk&
5!
37. By the Alternating Series Estimation Theorem, the error is less than
5
Ê kxk È
6 ‚ 10# ¸ 0.56968
38. If cos x œ 1
Ê kxk& a5!b a5 ‚ 10% b Ê kxk& 600 ‚ 10%
%
x#
#
and kxk 0.5, then the error is less than ¹ (.5)
24 ¹ œ 0.0026, by Alternating Series Estimation Theorem;
since the next term in the series is positive, the approximation 1
x#
#
is too small, by the Alternating Series Estimation
Theorem
39. If sin x œ x and kxk 10$ , then the error is less than
a10c$ b
3!
$
¸ 1.67 ‚ 1010 , by Alternating Series Estimation Theorem;
$
The Alternating Series Estimation Theorem says R# (x) has the same sign as x3! . Moreover, x sin x
Ê 0 sin x x œ R# (x) Ê x 0 Ê 10$ x 0.
40. È1 x œ 1
x
#
x#
8
x$
16
#
á . By the Alternating Series Estimation Theorem the kerrork ¹ 8x ¹
œ 1.25 ‚ 10&
c $
3Ð0Þ1Ñ (0.1)$
3!
c $
(0.1)$
3!
41. kR# (x)k œ ¹ e3!x ¹
42. kR# (x)k œ ¹ e3!x ¹
2x ‰
43. sin# x œ ˆ 1 cos
œ
#
Ê
d
dx
asin# xb œ
œ 2x
(2x)$
3!
d
dx
(2x)&
5!
"
#
1.87 ‚ 104 , where c is between 0 and x
œ 1.67 ‚ 10% , where c is between 0 and x
#
"
#
Š 2x
2!
(2x)(
7!
cos 2x œ
2$ x%
4!
2& x'
6!
"
#
"# Š1
(2x)#
2!
á ‹ œ 2x
(2x)%
4!
(2x)$
3!
(2x)'
6!
(2x)&
5!
á‹ œ
(2x)(
7!
2x#
#!
2$ x%
4!
2& x'
6!
á
á Ê 2 sin x cos x
á œ sin 2x, which checks
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
(0.01)#
8
Section 10.9 Convergence of Taylor Series
44. cos# x œ cos 2x sin# x œ Š1
œ1
#
2x
#!
$ %
2 x
4!
& '
2 x
6!
(2x)#
#!
(2x)%
4!
(2x)'
6!
á œ 1 x# "3 x%
2
45
(2x))
8!
x'
#
á ‹ Š 2x
#!
"
315
2$ x%
4!
2& x'
6!
2( x)
8!
625
á‹
x) á
45. A special case of Taylor's Theorem is f(b) œ f(a) f w (c)(b a), where c is between a and b Ê f(b) f(a) œ f w (c)(b a),
the Mean Value Theorem.
46. If f(x) is twice differentiable and at x œ a there is a point of inflection, then f ww (a) œ 0. Therefore,
L(x) œ Q(x) œ f(a) f w (a)(x a).
47. (a) f ww Ÿ 0, f w (a) œ 0 and x œ a interior to the interval I Ê f(x) f(a) œ
Ê f(x) Ÿ f(a) throughout I Ê f has a local maximum at x œ a
(b) similar reasoning gives f(x) f(a) œ
local minimum at x œ a
f ww (c# )
#
(x a)#
f ww (c# )
#
(x a)# Ÿ 0 throughout I
0 throughout I Ê f(x)
f(a) throughout I Ê f has a
48. f(x) œ (1 x)" Ê f w (x) œ (1 x)# Ê f ww (x) œ 2(1 x)$ Ê f Ð3Ñ (x) œ 6(1 x)%
Ê f Ð4Ñ (x) œ 24(1 x)& ; therefore
"
1 x
¸ 1 x x# x$ . kxk 0.1 Ê
&
%
Ð4Ñ
10
11
"
1 x
10
9
‰
Ê ¹ (1"x)& ¹ ˆ 10
9
&
%
‰ Ê the error e$ Ÿ ¹ max f 4! (x) x ¹ (0.1)% ˆ 10
‰ œ 0.00016935 0.00017, since ¹ f
Ê ¹ (1x x)& ¹ x% ˆ 10
9
9
Ð4Ñ
&
(x)
4! ¹
œ ¹ (1"x)& ¹ .
49. (a) f(x) œ (1 x)k Ê f w (x) œ k(1 x)k1 Ê f ww (x) œ k(k 1)(1 x)k2 ; f(0) œ 1, f w (0) œ k, and f ww (0) œ k(k 1)
Ê Q(x) œ 1 kx k(k # ") x#
"
(b) kR# (x)k œ ¸ 3†3!2†" x$ ¸ 100
Ê kx$ k
"
100
Ê 0x
"
100"Î$
or 0 x .21544
50. (a) Let P œ x 1 Ê kxk œ kP 1k .5 ‚ 10n since P approximates 1 accurate to n decimals. Then,
P sin P œ (1 x) sin (1 x) œ (1 x) sin x œ 1 (x sin x) Ê k(P sin P) 1k
œ ksin x xk Ÿ
kxk$
3!
0.125
3!
‚ 103n .5 ‚ 103n Ê P sin P gives an approximation to 1 correct to 3n decimals.
_
_
n œ0
n œk
51. If f(x) œ ! an xn , then f ÐkÑ (x) œ ! n(n 1)(n 2)â(n k 1)an xnk and f ÐkÑ (0) œ k! ak
Ê ak œ
f ÐkÑ (0)
k!
for k a nonnegative integer. Therefore, the coefficients of f(x) are identical with the corresponding
coefficients in the Maclaurin series of f(x) and the statement follows.
52. Note: f even Ê f(x) œ f(x) Ê f w (x) œ f w (x) Ê f w (x) œ f w (x) Ê f w odd;
f odd Ê f(x) œ f(x) Ê f w (x) œ f w (x) Ê f w (x) œ f w (x) Ê f w even;
also, f odd Ê f(0) œ f(0) Ê 2f(0) œ 0 Ê f(0) œ 0
(a) If f(x) is even, then any odd-order derivative is odd and equal to 0 at x œ 0. Therefore,
a" œ a$ œ a& œ á œ 0; that is, the Maclaurin series for f contains only even powers.
(b) If f(x) is odd, then any even-order derivative is odd and equal to 0 at x œ 0. Therefore,
a! œ a# œ a% œ á œ 0; that is, the Maclaurin series for f contains only odd powers.
53-58. Example CAS commands:
Maple:
f := x -> 1/sqrt(1+x);
x0 := -3/4;
x1 := 3/4;
# Step 1:
plot( f(x), x=x0..x1, title="Step 1: #53 (Section 10.9)" );
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
626
Chapter 10 Infinite Sequences and Series
# Step 2:
P1 := unapply( TaylorApproximation(f(x), x = 0, order=1), x );
P2 := unapply( TaylorApproximation(f(x), x = 0, order=2), x );
P3 := unapply( TaylorApproximation(f(x), x = 0, order=3), x );
# Step 3:
D2f := D(D(f));
D3f := D(D(D(f)));
D4f := D(D(D(D(f))));
plot( [D2f(x),D3f(x),D4f(x)], x=x0..x1, thickness=[0,2,4], color=[red,blue,green], title="Step 3: #57 (Section 9.9)" );
c1 := x0;
M1 := abs( D2f(c1) );
c2 := x0;
M2 := abs( D3f(c2) );
c3 := x0;
M3 := abs( D4f(c3) );
# Step 4:
R1 := unapply( abs(M1/2!*(x-0)^2), x );
R2 := unapply( abs(M2/3!*(x-0)^3), x );
R3 := unapply( abs(M3/4!*(x-0)^4), x );
plot( [R1(x),R2(x),R3(x)], x=x0..x1, thickness=[0,2,4], color=[red,blue,green], title="Step 4: #53 (Section 10.9)" );
# Step 5:
E1 := unapply( abs(f(x)-P1(x)), x );
E2 := unapply( abs(f(x)-P2(x)), x );
E3 := unapply( abs(f(x)-P3(x)), x );
plot( [E1(x),E2(x),E3(x),R1(x),R2(x),R3(x)], x=x0..x1, thickness=[0,2,4], color=[red,blue,green],
linestyle=[1,1,1,3,3,3], title="Step 5: #53 (Section 10.9)" );
# Step 6:
TaylorApproximation( f(x), view=[x0..x1,DEFAULT], x=0, output=animation, order=1..3 );
L1 := fsolve( abs(f(x)-P1(x))=0.01, x=x0/2 );
# (a)
R1 := fsolve( abs(f(x)-P1(x))=0.01, x=x1/2 );
L2 := fsolve( abs(f(x)-P2(x))=0.01, x=x0/2 );
R2 := fsolve( abs(f(x)-P2(x))=0.01, x=x1/2 );
L3 := fsolve( abs(f(x)-P3(x))=0.01, x=x0/2 );
R3 := fsolve( abs(f(x)-P3(x))=0.01, x=x1/2 );
plot( [E1(x),E2(x),E3(x),0.01], x=min(L1,L2,L3)..max(R1,R2,R3), thickness=[0,2,4,0], linestyle=[0,0,0,2],
color=[red,blue,green,black], view=[DEFAULT,0..0.01], title="#53(a) (Section 10.9)" );
abs(`f(x)`-`P`[1](x) ) <= evalf( E1(x0) );
# (b)
abs(`f(x)`-`P`[2](x) ) <= evalf( E2(x0) );
abs(`f(x)`-`P`[3](x) ) <= evalf( E3(x0) );
Mathematica: (assigned function and values for a, b, c, and n may vary)
Clear[x, f, c]
f[x_]= (1 x)3/2
{a, b}= {1/2, 2};
pf=Plot[ f[x], {x, a, b}];
poly1[x_]=Series[f[x], {x,0,1}]//Normal
poly2[x_]=Series[f[x], {x,0,2}]//Normal
poly3[x_]=Series[f[x], {x,0,3}]//Normal
Plot[{f[x], poly1[x], poly2[x], poly3[x]}, {x, a, b},
PlotStyle Ä {RGBColor[1,0,0], RGBColor[0,1,0], RGBColor[0,0,1], RGBColor[0,.5,.5]}];
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.10 The Binomial Series
627
The above defines the approximations. The following analyzes the derivatives to determine their maximum values.
f''[c]
Plot[f''[x], {x, a, b}];
f'''[c]
Plot[f'''[x], {x, a, b}];
f''''[c]
Plot[f''''[x], {x, a, b}];
Noting the upper bound for each of the above derivatives occurs at x = a, the upper bounds m1, m2, and m3 can be defined
and bounds for remainders viewed as functions of x.
m1=f''[a]
m2=-f'''[a]
m3=f''''[a]
r1[x_]=m1 x2 /2!
Plot[r1[x], {x, a, b}];
r2[x_]=m2 x3 /3!
Plot[r2[x], {x, a, b}];
r3[x_]=m3 x4 /4!
Plot[r3[x], {x, a, b}];
A three dimensional look at the error functions, allowing both c and x to vary can also be viewed. Recall that c must be a
value between 0 and x, so some points on the surfaces where c is not in that interval are meaningless.
Plot3D[f''[c] x2 /2!, {x, a, b}, {c, a, b}, PlotRange Ä All]
Plot3D[f'''[c] x3 /3!, {x, a, b}, {c, a, b}, PlotRange Ä All]
Plot3D[f''''[c] x4 /4!, {x, a, b}, {c, a, b}, PlotRange Ä All]
10.10 THE BINOMIAL SERIES
1. (1 x)"Î# œ 1 "# x
ˆ "# ‰ ˆ "# ‰ x#
2. (1 x)"Î$ œ 1 "3 x
ˆ "3 ‰ ˆ 23 ‰ x#
"Î#
8. a1 x# b
"Î$
œ 1 "# x$
œ 1 "3 x#
"Î#
9. ˆ1 1x ‰ œ 1 "# ˆ x1 ‰
"
16
x$ á
ˆ 3" ‰ ˆ 32 ‰ ˆ 53 ‰ x$
á œ 1 3" x 9" x#
5
81
x$ á
3!
#!
ˆ "# ‰ ˆ "# ‰ (2x)#
#!
(2)(3) ˆ x# ‰
4
6. ˆ1 x3 ‰ œ 1 4 ˆ x3 ‰
á œ 1 "# x "8 x#
ˆ "# ‰ ˆ 3# ‰ (x)#
4. (1 2x)"Î# œ 1 "# (2x)
#
5. ˆ1 x# ‰ œ 1 # ˆ x# ‰
ˆ "# ‰ ˆ "# ‰ ˆ 3# ‰ x$
3!
#!
3. (1 x)"Î# œ 1 "# (x)
7. a1 x$ b
#!
#
#!
(4)(3) ˆ x3 ‰
#!
#
Š "# ‹ Š "# ‹ Š #3 ‹ (2x)$
3!
$
3!
(4)(3)(2) ˆ x3 ‰
#!
ˆ "3 ‰ ˆ 43 ‰ ax# b#
#!
#!
3!
(2)(3)(4) ˆ x# ‰
ˆ "# ‰ ˆ 3# ‰ ax$ b#
ˆ "# ‰ ˆ "# ‰ ˆ 1x ‰#
ˆ "# ‰ ˆ 3# ‰ ˆ 5# ‰ (x)$
3!
$
5
16
x$ á
á œ 1 x 12 x# 12 x$ á
á œ 1 x 34 x# "# x$
(4)(3)(2)(1) ˆ x3 ‰
3!
ˆ 3" ‰ ˆ 43 ‰ ˆ 73 ‰ ax# b$
3!
ˆ "# ‰ ˆ "# ‰ ˆ 3# ‰ ˆ 1x ‰$
4
4!
ˆ #" ‰ ˆ #3 ‰ ˆ #5 ‰ ax$ b$
3!
á œ 1 "# x 38 x#
0 á œ 1 34 x 32 x2
á œ 1 "# x$ 38 x'
á œ 1 "3 x# 29 x%
á œ1
"
#x
1
8x#
"
16x$
x* á
5
16
14
81
x' á
á
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
4
27
x3
1 4
81 x
628
10.
Chapter 10 Infinite Sequences and Series
x
3
È
1x
(4)(3)x#
#!
11. (1 x)% œ 1 4x
$
12. a1 x# b œ 1 3x#
(4)(3)(2)x$
3!
#
(3)(2) ax# b
#!
$
(3)(2)(1) ax# b
3!
(3)(2)(2x)#
#!
%
14. ˆ1 #x ‰ œ 1 4 ˆ #x ‰
(4)(3) ˆ x# ‰
#
'00 2 sin x# dx œ '00 2 Šx# x3! x5!
Þ
Þ
kE k Ÿ
(
Þ
x
x#
4
'00 1 È "
x$
18
Š1 x
á“
kE k Ÿ
(0.1)&
10
œ 0.000001
'!!Þ#&
$È
1 x# dx œ '0
0Þ25
&
(0.25)
45
œ 1 3x# 3x% x'
œ 1 6x 12x# 8x$
(4)(3)(2) ˆ x# ‰
3!
$
(4)(3)(2)(1) ˆ x# ‰
$
á ‹ dx œ ’ x3
%
4!
x(
7†3!
!Þ#
á“
œ 1 2x 23 x# "# x$
$
¸ ’ x3 “
!
!Þ#
!
"
16
x%
¸ 0.00267 with error
x#
#!
x$
3!
x%
4!
á 1‹ dx œ '0 Š1
0 Þ2
¸ 0.19044 with error kEk Ÿ
(0.2)%
96
x%
2
á“
3x)
8
x#
3
Š1
á ‹ dx œ ’x
x%
9
x&
10
á ‹ dx œ ’x
x$
9
x#
6
x
#
x$
24
á ‹ dx
¸ 0.00002
!Þ"
¸ [x]!Þ"
! ¸ 0.1 with error
!
x&
45
á“
!Þ#&
!
¸ ’x
!Þ#&
x$
9 “!
¸ 0.25174 with error
'00 1 sinx x dx œ '00 1 Š1 x3! x5! x7! á ‹ dx œ ’x 3x†3! 5x†5! 7x†7! á “ !Þ" ¸ ’x 3x†3! 5x†5! “ !Þ"
Þ
#
%
'
$
&
(
$
&
!
(0.1)7
7†7!
Þ
%
(0.1)9
216
¸ 0.0996676643, kEk Ÿ
21. a1 x% b
Ê
"Î#
œ (1)"Î#
ˆ "# ‰ ˆ "# ‰ ˆ 3# ‰ ˆ 5# ‰
4!
'0
0Þ1
Š1
x%
#
x)
8
Š "# ‹
1
'
)
ˆ "# ‰ ˆ "# ‰
(1)"Î# ax% b
#!
%
x"#
16
$
x&
10
x(
42
á“
!Þ"
!
¸ ’x
x$
3
x&
10
¸ 4.6 ‚ 1012
(1)(Î# ax% b á œ 1
!
¸ 2.8 ‚ 1012
'00 1 exp ax# b dx œ '00 1 Š1 x# x2! x3! x4! á ‹ dx œ ’x x3
Þ
5x"'
128
x%
#
#
(1)$Î# ax% b
á ‹ dx ¸ ’x
x)
8
x"#
16
!Þ"
x&
10 “ !
ˆ "# ‰ ˆ "# ‰ ˆ 3# ‰
3!
5x"'
128
(1)&Î# ax% b
$
á
¸ 0.100001, kEk Ÿ
(0.1)9
72
¸ 1.39 ‚ 1011
"
x
'01 ˆ 1 xcos x ‰ dx œ '01 Š "# x4! x6! x8! 10!
á ‹ dx ¸ ’ x# 3x†4! 5x†6! 7x†8! 9†x10! “
#
%
'
)
$
&
(
#
¸ 0.4863853764, kEk Ÿ
23.
á
¸ 0.0000217
¸ 0.0999444611, kEk Ÿ
22.
!
0Þ1
Þ
20.
!Þ#
dx œ '0 Š1
kEk Ÿ
19.
"
x
1 x%
Þ
18.
"!
14 4
81 x
œ 1 4x 6x# 4x$ x%
(3)(2)(1)(2x)$
3!
á œ x 31 x# 92 x3
3!
¸ 0.0000003
(.2)
7†3!
œ ’x
17.
'
'00 2 ec x " dx œ '00 2
Þ
16.
#!
ˆ "3 ‰ ˆ 43 ‰ ˆ 73 ‰ x$
#!
(4)(3)(2)x%
4!
13. (1 2x)$ œ 1 3(2x)
15.
ˆ "3 ‰ ˆ 43 ‰ x#
œ xa1 xb"Î3 œ xŒ1 ˆc "3 ‰x
1
11†12!
!
10
¸ 1.9 ‚ 10
'01 cos t# dt œ '01 Š1 t# 4!t t6! á ‹ dt œ ’t 10t 9t†4! 13t †6! á “ "
%
)
*
"#
&
*
"$
!
Ê kerrork
"
13†6!
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
¸ .00011
!Þ"
x(
42 “ !
Section 10.10 The Binomial Series
24.
'01 cos Èt dt œ '01 Š1 #t 4!t 6!t 8!t á ‹ dt œ ’t t4 3t†4! 4t†6! 5t†8! á “ "
#
Ê kerrork
x
Ê kerrork
t'
3!
"
15†7!
t"!
5!
x&
5
x(
7†2!
x*
9†3!
x""
11†4!
"
33†34
t$
3
aex (1 x)b œ
"
t%
2x%
5!
"
)&
t#
6!
)
7!
ay tan" yb œ
"
t%
œ
9!
t#
#
’1
t%
8!
)%
"
y$
x#
#
$
%
&
t"&
15†7!
x
á“
x$
3
¸
!
x(
7†3!
t&
5
#
x
#
x#
##
t(
7†2!
t*
9†3!
t""
11†4!
t"$
13†5!
x""
11†5!
á“
x
!
¸ 0.00064
t%
1#
'
%
x
3†4
x
t'
30
x
5†6
x#
#
á“ ¸
t#
2 †2
t$
3 †3
x$
3#
x%
4#
!
)
x
7†8
x%
1#
Ê kerrork
t&
5 †5
(0.5)'
30
¸ .00052
$#
á (1)"&
t%
4 †4
x
31†32
x
á“ ¸ x
!
x#
##
x$
3#
x%
4#
x&
5#
x$"
31#
á (1)$"
x$
3!
á ‹ 1 x‹ œ
"
#
x
3!
x#
4!
á Ê lim
x%
4!
á ‹ Š1 x
x#
#!
x$
3!
x%
4!
á ‹“ œ
2x%
5!
2x'
7!
ex (1 x)
x#
xÄ0
"
#
x$
3!
xÄ0
x#
#!
"
13†5!
F(x) ¸ x
á Ê lim
sin )‹ œ
)Ä0
"
y$
2x'
7!
#
$
á ‹ dt œ ’t
á‹ œ
#
)$
6
t$
4
x#
4!
Š1 cos t t# ‹ œ
Š)
t""
11†5!
á ‹ dt œ ’ t3
#
ŠŠ1 x
’Š1 x
t(
7†3!
á ‹ dt œ ’ t2
"
x#
"
x
aex ex b œ
2x#
3!
t(
7
"
x
œ lim Š 5!"
34.
t#
3
tÄ0
33.
t
2
x
3!
œ lim Š 4!"
32.
œ lim Š "#
œ2
31.
t&
5
t"#
5!
Ê kerrork
(0.5)'
6# ¸ .00043
"
32# ¸ .00097 when
xÄ0
30.
t"!
4!
¸ .00089 when F(x) ¸
x
"
x#
$
á ‹ dt œ ’ t3
t)
3!
28. (a) F(x) œ '0 Š1
(b) kerrork
t"%
7!
x
Ê kerrork
t'
2!
27. (a) F(x) œ '0 Št
(b) kerrork
#
¸ 0.000013
x
x$
3
%
¸ 0.000004960
26. F(x) œ '0 Št# t%
¸
$
!
"
5†8!
25. F(x) œ '0 Št#
29.
629
ex e
x
x
t#
#
t%
4!
Š1
œ x lim
Š2
Ä_
t'
6!
2x#
3!
á ‹“ œ 4!"
t#
6!
"
x
Š2x
2x$
3!
2x&
5!
2x(
7!
á‹
y%
7
á‹ œ 2
t%
8!
#
á Ê lim
" cos t Š t# ‹
t%
tÄ0
"
á ‹ œ 24
"
)&
)$
6
Š)
á‹ œ
’y Šy
)
)$
3!
)&
5!
á‹ œ
"
5!
)#
7!
)%
9!
$
á Ê lim
sin ) ) Š )6 ‹
)&
)Ä0
"
1 #0
y$
3
y&
5
á ‹“ œ
"
3
y#
5
y%
7
á Ê lim
yÄ0
y tan " y
y$
œ lim Š 3"
yÄ0
y#
5
"
3
tanc" y sin y
y$ cos y
Ê lim
yÄ0
œ
Œy
y$
3
y&
5
á Œy
y$
tanc" y sin y
y$ cos y
œ lim
Œ
"
6
35. x# Š1 e1Îx ‹ œ x# ˆ1 1
ˆ1
œ x lim
Ä_
"
#x#
"
6x%
y&
5!
á
cos y
yÄ0
#
y$
3!
23y#
5!
á
cos y
"
x#
"
#x%
œ
Œ
y$
6
23y&
5!
y$
cos y
á
œ
"
Π6
23y#
5!
á
cos y
œ 6"
"
6x'
á ‰ œ 1
"
#x#
"
6x%
#
x# Še1Îx 1‹
á Ê x lim
Ä_
á ‰ œ 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
á‹
630
Chapter 10 Infinite Sequences and Series
36. (x 1) sin ˆ x " 1 ‰ œ (x 1) Š x " 1
"
3!(x 1)$
Ê x lim
(x 1) sin ˆ x " 1 ‰ œ x lim
Š1
Ä_
Ä_
%
37.
ln a1 x# b
1 cos x
38.
x# 4
ln (x 1)
'
x
x
#
Œx # 3 á
œ
x#
#!
1 Š1
x%
4!
á‹
(x c 2)#
’(x 2)
#
x 2
œ lim
x Ä 2 ’1 x c# 2 (x
2)#
3
39. sin 3x2 œ 3x2 92 x6
(x c 2)$
3
á“
81 10
40 x
10
3x2 92 x6 81
40 x . . .
2 2 x4 4 x6 . . .
2x
xÄ0
3
45
œ lim
40. lna1 x3 b œ x3
41. 1 1
x6
2
6
xÄ0
Š #"!
x#
4!
(x 2)(x 2)
œ
œ lim
œ
x#
#
Œ1
x3
1
2x
9
x9
3
á“
1
3x
1
4x
"
3!(x 1)#
á
á‹
x2
’1
x
#
2
(x c 2)#
3
"
5!(x 1)%
á
á‹ œ 1
x#
#
Š #"!
x#
4!
Œ1
œ lim
xÄ0
x%
3
á
á‹
x Ä 2 ln (x 1)
8
3 92 x4 81
40 x . . .
2 2
4 4
2
x
x
. . .
xÄ0
3
45
x12
4
œ
. . . and x sin x2 œ x3 16 x7
12
3
6
9
1 x2 x3 x4 . . .
1 8
1
120
x 5040
x12 Þ Þ Þ
x Ä 0 1
œ lim
4 6
45 x
sin 3x2
. . . Ê lim
x Ä 0 1 cos 2x
3
2
1 4
6x
1 11
120 x
1
15
5040 x
Þ Þ Þ Ê lim
xÄ0
œ1
Þ Þ Þ œ e1 œ e
3
4
5
3
2
42. ˆ 14 ‰ ˆ 14 ‰ ˆ 14 ‰ Þ Þ Þ œ ˆ 14 ‰ ”1 ˆ 14 ‰ ˆ 14 ‰ Þ Þ Þ • œ
43. 1
32
42 2x
34
44 4x
36
4 6 6x
ÞÞÞ œ 1
1 ˆ 3 ‰2
2x 4
1 ˆ 3 ‰4
4x 4
1 ˆ 3 ‰6
6x 4
1
1
64 1 1Î4
œ
1 4
64 3
œ
1
48
Þ Þ Þ œ cosˆ 34 ‰
44.
1
2
1
2†22
1
3†23
1
4†24
2
3
4
Þ Þ Þ œ ˆ 21 ‰ 21 ˆ #1 ‰ 31 ˆ #1 ‰ 41 ˆ #1 ‰ Þ Þ Þ œ lnˆ1 21 ‰ œ lnˆ 23 ‰
45.
1
3
13
33 3x
15
35 5x
17
37 7x
ÞÞÞ œ
46.
2
3
23
33 †3
25
35 †5
27
37 †7
3
5
7
Þ Þ Þ œ ˆ 32 ‰ 31 ˆ 32 ‰ 51 ˆ 32 ‰ 71 ˆ 32 ‰ Þ Þ Þ œ tan1 ˆ 23 ‰
1
3
1 ˆ 1 ‰3
3x 3
1 ˆ 1 ‰5
5x 3
1 ˆ 1 ‰7
7x 3
Þ Þ Þ œ sinˆ 13 ‰ œ
47. x3 x4 x5 x6 Þ Þ Þ œ x3 a1 x x2 x3 Þ Þ Þ b œ x3 ˆ 1 1 x ‰ œ
48. 1
32 x2
2x
34 x4
4x
36 x6
6x
ÞÞÞ œ 1
2
1
2x a3xb
4
1
4x a3xb
2
6
1
6x a3xb
22 x4
2x
23 x5
3x
24 x6
4x
Þ Þ Þ œ cosa3xb
3
Þ Þ Þ œ x2 Š1 2x
51. 1 2x 3x2 4x3 5x4 Þ Þ Þ œ
52. 1
x
2
x2
3
x3
4
x4
5
d
dx a1
Þ Þ œ 1x Šx
a2xb2
2x
a2xb3
3x
a2xb4
4x
x3
3
x4
4
x5
5
x3
1 + x2
Þ Þ Þ ‹ œ x2 e2x
x x2 x3 x4 x5 Þ Þ Þ b œ
x2
2
È3
2
x3
1x
49. x3 x5 x7 x9 Þ Þ Þ œ x3 Š1 x2 ax2 b ax2 b Þ Þ Þ ‹ œ x3 ˆ 1 +1x2 ‰ œ
50. x2 2x3
œ 2! œ 2
x# 4
Ê lim
á“
. . . and 1 cos 2x œ 2x2 23 x4
œ lim
"
5!(x 1)%
#
lim ln a1 x b
x Ä 0 1 cos x
Ê
"
3!(x 1)#
á‹ œ 1
œ4
x3 x2 x3 x4 . . .
1 11
1
16 x7 120
x 5040
x15 Þ Þ Þ
œ
x%
3
"
5!(x 1)&
d ˆ 1 ‰
dx 1 x
œ
Þ Þ ‹ œ 1x lna1 xb œ
1
a1 x b 2
lna1xb
x
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
lnˆ1 x3 ‰
x sin x2
Section 10.10 The Binomial Series
x‰
53. ln ˆ 11
x œ ln (1 x) ln (1 x) œ Šx
54. ln (1 x) œ x
"
n10n
"
10)
x$
3
x%
4
á
Ê n10n 10) when n
55. tan" x œ x
"
#n1
x#
#
"
10$
x$
3
x&
5
Ê n
56. tan" x œ x
x$
3
x*
9
(1)n 1 xn
n
x$
3
x%
4
á ‹ Šx
á Ê kerrork œ ¹ (")n
n 1 n
á
(")n 1 x2n
2n1
1
n œ1
¹œ
"
n10n
n 1 2n 1
x(
7
x*
9
á
(1)n1 x2n1
2n1
x%
4
á ‹ œ 2 Šx
x$
3
x&
5
á‹
when x œ 0.1;
¹œ
2n 1
x2n1 ¹
2n 1
á and n lim
†
¹x
Ä _ 2n 1
(1)n
2n1
n œ1
(1)nc1
2n1
x$
3
á Ê kerrork œ ¹ (1)2nx1
_
_
x
8 Ê 7 terms
Ê tan" x converges for kxk 1; when x œ 1 we have !
we have !
x#
#
"
#n 1
when x œ 1;
œ 500.5 Ê the first term not used is the 501st Ê we must use 500 terms
1001
#
x&
5
x(
7
x#
#
631
¸ 2n 1 ¸ œ x#
œ x# n lim
Ä _ #n 1
which is a convergent series; when x œ 1
which is a convergent series Ê the series representing tan" x diverges for kxk 1
(1)n 1 x2n 1
x$
x&
x(
x*
á and when the series representing 48
3 5 7 9 á
2n 1
"
'
error less than 3 † 10 , then the series representing the sum
" ‰
" ‰
48 tan" ˆ 18
32 tan" ˆ 57
20 tan" ˆ #"39 ‰ also has an error of magnitude less than 10' ;
" ‰
tan" ˆ 18
has an
57. tan" x œ x
thus
2nc1
kerrork œ 48
"
Š 18
‹
#n 1
"
3†10'
Ê n
4 using a calculator Ê 4 terms
58. ln (sec x) œ '0 tan t dt œ '0 Št
x
"Î#
x
t$
3
x#
3x%
# 8
2nb3
lim ¹ 1†3†5â(2n 1)(2n 1)x
†
n Ä _ 2†4†6â(2n)(2n 2)(2n 3)
59. (a) a1 x# b
¸1
2t&
15
á ‹ dt ¸
x#
#
x%
12
$
5x'
"
x ¸ x x6
16 Ê sin
2†4†6â(2n)(2n ")
1†3†5â(2n 1)x2nb1 ¹ 1 Ê
x'
45
3x&
40
á
5x(
112
; Using the Ratio Test:
(2n 1)(2n1)
x# n lim
¹
¹1
Ä _ (2n 2)(2n 3)
Ê kxk 1 Ê the radius of convergence is 1. See Exercise 69.
d
dx
(b)
acos" xb œ a1 x# b
60. (a) a1 t# b
œ1
"Î#
t#
#
term,
61.
"
1x
5x
112
Ê cos" x œ
¸ (1)"Î# ˆ "# ‰ (1)$Î# at# b
3t%
2# †2!
(b) sinh" ˆ 4" ‰ ¸
(
"Î#
"
4
3†5t'
2$ †3!
"
384
1
#
sin" x ¸
x
, evaluated at t œ
"
4
Šx
ˆ "# ‰ ˆ #3 ‰ (1) &Î# at# b#
#!
Ê sinh" x ¸ '0 Š1
3
40,960
1
#
t#
#
3t%
8
5t'
16 ‹
x$
6
3x&
40
5x(
112 ‹
¸
1
#
x
x$
6
3x&
40
ˆ #" ‰ ˆ #3 ‰ ˆ #5 ‰ (1) (Î# at# b$
3!
dt œ x
x$
6
3x&
40
5x(
112
œ 0.24746908; the error is less than the absolute value of the first unused
since the series is alternating Ê kerrork
œ 1 "(x) œ 1 x x# x$ á Ê
d
dx
ˆ 11x ‰ œ
"
1 x#
œ
d
dx
5 ˆ 4" ‰
112
(
¸ 2.725 ‚ 10'
a1 x x# x$ á b
œ 1 2x 3x# 4x$ á
62.
"
1 x#
œ 1 x# x% x' á Ê
d
dx
ˆ 1 " x# ‰ œ
2x
a1 x# b#
œ
d
dx
a1 x# x% x' á b œ 2x 4x$ 6x& á
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
5x(
112
632
Chapter 10 Infinite Sequences and Series
8â(2n 2)†(2n)
63. Wallis' formula gives the approximation 1 ¸ 4 ’ 3†23††45††45††67††67†â
(2n 1)†(2n 1) “ to produce the table
n
µ1
10
3.221088998
20
3.181104886
30
3.167880758
80
3.151425420
90
3.150331383
93
3.150049112
94
3.149959030
95
3.149870848
100
3.149456425
At n œ 1929 we obtain the first approximation accurate to 3 decimals: 3.141999845. At n œ 30,000 we still do
not obtain accuracy to 4 decimals: 3.141617732, so the convergence to 1 is very slow. Here is a Maple CAS
procedure to produce these approximations:
pie :=
proc(n)
local i,j;
a(2) := evalf(8/9);
for i from 3 to n do a(i) := evalf(2*(2*i2)*i/(2*i1)^2*a(i1)) od;
[[j,4*a(j)] $ (j = n5 .. n)]
end
_
_
_
64. (a) faxb œ 1 !ˆ mk ‰xk Ê f w axb œ !ˆ mk ‰k xk1 Ê a1 xb † f w axb œ a1 xb!ˆ mk ‰k xk1
kœ1
kœ1
_
_
kœ1
_
_
_
_
kœ2
kœ1
œ !ˆ mk ‰k xk 1 x † !ˆ mk ‰k xk1 œ !ˆ mk ‰k xk 1 !ˆ mk ‰k xk œ ˆ m1 ‰a1b x0 !ˆ mk ‰k xk1 !ˆ mk ‰k xk
kœ1
kœ1
_
œ m!
kœ2
ˆ mk ‰k xk 1
kœ1
_
!
kœ1
ˆ mk ‰k xk
kœ1
_
ˆ mk ‰k xk 1
Note that: !
kœ2
_
_
œ!
kœ1
_
ˆkm
‰
1 ak
1b xk .
_
_
k
! ˆ m ‰k xk
‰
Thus, a1 xb † f w axb œ m ! ˆ mk ‰k xk 1 ! ˆ mk ‰k xk œ m ! ˆ k m
1 ak 1b x
k
kœ2
_
œ m!
kœ1
‰
’ˆ k m
1 ak
1 b xk
kœ1
ˆ mk ‰k xk “
kœ1
_
œ m!
kœ1
m ‰
’ˆˆ k
1 ak
kœ1
1b
ˆ mk ‰k ‰xk “.
m†am "bâam ak 1b 1b
ak 1b m†am "bâk!am k 1b k
ak1b!
m†am "bâam k 1b
k œ m†am "bâk!am k 1b aam kb kb œ m m†am "bâk!am k 1b
k!
m ‰
ˆm‰
Note that: ˆ k
1 ak 1b k k œ
œ
m†am "bâam kb
k!
_
_
_
kœ1
kœ1
kœ1
œ mˆ mk ‰.
! ’ˆmˆ m ‰ ‰xk “ œ m m!ˆ m ‰xk
‰
ˆm‰ ‰ k
Thus, a1 xb † f w axb œ m ! ’ˆˆ k m
1 ak 1b k k x “ œ m
k
k
_
œ mŒ1 !ˆ mk ‰xk œ m † faxb Ê f w axb œ
kœ1
m†faxb
a1 x b
if " x 1.
(b) Let gaxb œ a1 xbm faxb Ê gw axb œ ma1 xbm1 faxb a1 xbm f w axb
œ ma1 xbm1 faxb a1 xbm †
m†faxb
a1xb
œ ma1 xbm1 faxb a1 xbm1 † m † faxb œ 0.
(c) gw axb œ 0 Ê gaxb œ c Ê a1 xbm faxb œ c Ê faxb œ
Ê fa0b œ 1
65. a1 x# b
"Î#
_
!ˆ m ‰a0bk
k
kœ1
œ a1 ax# bb
3!
_
œ ca1 xbm . Since faxb œ 1 !ˆ mk ‰xk
kœ1
m
m
œ 1 0 œ 1 Ê ca1 0b œ 1 Ê c œ 1 Ê faxb œ a1 xb .
"Î#
ˆ "# ‰ ˆ 3# ‰ ˆ 5# ‰ (1) (Î# ax# b$
c
a1xbcm
œ (1)"Î# ˆ "# ‰ (1)$Î# ax# b
á œ1
#
x
#
%
1†3x
2# †#!
1†3†5x
2$ †3!
'
ˆ "# ‰ ˆ 3# ‰ (1)c&Î# ax# b#
_
á œ 1!
n œ1
#!
1†3†5â(2n1)x2n
#n †n!
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 10.10 The Binomial Series
Ê sin" x œ '0 a1 t# b
x
dt œ '0 Œ1 !
_
x
"Î#
1†3†5â(2n 1)x2n
#n †n!
n œ1
_
dt œ x !
nœ1
1†3†5â(2n 1)x2nb1
#†4â(2n)(2n 1)
633
,
where kxk 1
_
œ 'x ˆ t"#
tan" x œ 'x
"
t%
1
#
Ê tan" x œ
œ
lim
b Ä _
_
1
#
_
66. ctan" td x œ
"
t'
"
t)
"
x
"t
"
3t$
_
t#
bÄ_
_
Š 1# ‹
œ 'x – t " — dt œ 'x
1Š ‹
á ‰ dt œ lim
"
3x$
dt
1 t#
"t
"
3t$
"
5t&
"
t#
ˆ1
"
7t(
"
t#
b
á ‘x œ
"
" x
td c_ œ tan" x 1#
5x& á , x 1; ctan
"
"
"
"
"
"
‘x
5t& 7t( á b œ x 3x$ 5x& 7x( á
œ
"
t%
"
t'
"
x
"
3x$
'
x
_
á ‰ dt
"
5x&
"
7x(
dt
1 t#
"
Ê tan
x œ 1#
á
"
x
"
3x$
"
30
x& á ;
"
5x&
á ,
x 1
67. (a) ei1 œ cos (1) i sin (1) œ 1 i(0) œ 1
(b) ei1Î4 œ cos ˆ 14 ‰ i sin ˆ 14 ‰ œ
"
È2
i
È2
œ Š È" ‹ (1 i)
2
(c) ei1Î2 œ cos ˆ 1# ‰ i sin ˆ 1# ‰ œ 0 i(1) œ i
68. ei) œ cos ) i sin ) Ê ei) œ ei()) œ cos ()) i sin ()) œ cos ) i sin );
ei) eci)
;
#
ei) eci)
œ #i
ei) ei) œ cos ) i sin ) cos ) i sin ) œ 2 cos ) Ê cos ) œ
ei) ei) œ cos ) i sin ) (cos ) i sin )) œ 2i sin ) Ê sin )
69. ex œ 1 x
x#
#!
x$
3!
(i))#
2!
ei) œ 1 i)
Ê
œ
ei) eci)
œ
#
%
)#
1 #! )4!
ei) eci)
#i
œ)
œ
)$
3!
)&
5!
Š1 i)
)'
6!
Š1 i)
(i))#
#!
(i))$
3!
(i))%
4!
œ 1 i)
(i))#
2!
á œ 1 i)
á‹ Š1 i)
(i))#
#!
(i))$
3!
(i))$
3!
(i))#
#!
(i))$
3!
(i))%
4!
(i))%
4!
á and
(i))%
4!
á
á‹
#
á œ cos );
(i))#
#!
)(
7!
x%
i)
4! á Ê e
(i))$
(i))%
3! 4!
$
%
#
$
%
))
))
))
))
(i3!
(i4!
á‹ Š1 i) (i#)!) (i3!
(i4!
á‹
#i
á œ sin )
70. ei) œ cos ) i sin ) Ê ei) œ eiÐ)Ñ œ cos ()) i sin ()) œ cos ) i sin )
ei) eci)
œ cosh i)
#
i)
c
i)
œ e 2e œ sinh i)
(a) ei) ei) œ (cos ) i sin )) (cos ) i sin )) œ 2 cos ) Ê cos ) œ
(b) ei) ei) œ (cos ) i sin )) (cos ) i sin )) œ 2i sin ) Ê i sin )
71. ex sin x œ Š1 x
x#
#!
"
6
#
œ (1)x (1)x ˆ
x$
3!
x%
4!
"‰ $
# x
á ‹ Šx
"
6
ˆ
x$
3!
"‰ %
6 x
x&
5!
x(
7!
á‹
ˆ 1#"0 1"#
x
" ‰ &
#4 x
x
á œ x x# "3 x$
ex † eix œ eÐ1iÑx œ ex (cos x i sin x) œ ex cos x i ae sin xb Ê e sin x is the series of the imaginary part
_
of eÐ1iÑx which we calculate next; eÐ1iÑx œ !
n œ0
œ 1 x ix
Ð1iÑx
of e
is x
(xix)n
n!
œ 1 (x ix)
(x ix)#
#!
(x ix)$
3!
(x ix)%
4!
"
"
"
"
"
#
$
$
%
&
&
'
#! a2ix b 3! a2ix 2x b 4! a4x b 5! a4x 4ix b 6! a8ix b á Ê the imaginary
2 #
2 $
4 &
8 '
" $
" &
" '
#
#! x 3! x 5! x 6! x á œ x x 3 x 30 x 90 x á in agreement with our
x
product calculation. The series for e sin x converges for all values of x.
72.
d
dx
ˆeÐaibÑ ‰ œ
á
d
dx
ceax (cos bx i sin bx)d œ aeax (cos bx i sin bx) eax (b sin bx bi cos bx)
œ aeax (cos bx i sin bx) bieax (cos bx i sin bx) œ aeÐaibÑx ibeÐaibÑx œ (a ib)eÐaibÑx
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
part
634
Chapter 10 Infinite Sequences and Series
73. (a) ei)" ei)# œ (cos )" i sin )" )(cos )# i sin )# ) œ (cos )" cos )# sin )" sin )# ) i(sin )" cos )# sin )# cos )" )
œ cos()" )# ) i sin()" )# ) œ eiÐ)" )# Ñ
) i sin ) ‰
"
"
(b) ei) œ cos()) i sin()) œ cos ) i sin ) œ (cos ) i sin )) ˆ cos
cos ) i sin ) œ cos ) i sin ) œ ei)
74.
a bi ÐabiÑx
C" iC# œ ˆ aa# bib# ‰ eax (cos bx i sin bx) C" iC#
a# b# e
ax
œ a# e b# (a cos bx ia sin bx ib cos bx b sin bx) C" iC#
ax
œ a# e b# [(a cos bx b sin bx) (a sin bx b cos bx)i] C" iC#
ax
ax
œ e (a cosa#bxb#b sin bx) C" ie (a sina#bxb#b cos bx) iC# ;
ÐabiÑx
ax ibx
ax
ax
ax
e
'e
œe e
ÐabiÑx
dx œ
œ e (cos bx i sin bx) œ e cos bx ie sin bx, so that given
a bi
a# b#
eÐabiÑx C" iC# we conclude that ' eax cos bx dx œ
and ' eax sin bx dx œ
e (a sin bx b cos bx)
a# b#
ax
eax (a cos bx b sin bx)
a# b#
C"
C#
CHAPTER 10 PRACTICE EXERCISES
1. converges to 1, since n lim
a œ n lim
Š1
Ä_ n
Ä_
2. converges to 0, since 0 Ÿ an Ÿ
2
Èn
(1)n
n ‹
œ1
, n lim
0 œ 0, n lim
Ä_
Ä_
œ 0 using the Sandwich Theorem for Sequences
2
Èn
ˆ 1 2n2 ‰ œ lim ˆ #"n 1‰ œ 1
3. converges to 1, since n lim
a œ n lim
Ä_ n
Ä_
nÄ_
n
4. converges to 1, since n lim
a œ n lim
c1 (0.9)n d œ 1 0 œ 1
Ä_ n
Ä_
5. diverges, since ˜sin
n1 ™
#
œ e0ß 1ß 0ß 1ß 0ß 1ß á f
6. converges to 0, since {sin n1} œ {0ß 0ß 0ß á }
7. converges to 0, since n lim
a œ n lim
Ä_ n
Ä_
ln n#
n
8. converges to 0, since n lim
a œ n lim
Ä_ n
Ä_
ln (2n")
n
œ 2 n lim
Ä_
Š "n ‹
1
Š 2n 2b 1 ‹
œ n lim
Ä_
1
ˆ n nln n ‰ œ lim
9. converges to 1, since n lim
a œ n lim
Ä_ n
Ä_
nÄ_
10. converges to 0, since n lim
a œ n lim
Ä_ n
Ä_
ln a2n$ 1b
n
œ0
1Š "n ‹
œ n lim
Ä_
1
Š
"
e
ˆ1 "n ‰cn œ lim
, since n lim
a œ n lim
Ä_ n
Ä_
nÄ_
œ1
6n#
‹
2n$ 1
1
ˆ n n 5 ‰n œ lim Š1
11. converges to ec5 , since n lim
a œ n lim
Ä_ n
Ä_
nÄ_
12. converges to
œ0
œ n lim
Ä_
n
(5)
n ‹
"
ˆ1 "n ‰n
œ
12n
6n#
œ n lim
Ä_
œ0
œ ec5 by Theorem 5
"
e
by Theorem 5
ˆ 3 ‰1În œ lim
13. converges to 3, since n lim
a œ n lim
Ä_ n
Ä_ n
nÄ_
3
n1În
œ
3
1
œ 3 by Theorem 5
ˆ 3 ‰1În œ lim
14. converges to 1, since n lim
a œ n lim
Ä_ n
Ä_ n
nÄ_
31În
n1În
œ
1
1
œ 1 by Theorem 5
n
2
n
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 10 Practice Exercises
21În 1
Š "n ‹
15. converges to ln 2, since n lim
a œ n lim
n a21În 1b œ n lim
Ä_ n
Ä_
Ä_
œ n lim
Ä_
–
Š 21În ln 2‹
n#
—
635
œ n lim
21În ln 2
Ä_
Š n#" ‹
œ 2! † ln 2 œ ln 2
2
n
È
16. converges to 1, since n lim
a œ n lim
2n 1 œ n lim
exp Š ln (2nn 1) ‹ œ n lim
exp Œ 2n1b 1 œ e! œ 1
Ä_ n
Ä_
Ä_
Ä_
17. diverges, since n lim
a œ n lim
Ä_ n
Ä_
(n 1)!
n!
œ n lim
(n 1) œ _
Ä_
18. converges to 0, since n lim
a œ n lim
Ä_ n
Ä_
(4)n
n!
Š "# ‹
Š "# ‹
19.
"
(2n 3)(2n 1)
œ
#n 3
Š "# ‹
Ê sn œ –
2n 1
Š "# ‹
"
Ê n lim
s œ n lim
Ä_ n
Ä _ –6
20.
2
n(n 1)
œ
2
n
2
n1
ˆ1
œ n lim
Ä_
21.
22.
9
(3n 1)(3n 2)
œ 3# 3n3#
œ
_
_
n œ0
nœ0
_
n œ1
ˆ 34 ‰
1ˆ c4" ‰
3
4n
"
en
3
3n 2
_
n œ1
—–
Š "# ‹
5
Š "# ‹
Ê sn œ ˆ #3 53 ‰ ˆ 53 83 ‰ ˆ 83
3 ‰
3n 2
œ
7
Š "‹
#
— á – #n 3
Š "# ‹
2n 1 —
œ
Š "# ‹
3
Š "# ‹
2n 1
2 ‰
n1
œ #2
2
n1
Ê n lim
s
Ä_ n
Ê sn œ ˆ 92
ˆ
œ n lim
Ä_
3 ‰
11
á ˆ 3n 3 1
3 ‰
3n 2
3
#
2 ‰
2 ‰
ˆ 2
13 13 17
2
2 ‰
2
9 4n1 œ 9
, a convergent geometric series with r œ
"
e
ˆ 172
2 ‰
21
á ˆ 4n2 3
and a œ 1 Ê the sum is
_
‰n a convergent geometric series with r œ 4" and a œ
œ ! ˆ 43 ‰ ˆ "
4
n œ0
"
1 Š "e ‹
3
4
œ
2 ‰
4n 1
e
e1
Ê the sum is
œ 35
25. diverges, a p-series with p œ
26. !
5
"
6
ˆ3
Ê n lim
s œ n lim
Ä_ n
Ä_ #
23. ! en œ !
Š "# ‹
œ 1
8
2
2
(4n 3)(4n 1) œ 4n 3 4n 1
œ 29 4n21 Ê n lim
s
Ä_ n
24. ! (1)n
œ
Ê sn œ ˆ #2 32 ‰ ˆ 32 42 ‰ á ˆ n2
2 ‰
n1
3
3n 1
2n 1 —
3
œ 0 by Theorem 5
5
n
_
œ 5 !
nœ1
"
x"Î#
27. Since f(x) œ
_
"
n,
"
#
diverges since it is a nonzero multiple of the divergent harmonic series
Ê f w (x) œ #x"$Î# 0 Ê f(x) is decreasing Ê an1 an , and
_
lim a œ lim
nÄ_ n nÄ_
1
Èn
œ 0, the
)
! È" diverges, the given series converges conditionally.
series ! ("
Èn converges by the Alternating Series Test. Since
n
n
nœ1
nœ1
28. converges absolutely by the Direct Comparison Test since
"
#n$
"
n$
for n
1, which is the nth term of a convergent
p-series
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
636
Chapter 10 Infinite Sequences and Series
29. The given series does not converge absolutely by the Direct Comparison Test since
the nth term of a divergent series. Since f(x) œ
Ê an1 an , and n lim
a œ
Ä_ n
"
lim
n Ä _ ln (n 1)
"
ln (x 1)
"
ln (n 1)
"
(ln (x 1))# (x 1)
w
Ê f (x) œ
"
n1
, which is
0 Ê f(x) is decreasing
œ 0, the given series converges conditionally by the Alternating
Series Test.
30.
'2_ x(ln" x)
#
dx œ lim
bÄ_
'2b
"
x(ln x)#
b
dx œ lim c(ln x)" d 2 œ lim ˆ ln"b
bÄ_
bÄ_
" ‰
ln 2
œ
"
ln #
Ê the series
converges absolutely by the Integral Test
31. converges absolutely by the Direct Comparison Test since
ln n
n$
n
n$
"
n#
œ
, the nth term of a convergent p-series
n
n
32. diverges by the Direct Comparison Test for en n Ê ln ˆen ‰ ln n Ê nn ln n Ê ln nn ln (ln n)
Ê n ln n ln (ln n) Ê
33. n lim
Ä_
Š
"
n
È n#
Š n"# ‹
34. Since f(x) œ
1
‹
ln n
ln (ln n)
œ Én lim
Ä_
3x#
x$ 1
n#
n# 1
Ê f w (x) œ
"
n
, the nth term of the divergent harmonic series
œ È1 œ 1 Ê converges absolutely by the Limit Comparison Test
3x a2 x$ b
ax$ 1b#
0 when x
2 Ê an1 an for n
2 and n lim
Ä_
3n#
n$ 1
œ 0, the
series converges by the Alternating Series Test. The series does not converge absolutely: By the Limit
#
Comparison Test, n lim
Ä_
Š n$3n 1 ‹
ˆ n" ‰
œ n lim
Ä_
3n$
n$ 1
œ 3. Therefore the convergence is conditional.
35. converges absolutely by the Ratio Test since n lim
’ n 2 †
Ä _ (n 1)!
36. diverges since n lim
a œ n lim
Ä_ n
Ä_
(")n an# 1b
2n# n 1
n!
n1“
œ n lim
Ä_
n2
(n 1)#
does not exist
nb1
37. converges absolutely by the Ratio Test since n lim
†
’ 3
Ä _ (n 1)!
n!
3n “
œ n lim
Ä_
3
n1
œ01
n 2 3
n
È
É
38. converges absolutely by the Root Test since n lim
an œ n lim
nn œ n lim
Ä_
Ä_
Ä_
n n
39. converges absolutely by the Limit Comparison Test since n lim
Ä_
40. converges absolutely by the Limit Comparison Test since n lim
Ä_
nb1
41. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (x 4) †
Ä_
Ä _ (n 1)3nb1
œ01
n3n
(x 4)n ¹
1 Ê
"
Š $Î#
‹
n
Š Èn(n "1)(n
Š n"# ‹
Š È "#
n n
kx 4 k
lim
3
nÄ_
1
‹
‹
2)
ˆ n n 1 ‰ 1 Ê
n œ1
n
3
n3n
n(n 1)(n 2)
n$
n # an # 1 b
n%
œ Én lim
Ä_
_
harmonic series, which converges conditionally; at x œ 1 we have!
œ01
œ Én lim
Ä_
Ê kx 4k 3 Ê 3 x 4 3 Ê 7 x 1; at x œ 7 we have !
_
6
n
n œ1
_
œ!
n œ1
kx 4 k
3
(1)n 3n
n3n
"
n
œ1
œ1
1
_
œ ! ("n ) , the alternating
n
n œ1
, the divergent harmonic series
(a) the radius is 3; the interval of convergence is 7 Ÿ x 1
(b) the interval of absolute convergence is 7 x 1
(c) the series converges conditionally at x œ 7
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 10 Practice Exercises
42. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (x1) † (2n1)! ¹ 1 Ê (x 1)# n lim
Ä_
Ä _ (2n1)! (x1)2nc2
Ä_
(a) the radius is _; the series converges for all x
(b) the series converges absolutely for all x
(c) there are no values for which the series converges conditionally
2n
nb1
43. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (3x(n1)1)# †
Ä_
Ä_
n#
(3x 1)n ¹
1 Ê k3x 1k n lim
Ä_
Ê 1 3x 1 1 Ê 0 3x 2 Ê 0 x
_
"
n#
œ !
n œ1
_
have !
n œ1
"
(#n)(2n1)
2
3
n#
(n 1)#
_
œ 0 1, which holds for all x
1 Ê k3x 1k 1
nc1
_
2nc1
; at x œ 0 we have ! (1) n# (1) œ ! ("n)#
n
n œ1
n œ1
, a nonzero constant multiple of a convergent p-series, which is absolutely convergent; at x œ
_
(1)n 1 (1)n
n#
(a) the radius is
œ ! ("n)#
n 1
2
3
we
, which converges absolutely
nœ1
"
3
; the interval of convergence is 0 Ÿ x Ÿ
(b) the interval of absolute convergence is 0 Ÿ x Ÿ
2
3
2
3
(c) there are no values for which the series converges conditionally
44. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ n2 †
Ä_
Ä _ 2n 3
Ê
_
!
n œ1
k2x 1k
#
n1
2n 1
†
(2x 1)nb1
2nb1
†
2n 1
n1
†
2n
(2x 1)n ¹
1 Ê
k2x 1k
lim
2
nÄ_
n2
¸ 2n
3 †
2n " ¸
n1
(1) 1 Ê k2x 1k 2 Ê 2 2x 1 2 Ê 3 2x 1 Ê #3 x
(2)n
#n
_
œ!
lim ˆ n 1 ‰ œ
n Ä _ 2n 1
n œ1
"
#
(")n (n1)
2n 1
"
#
1
; at x œ #3 we have
which diverges by the nth-Term Test for Divergence since
Á 0; at x œ
"
#
_
n1
we have ! 2n
1 †
n œ1
2n
#n
_
n"
œ ! 2n
1 , which diverges by the nth-Term Test
n œ1
(a) the radius is 1; the interval of convergence is 3# x
(b) the interval of absolute convergence is 3# x
"
#
"
#
(c) there are no values for which the series converges conditionally
nb1
45. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹ x
Ä_
Ä _ (n 1)nb1
Ê
kx k
e
nn
xn ¹
¸ˆ n ‰n ˆ n " 1 ‰¸ 1 Ê
1 Ê kxk n lim
Ä _ n1
kx k
e n lim
Ä_
ˆ n " 1 ‰ 1
† 0 1, which holds for all x
(a) the radius is _; the series converges for all x
(b) the series converges absolutely for all x
(c) there are no values for which the series converges conditionally
nb1
46. n lim
†
¹ uunbn 1 ¹ 1 Ê n lim
¹ x
Ä_
Ä _ Èn 1
_
Èn
xn ¹
n
1 Ê kxk n lim
1 Ê kxk 1; when x œ 1 we have
Ä _ Én1
_
! (È1) , which converges by the Alternating Series Test; when x œ 1 we have !
n
n
nœ1
n œ1
"
Èn
, a divergent p-series
(a) the radius is 1; the interval of convergence is 1 Ÿ x 1
(b) the interval of absolute convergence is 1 x 1
(c) the series converges conditionally at x œ 1
2nb1
47. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (n 32)x
nb1
Ä_
Ä_
_
_
n œ1
n œ1
the series ! nÈ31 and !
n1
È3
†
3n
(n 1)x2n
1
¹1 Ê
x#
3 n lim
Ä_
2‰
È
È3;
ˆ nn
1 1 Ê 3 x
, obtained with x œ „ È3, both diverge
(a) the radius is È3; the interval of convergence is È3 x È3
(b) the interval of absolute convergence is È3 x È3
(c) there are no values for which the series converges conditionally
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
637
638
Chapter 10 Infinite Sequences and Series
2nb3
48. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ (x 2n1)x
3
Ä_
Ä_
2n 1
(x 1)2nb1 ¹
†
ˆ 2n 1 ‰ 1 Ê (x 1)# (1) 1
1 Ê (x 1)# n lim
Ä _ 2n 3
_
Ê (x 1)# 1 Ê kx 1k 1 Ê 1 x 1 1 Ê 0 x 2; at x œ 0 we have ! (1)#n(1)1
n
2nb1
n œ1
_
œ!
n œ1
_
(1)3nb1
2n 1
that !
n œ1
"
2n 1
_
œ!
n œ1
(1)nc1
2n 1
which converges conditionally by the Alternating Series Test and the fact
_
(1)n (1)2nb1
2n 1
diverges; at x œ 2 we have !
nœ1
_
œ!
nœ1
(1)n
2n 1
, which also converges conditionally
(a) the radius is 1; the interval of convergence is 0 Ÿ x Ÿ 2
(b) the interval of absolute convergence is 0 x 2
(c) the series converges conditionally at x œ 0 and x œ 2
(n 1)x
49. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹ cschcsch
(n)xn
Ä_
Ä_
c"
Ê kxk n lim
¹ e1 ee 2n
Ä_
2n 1
2
¹1 Ê
nb1
¹ 1 Ê kxk n lim
Ä_ »
Š enb1 c2ecnc1 ‹
ˆ en c2ecn ‰
»1
_
kx k
e
1 Ê e x e; the series ! a „ ebn csch n, obtained with x œ „ e,
n œ1
both diverge since n lim
a „ e) csch n Á 0
Ä_
n
(a) the radius is e; the interval of convergence is e x e
(b) the interval of absolute convergence is e x e
(c) there are no values for which the series converges conditionally
50. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹x
Ä_
Ä_
nb1
coth (n 1)
xn coth (n) ¹
c2nc2
1 Ê kxk n lim
†
¹1e
Ä _ 1 ec2nc2
1 ec2n
1 ec2n ¹
1 Ê kxk 1
_
Ê 1 x 1; the series ! a „ 1bn coth n, obtained with x œ „ 1, both diverge since n lim
a „ 1bn coth n Á 0
Ä_
n œ1
(a) the radius is 1; the interval of convergence is 1 x 1
(b) the interval of absolute convergence is 1 x 1
(c) there are no values for which the series converges conditionally
51. The given series has the form 1 x x# x$ á (x)n á œ
52. The given series has the form x
ln ˆ 53 ‰ ¸ 0.510825624
x#
#
x$
3
á (1)n1
53. The given series has the form x
x$
3!
x&
5!
á (1)n
x2n 1
(2n 1)!
54. The given series has the form 1
x#
2!
x%
4!
á (1)n
x2n
(2n)!
55. The given series has the form 1 x
56. The given series has the form x
tan" Š È"3 ‹ œ
57. Consider
"
1 2x
x$
3
x#
2!
x&
5
x#
3!
á
xn
n!
á (1)n
xn
n
"
1x
, where x œ
"
4
; the sum is
á œ ln (1 x), where x œ
2
3
n œ0
nœ0
1
3
á œ ex , where x œ ln 2; the sum is eln Ð2Ñ œ 2
á œ tan" x, where x œ
as the sum of a convergent geometric series with a œ 1 and r œ 2x Ê
_
4
5
á œ sin x, where x œ 1; the sum is sin 1 œ 0
á œ cos x, where x œ 13 ; the sum is cos
x2n 1
(2n 1)
œ
; the sum is
"
È3
1
6
_
"
1 ˆ "4 ‰
œ 1 (2x) (2x)# (2x)$ á œ ! (2x)n œ ! 2n xn where k2xk 1 Ê kxk
"
1 2x
"
#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
; the sum is
œ
"
#
Chapter 10 Practice Exercises
"
1 x$
58. Consider
"
1 x$
as the sum of a convergent geometric series with a œ 1 and r œ x$ Ê
œ
639
"
1 ax$ b
_
#
$
œ 1 ax$ b ax$ b ax$ b á œ ! (1)n x3n where kx$ k 1 Ê kx$ k 1 Ê kxk 1
n œ0
_
59. sin x œ !
nœ0
_
60. sin x œ !
nœ0
_
61. cos x œ !
n œ0
_
62. cos x œ !
n œ0
_
63. ex œ !
n œ0
_
64. ex œ !
n œ0
_
(1)n x2nb1
(2n 1)!
nœ0
Ê sin
_
(1)n x2n
(2n)!
3
Ê cosŠ Èx 5 ‹ œ !
_ ˆ 1 x ‰n
Ê eÐ1xÎ2Ñ œ !
xn
n!
#
Ê ex œ !
#
n!
n œ0
_
a x # b
n!
nœ0
œ
3
8
69.
"
x1 œ
œ 4"# ,
n œ0
1 n xn
#n n!
(1)n x2n
n!
Ê f w (x) œ x a3 x# b
9
32
2†1!
"
1x
"
#% †4
¸
"
#
"
9 †2 $
a3 x# b
2 †2!
"
#
œ
"Î#
3
8
,
2 †3!
(x 1)" Ê f w (x) œ (x 1)# Ê f ww (x) œ 2(x 1)$ Ê f www (x) œ 6(x 1)% ; f(3) œ
ww
f (3) œ
Ê
x""
11†3!
tanc" x
x
$Î#
œ 1 (x 2) (x 2)# (x 2)$ á
2
4$
6
4%
www
, f (2) œ
"
x
"
a
œ
"
a#
"
x 1
Ê
"
a$
(x a)
"
#( †7†2!
"
2"! †10†3!
"
"
4
œ
(x a)#
x"(
17†5!
x#$
23†7!
dx œ '1 Š1
1Î2
"
5# †#&
"
7# †#(
x#
3
*
2"$ †13†4!
*
'11Î2
Ê f ww (x) œ x# a3 x# b
$Î#
'01 x sin ax$ b dx œ '01 x Šx$ x3! x5!
&
"Î#
3x a3 x# b
; f(1) œ 2, f w (1) œ "# , f ww (1) œ 8"
#
$
Ê È3 x# œ 2 (x 1) 3(x$ 1) 9(x& 1) á
'
"
#
(1)n x6n
5n (#n)!
"
4#
(x 3)
"
4$
"
4%
#
(x 3)
œ x" Ê f w (x) œ x# Ê f ww (x) œ 2x$ Ê f www (x) œ 6x% ; f(a) œ
6
a%
œ ’ x5
71.
_
œ!
nœ0
'01Î2 exp ax$ b dx œ '01Î2 Š1 x$ x2! x3! x4!
¸
70.
_
œ!
n œ0
n
_
œ!
(2n)!
(1)n x10nÎ3
(#n)!
œ (1 x)" Ê f w (x) œ (1 x)# Ê f ww (x) œ 2(1 x)$ Ê f www (x) œ 6(1 x)% ; f(2) œ 1, f w (2) œ 1,
67. f(x) œ
f www (a) œ
nœ0
5
&Î#
f ww (2) œ 2, f www (2) œ 6 Ê
"
x
œ!
_ (1)n Š x3 ‹
È
nœ0
(1)n 22nb1 x2nb1
32nb1 (#n 1)!
_
2n
2n
3
f www (1) œ 32
68. f(x) œ
nœ0
(1)n ˆx5Î3 ‰
(2n)!
nœ0
Ê f www (x) œ 3x$ a3 x# b
f (3)
nœ0
Ê cos ˆx5Î3 ‰ œ !
"Î#
w
_
œ!
(2n 1)!
(1)n x2n
(2n)!
xn
n!
"
1x
nœ0
_ (1)n Š 2x ‹
3
œ!
2x
3
(1)n 12nb1 x2nb1
(#n 1)!
œ!
2nb1
(1)n x2nb1
(2n 1)!
65. f(x) œ È3 x# œ a3 x# b
66. f(x) œ
_
(1)n (1x)2nb1
(2n 1)!
Ê sin 1x œ !
"&
x#*
29†9!
x%
5
"
9# †2*
"
x#"
7!
x#(
9!
x%
4
, f w (a) œ a"# , f ww (a) œ
x(
7†2!
x"!
10†3!
x"!
3!
x"'
5!
x##
7!
x&
25
x(
49
x*
81
"
21# †##"
x"$
13†4!
á“
"Î#
!
¸ 0.484917143
á ‹ dx œ '0 Šx%
1
x#)
9!
á ‹ dx
"
!
x'
7
"
11# †2""
x)
9
x"!
11
"
13# †2"$
á ‹ dx œ ’x
"
15# †2"&
"
17# †#"(
x$
9
"
19# †#"*
2
a$
,
(x 3) á
(x a)$ á
á ‹ dx œ ’x
2"' †16†5!
"
a
,
$
á “ ¸ 0.185330149
"#
"
a%
"
4
x""
121
á“
"Î#
!
¸ 0.4872223583
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
640
72.
Chapter 10 Infinite Sequences and Series
'01Î64
dx œ '0
1Î64
tan " x
Èx
œ 23 x$Î#
2
#1
x(Î#
"
Èx
2
55
x$
3
Šx
x""Î#
2
105
$
7 sin x
2x
x Ä 0 e 1
73. lim
74.
x Ä 0 Š2x
œ lim
5!
#
á ‹ dx œ '0
1Î64
"Î'%
&
2$ x $
3!
œ ˆ 3†28$
#
á‹
x Ä 0 Š2
2# x
#!
á‹
75.
œ lim
tÄ0
76. lim
"‰
t#
œ
#
"
2 Š 4!
t6! á‹
#
Š1 2t4! á‹
Š sinh h ‹ cos h
#
h#
Π#!
$
œ lim
h%
5!
#
lim t # 2 2 cos t
t Ä 0 2t (1 cos t)
" cos# z
z Ä 0 ln (1 z) sin z
77. lim
2
105†8"&
á ‰ ¸ 0.0013020379
œ
7
#
$
&
$
&
2 Š )3! )5! á‹
œ lim
$
&
Š )3! )5! á‹
)Ä0
œ lim
h%
4!
%
t
á
t# 2 2 Œ1 t# 4!
t Ä 0 2t# Š1 1
t#
#
t%
4!
á‹
%
t'
6!
2 Πt4!
œ lim
tÄ0
Št%
2t'
4!
á
á‹
"
1#
œ
Œ1
h#
3!
%
#
%
h5! á Œ1 h#! h4! á
h#
h'
6!
h'
7!
á
h#
hÄ0
2
55†8""
œ2
hÄ0
h#
3!
á‹
) Š) )3! )5! á‹
œ lim
h#
hÄ0
2
21†8(
%
2$ x #
3!
#
lim ˆ "
t Ä 0 # 2 cos t
ˆx"Î# "3 x&Î# "5 x*Î# 7" x"$Î# á ‰ dx
7 Š1 x3! x5! á‹
œ lim
$
)Ä0
Š 3!" )5! á‹
)Ä0
2# x #
2!
x(
7
Š1 ) )#! )3! á‹ Š1 ) )#! )3! á‹ 2)
œ lim
)#
x"&Î# á ‘ !
#
)
c)
lim e )e sin)2)
)Ä0
x&
5
7 Šx x3! x5! á‹
œ lim
2 Š 3!"
œ lim
hÄ0
1 Š1 z #
œ lim
#
z%
3
Š #"!
"
3!
h#
5!
$
h#
4!
h%
6!
h%
7!
á‹ œ
"
3
%
á‹
$
&
z Ä 0 Šz z# z3 á‹ Šz z3! z5! á‹
Šz# z3 á‹
œ lim
#
$
%
z Ä 0 Š z# 2z3 z4 á‹
#
Š1 z3 á‹
œ lim
#
z
z Ä 0 Š "# 2z
3 4 á‹
y#
cos
y
cosh y
yÄ0
78. lim
"
œ lim
œ 2
y#
œ lim
y Ä 0 Œ1
y Ä 0 Œ1 2y% á
6!
y#
#
y%
4!
y'
6!
á Œ1
xÄ0
Ê
r
x#
3
x#
r
x#
y%
4!
y'
6!
á
y#
œ lim
y Ä 0 Œ
2y#
#
'
2y6! á
œ 1
$
79. lim ˆ sinx$3x
y#
#!
s‰ œ lim –
xÄ0
œ 0 and s
9
#
80. The approximation sin x ¸
&
(3x)
Š3x (3x)
6 120 á‹
x$
œ 0 Ê r œ 3 and s œ
6x
6 x#
r
x#
s— œ lim Š x3#
xÄ0
9
#
81x#
40
á
9
#
is better than sin x ¸ x.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
r
x#
s‹ œ 0
Chapter 10 Practice Exercises
nb1
(3n 1)(3n 2)x
â(2n)
¸ 3n 2 ¸ 1 Ê kxk
81. n lim
† #†52†8†4â†6(3n
¹ 2†5†8#â
†4†6â(2n)(2n 2)
1)xn ¹ 1 Ê kxk n lim
Ä_
Ä _ 2n 2
Ê the radius of convergence is 23
2
3
nb1
(2n1)(2n3)(x1)
¸ 2n 3 ¸ 1 Ê kxk
82. n lim
† 34††59††714ââ(2n(5n1)x1)n ¹ 1 Ê kxk n lim
¹ 3†5†47†â
9†14â(5n1)(5n4)
Ä_
Ä _ 5n 4
Ê the radius of convergence is 52
n
"‰
k#
83. ! ln ˆ1
kœ2
n
n
kœ2
kœ2
5
2
œ ! ln ˆ1 "k ‰ ln ˆ1 "k ‰‘ œ ! cln (k 1) ln k ln (k 1) ln kd
œ cln 3 ln 2 ln 1 ln 2d cln 4 ln 3 ln 2 ln 3d cln 5 ln 4 ln 3 ln 4d cln 6 ln 5 ln 4 ln 5d
á cln (n 1) ln n ln (n 1) ln nd œ cln 1 ln 2d cln (n 1) ln nd
after cancellation
n
"‰
k#
Ê ! ln ˆ1
k œ2
n
84. !
k œ2
"
k# 1 -
ˆ n " 1
_
Ê !
k œ2
"
#
œ
n
!ˆ
k œ2
" ‰‘
n1
"
k # 1
_
1 ‰
œ ln ˆ n2n
Ê ! ln ˆ1
"‰
k#
k œ2
"
k1
œ
"
#
" ‰
k1
ˆ 1"
"
#
"ˆ3
œ n lim
Ä_ # 2
œ
1
n
"
n
"
#
1‰
œ n lim
ln ˆ n 2n
œ ln
Ä_
" ‰
n1
1 ‰
n1
œ
œ
"
#
ˆ #3
"
n
" ‰
n1
œ
"
#
2(n 1) 2n
’ 3n(n 1)2n(n
“œ
1)
1†4†7â(3n 2)
(3n)!
n œ1
_
dy
dx
_
œ!
n œ1
_
1†4†7â(3n 2)
(3n 1)!
x3nc1
1†4†7â(3n5)
(3n3)!
x3nc2
(3n ")
(3n 1)(3n 2)(3n 3)
1†4†7â(3n 2)
(3n 2)!
x3nc2 œ x !
œ x Œ1 !
1†4†7â(3n 2)
(3n)!
x œ xy 0 Ê a œ 1 and b œ 0
x#
1x
œ x# x# (x) x# (x)# x# (x)$ á œ x# x$ x% x& á œ ! (1)n xn which
Ê
d# y
dx#
œ!
n œ1
_
n œ1
86. (a)
x3n Ê
3n# n 2
4n(n 1)
3
4
3nb3
_
is the sum
ˆ 11 3" ‰ ˆ #" 4" ‰ ˆ "3 5" ‰ ˆ 4" 6" ‰ á ˆ n " # n" ‰
1)x
$
85. (a) n lim
† 1†4†7â(3n)!
¹ 1†4†7â(3n(3n2)(3n
3)!
(3n 2)x3n ¹ 1 Ê kx k n lim
Ä_
Ä_
œ kx$ k † 0 1 Ê the radius of convergence is _
(b) y œ 1 !
"
#
œ
x#
1 (x)
nœ2
3n
_
n œ2
converges absolutely for kxk 1
_
_
n œ2
nœ2
(b) x œ 1 Ê ! (1)n xn œ ! (1)n which diverges
_
_
87. Yes, the series ! an bn converges as we now show. Since ! an converges it follows that an Ä 0 Ê an 1
n œ1
nœ1
_
_
n œ1
n œ1
for n some index N Ê an bn bn for n N Ê ! an bn converges by the Direct Comparison Test with ! bn
_
88. No, the series ! an bn might diverge (as it would if an and bn both equaled n) or it might converge (as it would if
n œ1
an and bn both equaled "n ).
_
_
n œ1
k œ1
!(xk1 xk ) œ lim (xn1 x" ) œ lim (xn1 ) x" Ê both the series and
89. ! (xn1 xn ) œ n lim
Ä_
nÄ_
nÄ_
sequence must either converge or diverge.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
641
642
Chapter 10 Infinite Sequences and Series
Š 1 banan ‹
90. It converges by the Limit Comparison Test since n lim
Ä_
an
"
1 a n
œ n lim
Ä_
_
œ 1 because ! an converges
n œ1
and so an Ä 0.
_
91. !
n œ1
œ a"
an
n
ˆ 9"
92. an œ
œ
"
10
"
ln n
"
ln #
a#
#
"
11
á
"
#
a%
4
"
3
a" ˆ #" ‰ a# ˆ "3 "4 ‰ a% ˆ "5
á
" ‰
16 a"'
2 Ê a#
for n
ˆ1
a$
3
"
#
á
a$
"
7
"8 ‰ a)
(a# a% a) a"' á ) which is a divergent series
á , and
a%
"
6
"
ln #
"
ln 4
_
á ‰ which diverges so that 1 !
n œ2
"
ln 8
"
n ln n
á œ
"
ln #
"
# ln 2
"
3 ln 2
á
diverges by the Integral Test.
CHAPTER 10 ADDITIONAL AND ADVANCED EXERCISES
1. converges since
"
(3n #)Ð2n 1ÑÎ2
"
Š $Î#
‹
n
lim
nÄ_
Š
"
(3n
‹
2)$Î#
"
(3n 2)$Î#
$
1$
192 ‹
œ
n œ1
"
(3n 2)$Î#
converges by the Limit Comparison Test:
ˆ 3n n 2 ‰$Î# œ 3$Î#
œ n lim
Ä_
2. converges by the Integral Test:
1
œ Š 24
_
and !
'1_ atan" xb# x dx1 œ
#
" x b$
lim ’ atan 3
bÄ_
" bb$
b
tan
“ œ lim ’ a 3
bÄ_
"
1$
192 “
71 $
192
c2n
e
3. diverges by the nth-Term Test since n lim
a œ n lim
(1)n tanh n œ lim (1)n Š 11
(1)n
ec2n ‹ œ n lim
Ä_ n
Ä_
Ä_
bÄ_
does not exist
4. converges by the Direct Comparison Test: n! nn Ê ln (n!) n ln (n) Ê
Ê logn (n!) n Ê
logn (n!)
n$
"
n#
12
(3)(5)(4)#
, a% œ ˆ 35††64 ‰ ˆ 24††53 ‰ ˆ 13††24 ‰ œ
given series and
12
(n 1)(n 3)(n 2)#
6. converges by the Ratio Test: n lim
Ä_
"2
(4)(6)(5)#
12
n%
12
(1)(3)(2)#
, a# œ
_
,á Ê 1!
n œ1
1†2
3†4
_
n œ1
"
32nb1
cos x œ
"
#
È3
#
1
3
12
(2)(4)(3)#
, a$ œ ˆ 42††53 ‰ ˆ 31††42 ‰
represents the
, which is the nth-term of a convergent p-series
anb1
an
œ n lim
Ä_
n
(n 1)(n 1)
_
and !
n œ1
2n
32n
œ01
È3
1#
ˆx
n
n n
È
2È
È3
ww
# ,f
1 ‰$
á
3
Ê f ˆ 13 ‰ œ 0.5, f w ˆ 13 ‰ œ
ˆx 13 ‰ 4" ˆx 13 ‰#
"
1L
Ê L# L 1 œ 0 Ê L œ
; the first subseries is a convergent geometric series and the
n 2n
É
second converges by the Root Test: n lim
32n œ n lim
Ä_
Ä_
9. f(x) œ cos x with a œ
œ
12
(n 1)(n 3)(n 2)#
7. diverges by the nth-Term Test since if an Ä L as n Ä _, then L œ
8. Split the given series into !
n
, which is the nth-term of a convergent p-series
5. converges by the Direct Comparison Test: a" œ 1 œ
œ
ln (n!)
ln (n)
9
œ
"†1
9
œ
"
9
1
ˆ 13 ‰ œ 0.5, f www ˆ 13 ‰ œ
È3
#
, f Ð4Ñ ˆ 13 ‰ œ 0.5;
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1 „ È5
#
Á0
Chapter 10 Additional and Advanced Exercises
643
10. f(x) œ sin x with a œ 21 Ê f(21) œ 0, f w (21) œ 1, f ww (21) œ 0, f www (21) œ 1, f Ð4Ñ (21) œ 0, f Ð5Ñ (21) œ 1,
f Ð6Ñ (21) œ 0, f Ð7Ñ (21) œ 1; sin x œ (x 21)
11. ex œ 1 x
x#
#!
x$
3!
(x 21)$
3!
(x 21)&
5!
(x 21)(
7!
á
á with a œ 0
12. f(x) œ ln x with a œ 1 Ê f(1) œ 0, f w (1) œ 1, f ww (1) œ 1,f www (1) œ 2, f Ð4Ñ (1) œ 6;
(x 1)#
#
ln x œ (x 1)
(x 1)$
3
(x 1)%
4
á
13. f(x) œ cos x with a œ 221 Ê f(221) œ 1, f w (221) œ 0, f ww (221) œ 1, f www (221) œ 0, f Ð4Ñ (221) œ ",
f Ð5Ñ (221) œ 0, f Ð6Ñ (221) œ 1; cos x œ 1 "# (x 221)# 4!" (x 221)% 6!" (x 221)' á
14. f(x) œ tan" x with a œ 1 Ê f(1) œ
tan
"
1
4
xœ
(x 1)
2
(x 1)#
4
1
4
(x 1)$
12
, f w (1) œ
"
#
, f ww (1) œ "# , f www (1) œ
ˆ ba ‰n ln ˆ ba ‰
a n
nÄ_ ˆ b ‰ 1
16. 1
2
10
3
10#
_
œ1!
n œ0
œ1
200
999
n 1
17. sn œ !
k œ0
7
10$
'kkb1
2
10%
_
2
103n1
!
30
999
dx
1 x#
7
999
3
10&
7
10'
_
3
103n2
n œ0
0†ln ˆ ba ‰
01
œ ln b
!
999237
999
œ
Ê sn œ '0
1
_
á œ1!
n œ1
7
œ
1 În
n
lnˆˆ ba ‰ 1‰
n
nÄ_
Ê n lim
c œ ln b lim
Ä_ n
œ ln b since 0 a b. Thus, n lim
c œ eln b œ b.
Ä_ n
103n3
n œ0
;
á
n
15. Yes, the sequence converges: cn œ aan bn b1În Ê cn œ b ˆˆ ba ‰ 1‰
œ ln b lim
"
#
œ1
2 ‰
ˆ 10
2
103n
$
1ˆ " ‰
10
_
2
!
n œ1
Š 103# ‹
$
1ˆ " ‰
3
103n
_
1
!
n œ1
Š 107$ ‹
"‰
1 ˆ 10
10
7
103n
$
412
333
dx
1 x#
'1
2
dx
1 x#
Ê n lim
s œ n lim
atan" n tan" 0b œ
Ä_ n
Ä_
nb1
á 'nc1 1 dxx# Ê sn œ '0
n
n
dx
1 x#
1
#
#
1)
18. n lim
† (n 1)(2x
¹ uunbn 1 ¹ œ n lim
¹ (n 1)x
¹ œ n lim
¹ x † (n 1) ¹ œ ¸ 2x x 1 ¸ 1
nxn
Ä_
Ä _ (n 2)(2x 1)n 1
Ä _ 2x 1 n(n 2)
Ê kxk k2x 1k ; if x 0, kxk k2x 1k Ê x 2x 1 Ê x 1; if "# x 0, kxk k2x 1k
n
Ê x 2x 1 Ê 3x 1 Ê x "3 ; if x #" , kxk k2x 1k Ê x 2x 1 Ê x 1. Therefore,
the series converges absolutely for x 1 and x "3 .
19. (a) No, the limit does not appear to depend on the value of the constant a
(b) Yes, the limit depends on the value of b
(c) s œ Š1
œ n lim
Ä_
cos ˆ na ‰ n
n ‹
a
n
lim Š1
nÄ_
Ê ln s œ
sin ˆ na ‰ cos ˆ na ‰
1
cos ˆ na ‰
n
cos ˆ na ‰ n
bn ‹
œ
ln Œ1
cos ˆ na ‰
n
ˆ n" ‰
01
10
Ê n lim
ln s œ
Ä_
"
cos ˆ na ‰ Œ
n
Š
a
a
a
n sin ˆ n ‰ cos ˆ n ‰
n#
"‹
n#
œ e1Îb
1În
_
"
#
1
œ 1 Ê n lim
s œ e" ¸ 0.3678794412; similarly,
Ä_
an ‰n
20. ! an converges Ê n lim
a œ 0; n lim
’ˆ 1 sin
“
#
Ä_ n
Ä_
n œ1
œ
an ‰
ˆ 1sin
œ n lim
œ
#
Ä_
Ä_
1sin Šnlim an ‹
#
Ê the series converges by the nth-Root Test
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
œ
1sin 0
#
644
Chapter 10 Infinite Sequences and Series
nb1 nb1
21. n lim
¹ uunbn 1 ¹ 1 Ê n lim
¹b x †
Ä_
Ä _ ln (n 1)
ln n
bn xn ¹
1 Ê kbxk 1 Ê b" x
"
b
œ5 Ê bœ „
"
5
22. A polynomial has only a finite number of nonzero terms in its Taylor series, but the functions sin x, ln x and
ex have infinitely many nonzero terms in their Taylor expansions.
sin (ax) sin xx
x$
xÄ0
Šax
œ lim
23. lim
a$ x$
3!
á‹ Šx
xÄ0
xÄ0
24.
25. (a)
(b)
26.
un
unb1
un
unb1
œ
(n 1)#
n#
un
unb1
œ
n1
n
œ
2n(2n 1)
(2n 1)#
Ê Cœ
3
#
Œ1
œ 1 Ê lim
œ1
œ1
œ
a# x#
#
1
n
œ lim ’ a x# 2
4n# 2n
4n# 4n 1
á b
"
n#
n œ1
_
"
n
n œ1
n
5n#
4n# 4n 1
œ
Š4
5
4
n
_
_
_
n œ1
n œ1
nœ1
&
Š a5!
"
#
5! ‹ x
á“
a#
4
a# x#
48
á ‹ œ 1
converges
diverges
œ1
5
4n# 4n 1
"
3!
œ 76
xÄ0
_
Š 64 ‹
"
3!
a$
3!
œ 1 Ê lim Š "2x#b
Ê C œ 2 1 and !
œ1
1 and kf(n)k œ
œ 23!
Ê C œ 1 Ÿ 1 and !
0
n#
xÄ0
$
#x#
"
n#
2
n
a% x%
4!
xÄ0
Ê b œ 1 and a œ „ 2
á‹ x
sin 2x sin x x
x$
is finite if a 2 œ 0 Ê a œ 2; lim
lim cos#axx# b
xÄ0
x$
3!
x$
Š 3# ‹
n
5n#
– Š4n# c 4n b 1‹ —
n#
after long division
_
"
‹
n#
Ÿ 5 Ê ! un converges by Raabe's Test
n œ1
27. (a) ! an œ L Ê an# Ÿ an ! an œ an L Ê ! an# converges by the Direct Comparison Test
(b) converges by the Limit Comparison Test: n lim
Ä_
an
Š1c
an ‹
an
œ n lim
Ä_
"
1 an
_
œ 1 since ! an converges and
n œ1
therefore x lim
a œ0
Ä_ n
28. If 0 an 1 then kln (1 an )k œ ln (1 an ) œ an
a#n
#
an$
3
á an an# an$ á œ
an
1 an
,
a positive term of a convergent series, by the Limit Comparison Test and Exercise 27b
_
29. (1 x)" œ 1 ! xn where kxk 1 Ê
n œ1
#
"
(1x)#
$
4 œ 1 2 ˆ "# ‰ 3 ˆ "# ‰ 4 ˆ "# ‰ á n ˆ "# ‰
_
30. (a) ! xn1 œ
nœ1
_
Ê !
n œ1
_
(b) x œ !
n œ1
x#
1 x
n(n 1)
xn
n(n ")
xn
_
Ê ! (n 1)xn œ
nœ1
œ
2
x
$
Š1 x" ‹
Ê xœ
œ
2x#
(x 1)$
2x#
(x 1)$
2xx#
(1x)#
œ
n 1
d
dx
_
(1 x)" œ ! nxn1 and when x œ
nœ1
"
#
we have
á
_
Ê ! n(n 1)xn1 œ
nœ1
2
(1x)$
_
Ê ! n(n 1)xn œ
nœ1
, kxk 1
Ê x$ 3x# x 1 œ 0 Ê x œ 1 Š1
È57 "Î$
9 ‹
Š1
¸ 2.769292, using a CAS or calculator
31. (a)
"
(1x)#
œ
d
dx
ˆ 1" x ‰ œ
d
dx
(b) from part (a) we have
_
a1 x x# x$ á b œ 1 2x 3x# 4x$ á œ ! nxn1
_
! n ˆ 5 ‰n1
6
n œ1
2x
(1x)$
n œ1
ˆ "6 ‰ œ ˆ "6 ‰ ’
2
"
“
1 ˆ 56 ‰
œ6
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
È57 "Î$
9 ‹
Chapter 10 Additional and Advanced Exercises
_
(c) from part (a) we have ! npn1 q œ
n œ1
_
_
32. (a) ! pk œ ! 2k œ
k œ1
k œ1
ˆ "# ‰
1ˆ "# ‰
q
(1 p)#
œ
q
q#
"
q
œ
_
_
k œ1
kœ1
œ 1 and E(x) œ ! kpk œ ! k2k œ
"
#
_
! k21k œ ˆ " ‰
#
kœ1
"
1ˆ "# ‰‘#
œ2
by Exercise 31(a)
_
_
k œ1
k œ1
(b) ! pk œ !
œ ˆ "6 ‰
"
1ˆ 56 ‰‘#
_
_
(c) ! pk œ !
kœ1
kœ1
_
"
k1
œ!
k œ1
5kc1
6k
œ
_
5
e
"
k(k1)
k œ1
6
_
œ ! ˆ k"
kœ1
" ‰
k1
œ
lim ˆ1
kÄ_
C! e
kt!
" ‰
k1
ˆ1 e
1e kt!
kœ1
"
6
œ
_
_
kœ1
kœ1
nkt! ‰
Ê Rœ
lim R œ
nÄ_ n
¸ 0.58195028;
e " a1 e "! b
1e "
R R"!
0.0001
R
C! e kt!
1 e kt!
"
e1 ¸ 0.58197671; R R"! ¸ 0.00002643 Ê
Þ1n
e Þ1 ˆ1 e Þ1n ‰ R
Rn œ
, # œ #" ˆ eÞ1 " 1 ‰ ¸ 4.7541659; Rn R# Ê 1eÞ1 e 1 ˆ #" ‰ ˆ eÞ1 " 1 ‰
1 e Þ1
n
n
Ê 1 enÎ10 "# Ê enÎ10 "# Ê 10
ln ˆ "# ‰ Ê 10
ln ˆ "# ‰ Ê n 6.93
34. (a) R œ
(b) t! œ
C!
ekt! 1
"
0.05
_
! k ˆ 5 ‰k1
6
k œ1
œ 1 and E(x) œ ! kpk œ ! k Š k(k " 1) ‹
Ê R" œ e" ¸ 0.36787944 and R"! œ
Rœ
(c)
k œ1
5kc1
6k
, a divergent series so that E(x) does not exist
a1 e n b
1e "
1
_
œ6
33. (a) Rn œ C! ekt! C! e2kt! á C! enkt! œ
(b) Rn œ
_
! ˆ 5 ‰k œ ˆ " ‰ ’ ˆ 6 ‰5 “ œ 1 and E(x) œ ! kpk œ ! k
6
5
1ˆ ‰
"
5
Ê Rekt! œ R C! œ CH Ê ekt! œ
CH
CL
Ê t! œ
"
k
œ
C!
ekt! 1
Ê nœ7
ln Š CCHL ‹
ln e œ 20 hrs
(c) Give an initial dose that produces a concentration of 2 mg/ml followed every t! œ
"
0.0#
2 ‰
ln ˆ 0.5
¸ 69.31 hrs
by a dose that raises the concentration by 1.5 mg/ml
0.1 ‰
"
‰
(d) t! œ 0.2
ln ˆ 0.03
œ 5 ln ˆ 10
3 ¸ 6 hrs
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
645
646
Chapter 10 Infinite Sequences and Series
NOTES:
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
CHAPTER 11 PARAMETRIC EQUATIONS AND
POLAR COORDINATES
11.1 PARAMETRIZATIONS OF PLANE CURVES
1. x œ 3t, y œ 9t# , _ t _ Ê y œ x#
2. x œ Èt , y œ t, t
0 Ê x œ È y
#
or y œ x , x Ÿ 0
3. x œ 2t 5, y œ 4t 7, _ t _
Ê x 5 œ 2t Ê 2(x 5) œ 4t
Ê y œ 2(x 5) 7 Ê y œ 2x 3
5. x œ cos 2t, y œ sin 2t, 0 Ÿ t Ÿ 1
Ê cos# 2t sin# 2t œ 1 Ê x# y# œ 1
4. x œ 3 3t, y œ 2t, 0 Ÿ t Ÿ 1 Ê y# œ t
Ê x œ 3 3 ˆ y# ‰ Ê 2x œ 6 3y
Ê y œ 2 23 x, ! Ÿ x Ÿ $
6. x œ cos (1 t), y œ sin (1 t), 0 Ÿ t Ÿ 1
Ê cos# (1 t) sin# (1 t) œ 1
Ê x# y# œ 1, y !
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
648
Chapter 11 Parametric Equations and Polar Coordinates
7. x œ 4 cos t, y œ 2 sin t, 0 Ÿ t Ÿ 21
Ê
16 cos# t
16
4 sin# t
4
œ1 Ê
x#
16
9. x œ sin t, y œ cos 2t, 12 Ÿ t Ÿ
y#
4
8. x œ 4 sin t, y œ 5 cos t, 0 Ÿ t Ÿ 21
œ1
1
2
Ê y œ cos 2t œ 1 2sin# t Ê y œ 1 2x2
11. x œ t2 , y œ t6 2t4 , _ t _
2 3
2 2
Ê y œ at b 2at b Ê y œ x3 2x2
13. x œ t, y œ È1 t# , 1 Ÿ t Ÿ 0
Ê y œ È1 x#
Ê
16 sin# t
16
25 cos# t
25
œ1 Ê
x#
16
y#
#5
œ1
10. x œ 1 sin t, y œ cos t 2, 0 Ÿ t Ÿ 1
Ê sin# t cos# t œ 1 Ê ax 1b# ay 2b# œ 1
12. x œ
t
t1,
Ê tœ
yœ
x
x1
t2
t1,
1 t 1
Êyœ
2x
2x 1
14. x œ Èt 1, y œ Èt, t 0
Ê y# œ t Ê x œ Èy# 1, y
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
0
Section 11.1 Parametrizations of Plane Curves
15. x œ sec# t 1, y œ tan t, 1# t
Ê sec# t 1 œ tan# t Ê x œ y#
1
#
16. x œ sec t, y œ tan t, 1# t
649
1
#
#
Ê sec# t tan# t œ 1 Ê x# y œ 1
17. x œ cosh t, y œ sinh t, _ 1 _
Ê cosh# t sinh# t œ 1 Ê x# y# œ 1
18. x œ 2 sinh t, y œ 2 cosh t, _ t _
Ê 4 cosh# t 4 sinh# t œ 4 Ê y# x# œ 4
19. (a) x œ a cos t, y œ a sin t, 0 Ÿ t Ÿ 21
20. (a) x œ a sin t, y œ b cos t,
(b) x œ a cos t, y œ a sin t, 0 Ÿ t Ÿ 21
(c) x œ a cos t, y œ a sin t, 0 Ÿ t Ÿ 41
(d) x œ a cos t, y œ a sin t, 0 Ÿ t Ÿ 41
1
#
ŸtŸ
51
#
(b) x œ a cos t, y œ b sin t, 0 Ÿ t Ÿ 21
(c) x œ a sin t, y œ b cos t, 1# Ÿ t Ÿ 9#1
(d) x œ a cos t, y œ b sin t, 0 Ÿ t Ÿ 41
21. Using a"ß $b we create the parametric equations x œ " at and y œ $ bt, representing a line which goes
through a"ß $b at t œ !. We determine a and b so that the line goes through a%ß "b when t œ ".
Since % œ " a Ê a œ &. Since " œ $ b Ê b œ %. Therefore, one possible parameterization is x œ " &t,
y œ $ %t, 0 Ÿ t Ÿ ".
22. Using a"ß $b we create the parametric equations x œ " at and y œ $ bt, representing a line which goes through
a"ß $b at t œ !. We determine a and b so that the line goes through a$ß #b when t œ ". Since $ œ " a Ê a œ %.
Since # œ $ b Ê b œ &. Therefore, one possible parameterization is x œ " %t, y œ $ &t, 0 Ÿ t Ÿ ".
23. The lower half of the parabola is given by x œ y# " for y Ÿ !. Substituting t for y, we obtain one possible
parameterization x œ t# ", y œ t, t Ÿ 0Þ
24. The vertex of the parabola is at a"ß "b, so the left half of the parabola is given by y œ x# #x for x Ÿ ". Substituting
t for x, we obtain one possible parametrization: x œ t, y œ t# #t, t Ÿ ".
25. For simplicity, we assume that x and y are linear functions of t and that the pointax, yb starts at a#ß $b for t œ ! and passes
through a"ß "b at t œ ". Then x œ fatb, where fa!b œ # and fa"b œ ".
Since slope œ ??xt œ "#
"! œ $, x œ fatb œ $t # œ # $t. Also, y œ gatb, where ga!b œ $ and ga"b œ ".
Since slope œ
?y
?t
œ
"3
"!
œ 4. y œ gatb œ %t $ œ $ %t.
One possible parameterization is: x œ # $t, y œ $ %t, t
!.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
650
Chapter 11 Parametric Equations and Polar Coordinates
26. For simplicity, we assume that x and y are linear functions of t and that the pointax, yb starts at a"ß #b for t œ ! and
passes through a!ß !b at t œ ". Then x œ fatb, where fa!b œ " and fa"b œ !.
Since slope œ
Since slope œ
?x
?t
?y
?t
œ
œ
! a"b
"!
!#
"! œ
œ ", x œ fatb œ "t a"b œ " t. Also, y œ gatb, where ga!b œ # and ga"b œ !.
#. y œ gatb œ #t # œ # #t.
One possible parameterization is: x œ " t, y œ # #t, t
27. Since we only want the top half of a circle, y
!.
0, so let x œ 2cos t, y œ 2lsin tl, 0 Ÿ t Ÿ 41
28. Since we want x to stay between 3 and 3, let x œ 3 sin t, then y œ a3 sin tb2 œ 9sin# t, thus x œ 3 sin t, y œ 9sin# t,
0Ÿt_
29. x# y# œ a# Ê 2x 2y
y# t# y# œ a# Ê y œ
dy
dx œ 0
a
È1t# and
Ê
x
dy
dy
x
dx œ y ; let t œ dx Ê
œ È1att , _ t _
xy œ t Ê x œ yt. Substitution yields
30. In terms of ), parametric equations for the circle are x œ a cos ), y œ a sin ), 0 Ÿ ) 21. Since ) œ as , the arc
length parametrizations are: x œ a cos as , y œ a sin as , and 0 Ÿ
s
a
21 Ê 0 Ÿ s Ÿ 21a is the interval for s.
31. Drop a vertical line from the point ax, yb to the x-axis, then ) is an angle in a right triangle, and from trigonometry we
know that tan ) œ yx Ê y œ x tan ). The equation of the line through a0, 2b and a4, 0b is given by y œ 12 x 2. Thus
x tan ) œ 12 x 2 Ê x œ
4
2 tan ) 1
and y œ
4 tan )
2 tan ) 1
where 0 Ÿ ) 12 .
32. Drop a vertical line from the point ax, yb to the x-axis, then ) is an angle in a right triangle, and from trigonometry we
know that tan ) œ yx Ê y œ x tan ). Since y œ Èx Ê y2 œ x Ê ax tan )b2 œ x Ê x œ cot2 ) Ê y œ cot ) where
0 ) Ÿ 12 .
33. The equation of the circle is given by ax 2b2 y2 œ 1. Drop a vertical line from the point ax, yb on the circle to the
x-axis, then ) is an angle in a right triangle. So that we can start at a1, 0b and rotate in a clockwise direction, let
x œ 2 cos ), y œ sin ), 0 Ÿ ) Ÿ 21.
34. Drop a vertical line from the point ax, yb to the x-axis, then ) is an angle in a right triangle, whose height is y and whose
base is x 2. By trigonometry we have tan ) œ x y 2 Ê y œ ax 2b tan ). The equation of the circle is given by
x2 y2 œ 1 Ê x2 aax 2btan )b2 œ 1 Ê x2 sec2 ) 4x tan2 ) 4tan2 ) 1 œ 0. Solving for x we obtain
xœ
4tan2 ) „ Éa4tan2 )b2 4 sec2 ) a4tan2 ) 1b
2 sec2 )
œ
4tan2 ) „ 2È1 3tan2 )
2 sec2 )
œ 2sin2 ) „ cos )Ècos2 ) 3sin2 )
œ 2 2cos2 ) „ cos )È4cos2 ) 3 and y œ Š2 2cos2 ) „ cos )È4cos2 ) 3 2‹ tan )
œ 2sin ) cos ) „ sin )È4cos2 ) 3. Since we only need to go from a1, 0b to a0, 1b, let
x œ 2 2cos2 ) cos )È4cos2 ) 3, y œ 2sin ) cos ) sin )È4cos2 ) 3, 0 Ÿ ) Ÿ tan1 ˆ 1 ‰.
2
To obtain the upper limit for ), note that x œ 0 and y œ 1, using y œ ax 2b tan ) Ê 1 œ 2 tan ) Ê ) œ tan1 ˆ 12 ‰.
35. Extend the vertical line through A to the x-axis and let C be the point of intersection. Then OC œ AQ œ x
2
2
and tan t œ OC
œ x2 Ê x œ tan2 t œ 2 cot t; sin t œ OA
Ê OA œ sin2 t ; and (AB)(OA) œ (AQ)# Ê AB ˆ sin2 t ‰ œ x#
#
2
2
2
sin
t
Ê AB ˆ sin t ‰ œ ˆ tan t ‰ Ê AB œ tan# t . Next y œ 2 AB sin t Ê y œ 2 ˆ 2tansin# tt ‰ sin t œ
2
2 sin# t
tan# t
œ 2 2 cos# t œ 2 sin# t. Therefore let x œ 2 cot t and y œ 2 sin# t, 0 t 1.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.1 Parametrizations of Plane Curves
36. Arc PF œ Arc AF since each is the distance rolled and
Arc PF
œ nFCP Ê Arc PF œ b(nFCP); ArcaAF œ )
b
Ê Arc AF œ a) Ê a) œ b(nFCP) Ê nFCP œ
nOCG œ
1
#
a
b
);
); nOCG œ nOCP nPCE
œ nOCP ˆ 1# !‰ . Now nOCP œ 1 nFCP
œ 1 ba ). Thus nOCG œ 1 ba )
œ 1 ba )
1
#
1
#
! Ê
! Ê ! œ 1 ba ) ) œ 1
1
# )
ˆ ab b )‰ .
Then x œ OG BG œ OG PE œ (a b) cos ) b cos ! œ (a b) cos ) b cos ˆ1
œ (a b) cos ) b cos ˆ a b b )‰ . Also y œ EG œ CG CE œ (a b) sin ) b sin !
ab
b
)‰
œ (a b) sin ) b sin ˆ1 a b b )‰ œ (a b) sin ) b sin ˆ a b b )‰ . Therefore
x œ (a b) cos ) b cos ˆ a b b )‰ and y œ (a b) sin ) b sin ˆ a b b )‰ .
If b œ 4a , then x œ ˆa 4a ‰ cos )
œ
œ
œ
œ
3a
4
3a
4
3a
4
3a
4
cos )
œ
œ
œ
3a
4
3a
4
3a
4
3a
4
cos 3) œ
3a
4
cos Š
a ˆ 4a ‰
ˆ 4a ‰
)‹
cos ) 4a (cos ) cos 2) sin ) sin 2))
cos ) a(cos )) acos# ) sin# )b (sin ))(2 sin ) cos ))b
a
2a
#
#
4 cos ) sin ) 4 sin ) cos )
#
$
) cos$ ) 3a
4 (cos )) a1 cos )b œ a cos );
a
a ˆ4‰
a‰
a
3a
a
3a
4 sin ) 4 sin Š ˆ 4a ‰ )‹ œ 4 sin ) 4 sin 3) œ 4
cos )
cos
y œ ˆa
œ
a
4
a
4
a
4
a
4
a
4
cos$ )
sin ) 4a (sin ) cos 2) cos ) sin 2))
sin ) 4a a(sin )) acos# ) sin# )b (cos ))(2 sin ) cos ))b
sin )
sin )
sin )
a
4
3a
4
3a
4
sin ) cos# )
sin ) cos# )
a
4
a
4
#
sin$ )
2a
4
cos# ) sin )
sin$ )
(sin )) a1 sin )b
a
4
sin$ ) œ a sin$ ).
37. Draw line AM in the figure and note that nAMO is a right
angle since it is an inscribed angle which spans the diameter
of a circle. Then AN# œ MN# AM# . Now, OA œ a,
AN
AM
a œ tan t, and a œ sin t. Next MN œ OP
Ê OP# œ AN# AM# œ a# tan# t a# sin# t
Ê OP œ Èa# tan# t a# sin# t
œ (a sin t)Èsec# t 1 œ
x œ OP sin t œ
a sin$ t
cos t œ
#
a sin# t
cos t
#
. In triangle BPO,
a sin t tan t and
y œ OP cos t œ a sin t Ê x œ a sin# t tan t and y œ a sin# t.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
651
652
Chapter 11 Parametric Equations and Polar Coordinates
38. Let the x-axis be the line the wheel rolls along with the y-axis through a low point of the trochoid
(see the accompanying figure).
Let ) denote the angle through which the wheel turns. Then h œ a) and k œ a. Next introduce xw yw -axes
parallel to the xy-axes and having their origin at the center C of the wheel. Then xw œ b cos ! and
yw œ b sin !, where ! œ 3#1 ). It follows that xw œ b cos ˆ 3#1 )‰ œ b sin ) and yw œ b sin ˆ 3#1 )‰
œ b cos ) Ê x œ h xw œ a) b sin ) and y œ k yw œ a b cos ) are parametric equations of the trochoid.
#
#
#
39. D œ É(x 2)# ˆy "# ‰ Ê D# œ (x 2)# ˆy "# ‰ œ (t 2)# ˆt# "# ‰ Ê D# œ t% 4t
Ê
d aD # b
dt
17
4
œ 4t$ 4 œ 0 Ê t œ 1. The second derivative is always positive for t Á 0 Ê t œ 1 gives a local
minimum for D# (and hence D) which is an absolute minimum since it is the only extremum Ê the closest
point on the parabola is (1ß 1).
#
#
40. D œ Ɉ2 cos t 34 ‰ (sin t 0)# Ê D# œ ˆ2 cos t 34 ‰ sin# t Ê
d aD # b
dt
œ 2 ˆ2 cos t 34 ‰ (2 sin t) 2 sin t cos t œ (2 sin t) ˆ3 cos t 3# ‰ œ 0 Ê 2 sin t œ 0 or 3 cos t
Ê t œ 0, 1 or t œ
1
3
,
51
3
. Now
#
#
d aD b
dt#
#
#
œ 6 cos t 3 cos t 6 sin t so that
#
#
d aD b
dt#
3
#
œ0
(0) œ 3 Ê relative
#
#
#
#
maximum, d dtaD# b (1) œ 9 Ê relative maximum, d dtaD# b ˆ 13 ‰ œ 92 Ê relative minimum, and
d # aD # b ˆ 5 1 ‰
œ 9# Ê relative minimum. Therefore both t œ 13 and t œ 531 give points on the ellipse
dt#
3
È
È
the point ˆ 34 ß !‰ Ê Š1ß #3 ‹ and Š1ß #3 ‹ are the desired points.
41. (a)
(b)
(c)
42. (a)
(b)
(c)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
closest to
Section 11.1 Parametrizations of Plane Curves
43.
44. (a)
(b)
(c)
45. (a)
(b)
46. (a)
(b)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
653
654
Chapter 11 Parametric Equations and Polar Coordinates
47. (a)
(b)
(c)
48. (a)
(b)
(c)
(d)
11.2 CALCULUS WITH PARAMETRIC CURVES
1. t œ
Ê
1
4 Ê
dy
dx ¹ tœ 1
x œ 2 cos
d# y
dx#
dyw /dt
dx/dt
œ cot
1
4
1
4
œ È2, y œ 2 sin
1
4
œ È2;
dx
dt
œ 2 sin t,
dy
dt
œ 2 cos t Ê
œ 1; tangent line is y È2 œ 1 Šx È2‹ or y œ x
dy
dx
œ
œ
dy/dt
dx/dt
w
2È2 ; dy
dt
2 cos t
2 sin t
œ cot t
œ csc# t
4
Ê
œ
œ
csc# t
2 sin t
"
œ 2 sin
$t Ê
d# y
dx# ¹ tœ 1
œ È 2
4
2. t œ 6" Ê x œ sin ˆ21 ˆ 6" ‰‰ œ sin ˆ 13 ‰ œ
dy
dt
œ 21 sin 21t Ê
tangent line is y
œ cos$"21t Ê
"
#
dy
dx
œ
21 sin 21t
21 cos 21t
œ È3 ’x Š
d# y
dx# ¹ tœc 1
È3
#
œ tan 21t Ê
, y œ cos ˆ21 ˆ 6" ‰‰ œ cos ˆ 13 ‰ œ
dy
dx ¹ tœc 1
œ tan ˆ21 ˆ
" ‰‰
6
"
#
;
dx
dt
œ tan ˆ
œ 21 cos 21t,
1‰
3
œ È3;
6
È3
# ‹“
or y œ È3x 2;
dyw
dt
œ 21 sec# 21t Ê
d# y
dx#
œ
21 sec# 21t
21 cos 21t
œ 8
6
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.2 Calculus With Parametric Curves
1
4
3. t œ
œ "# tan t Ê
dyw
dt
1
4
Ê x œ 4 sin
21
3 Ê
dy
dx ¹ tœ 21
4. t œ
Ê
œ "# tan
dy
dx ¹ tœ 1
4
d# y
dx#
œ "# sec# t Ê
dyw /dt
dx/dt
œ
1
4
"# sec# t
4 cos t
œ
œ 4 cos t,
dx
dt
œ 2 sin t Ê
dy
dt
dy
dx
œ
dy/dt
dx/dt
œ
2 sin t
4 cos t
d# y
dx# ¹ tœ 1
"
œ 8 cos
$t Ê
œ
È2
4
4
21
3
x œ cos
È3 dx
21
3 œ # ; dt
È3
È 3 x
# ‹œ
œ "# , y œ È3 cos
œ È3 ; tangent line is y Š
d# y
dx# ¹ tœ 21
œ È2;
œ "# ; tangent line is y È2 œ "# Šx 2È2‹ or y œ "# x 2È2 ;
3
Ê
1
4
œ 2È2, y œ 2 cos
œ sin t,
dy
dt
œ È3 sin t Ê
dy
dx
œ
È3 sin t
sin t
œ È3
d# y
dx#
œ
œ0
dyw
dt
œ0 Ê
"
œ 1; tangent line is
ˆ #" ‰‘ or y œ È3 x;
0
sin t
œ0
3
5. t œ
Ê xœ
1
4
y
"
#
1
4
,yœ
"
#
œ 1,
dx
dt
;
dy
dt
œ 1 † ˆx 4" ‰ or y œ x 4" ;
"
#Èt
œ
dyw
dt
Ê
œ
dy
dx
œ
dy/dt
dx/dt
d# y
dx#
œ "4 t$Î# Ê
#É "4
d# y
dx# ¹ tœ 1
œ 4" t$Î# Ê
œ 2
4
dy
dx
œ
sec# t
2 sec# t tan t
"
2 tan t
œ
"
#
œ
cot t Ê
d# y
dx# ¹ tœc 1
œ
œ
dy
dx ¹ tœc 1
4
y (1) œ "# (x 1) or y œ "# x "# ;
Ê
œ
dy
dx ¹ tœ 1
4
dyw /dt
dx/dt
œ
6. t œ 14 Ê x œ sec# ˆ 14 ‰ 1 œ 1, y œ tan ˆ 14 ‰ œ 1;
Ê
Ê
1
2È t
dyw
dt
"
#
œ 2 sec# t tan t,
dx
dt
œ sec# t
dy
dt
cot ˆ 14 ‰ œ "# ; tangent line is
d# y
dx#
œ "# csc# t Ê
œ
"# csc# t
2 sec# t tan t
œ "4 cot$ t
"
4
4
7. t œ
œ
1
6
Ê x œ sec
sec# t
sec t tan t
1
6
y œ tan
1
6
œ
dy
dx ¹ tœ 1
œ csc
1
6
œ 2; tangent line is y
d# y
dx#
dyw /dt
dx/dt
œ
œ csc t Ê
2
È3 ,
"
È3
;
dx
dt
œ sec t tan t,
6
dyw
dt
œ csc t cot t Ê
œ
csc t cot t
sec t tan t
œ
dy
dt
œ sec# t Ê
"
È3
œ 2 Šx
d# y
dx# ¹ tœ 1
œ cot$ t Ê
œ
dy
dx
2
È3 ‹
dy/dt
dx/dt
or y œ 2x È3 ;
œ 3È3
6
8. t œ 3 Ê x œ È3 1 œ 2, y œ È3(3) œ 3;
È
œ 3 Èt3t 1 œ
dyw
dt
Ê
œ
dy
dx ¹ tœ3
3 È 3 1
È3(3)
œ
dx
dt
œ 4t,
y 1 œ 1 † (x 5) or y œ x 4;
10. t œ 1 Ê x œ 1, y œ 2;
œ
dx
dt
dy
dt
œ
Ê xœ
sin t
1 cos t
d# y
dx#
œ
dy/dt
dx/dt
œ
Ê
3
2tÈ3t Èt1
dyw
dt
œ t"# ,
œ 4t$ Ê
œ 2t Ê
dy
dt
y (2) œ 1(x 1) or y œ x 1;
1
3
œ
3
#
(3t)"Î# Ê
dy
dx
œ
ˆ 3# ‰ (3t) "Î#
ˆ "# ‰ (t1) "Î#
Š 2tÈ3t3Èt 1 ‹
Š 2Èt1 1 ‹
œ tÈ33t
œ "3
9. t œ 1 Ê x œ 5, y œ 1;
11. t œ
dy
dt
œ 2; tangent line is y 3 œ 2[x (2)] or y œ 2x 1;
È3t 3 (t 1) "Î# ‘3Èt 1 3 (3t) "Î# ‘
#
#
3t
d# y
dx# ¹ tœ3
œ "# (t 1)"Î# ,
dx
dt
Ê
1
3
sin
dy
dx ¹ tœ 1
3
œ
1
3
œ
1
3
sin ˆ 13 ‰
1cos ˆ 13 ‰
œ
dyw
dt
"
t
dy
dx
d# y
dx#
Ê
œ
œ
dyw /dt
dx/dt
dy
dx
œ
œ 1 Ê
œ
ˆ "t ‰
Š t"# ‹
d# y
dx#
œ
4t$
4t
2t
4t
œ t# Ê
œ
"
#
Ê
œ t Ê
1
Š t"# ‹
dy
dx ¹ tœc1
d# y
dx# ¹ tœc1
dy
dx ¹ tœ1
œ t# Ê
œ (1)# œ 1; tangent line is
œ
"
#
œ 1; tangent line is
d# y
dx# ¹ tœ1
œ1
È3
#
dy
, y œ 1 cos 13 œ 1 #" œ #" ; dx
dt œ 1 cos t, dt œ sin t Ê
È
Š #3 ‹
È
œ ˆ " ‰ œ È3 ; tangent line is y "# œ È3 Šx 13 #3 ‹
#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
dy
dx
œ
dy/dt
dx/dt
655
656
Chapter 11 Parametric Equations and Polar Coordinates
1È3
3
Ê y œ È3x
1
(1 cos t)#
œ
12. t œ
Ê
1
2 Ê
dy
dx ¹ tœ 1
2;
d# y
dx# ¹ tœ 1
Ê
dyw
dt
(1 cos t)(cos t) (sin t)(sin t)
(1cos t)#
œ
1
1 cos t
œ
d# y
dx#
Ê
œ
dyw /dt
dx/dt
œ
œ
cos t
sin t
1 ‰
ˆ 1 cos
t
1 cos t
œ 4
3
1
2
x œ cos
œ cot
2
13. t œ 2 Ê x œ
œ 0, y œ 1 sin
1
#
œ 2;
œ sin t,
dx
dt
w
œ 0; tangent line is y œ 2;
œ 31 , y œ
1
21
1
2
tangent line is y œ 9x 1;
œ 2;
2
21
dyw
dt
œ
dx
dt
14. t œ 0 Ê x œ 0 e0 œ 1, y œ 1 e0 œ 0;
dyw
dt
dx
dt
2y$ 3t# œ 4 Ê 6y#
œ csc t Ê
œ
4 at 1 b3
a t 1b 3
œ 1 et ,
dx
dt
d# y
dx#
4t œ 0 Ê 3x2
dx
dt
œ 4t Ê
œ
6t
6y#
6t œ 0 Ê
dy
dt
dy
dt
e t
a 1 e t b3
œ
œ
t
y#
csc t
sin t
d# y
dx# ¹ tœ2
œ
4a 2 1b3
a 2 1b 3
d# y
dx# ¹ tœ0
Ê
dx
dt
œ
4t
3x2
;
; thus
dy
dx
œ
dy/dt
dx/dt
œ
œ
œ 1
œ
a2 1b2
a2 1 b2
œ 9;
œ
e 0
1 e0
2
at 1b2
at 1 b 2
Ê
e t
1 et
œ
d# y
dx# ¹ tœ 1
œ csc t Ê
œ
dy
dx
œ cot t
$
dy
dx
Ê
Ê
dy
dx
#
œ
d y
dx#
œ et Ê
dy
dt
Ê
œ
1
at 1b2
œ
et
a 1 e t b2
tangent line is y œ 12 x 12 ;
15. x3 2t# œ 9 Ê 3x2
d# y
dx#
#
#
dy
dt
1
, dy
at 1b2 dt
t 1b
œ 4ata
Ê
1 b3
œ cos t Ê
dy
dt
œ 108
Ê
e 0
a 1 e 0 b3
Š yt# ‹
dy
dx ¹ tœ0
œ 12 ;
œ 18
œ
Š c4t ‹
dy
dx ¹ tœ2
3x2
t(3x2 )
y# (4t)
œ
3x2
4y#
;tœ2
Ê x3 2(2)# œ 9 Ê x3 8 œ 9 Ê x3 œ 1 Ê x œ 1; t œ 2 Ê 2y$ 3(2)# œ 4
Ê 2y$ œ 16 Ê y$ œ 8 Ê y œ 2; therefore
16. x œ É5 Èt Ê
Ê at 1b
dy
dt
œ
"
#È t
therefore,
dy
dx ¹ tœ4
œ
dy
dt
dy
dt
"Î#
"
Èt y
œ #at
1b œ
" #yÈt
#tÈt 2Èt
œ
œ
"4
dx
dt
dy
t
Èy ‹ dt
3x"Î#
œ
y
2Èt1
Œ 2Èy (t b 1) b 2tÈt b 1
Š
"
2t b 1
‹
1 b 3x"Î#
dx
dt
œ 2t 1 Ê ˆ1 3x"Î# ‰
2Èy Ê
œ
dy
dt
dy
dt
dx
dt
1
#
; therefore
Èt É5 Èt
" #yÈt
#Ètat" b
sin t x cos t 2
dy
dx ¹ tœ1
œ
†
4Èt É5 Èt
"
2
3
Š 2Èct yb 1 2Èy‹
ŠÈt 1 Èy ‹
t
dy
dt
œ
dx
dt
Èt 1
œ
2t1
13x"Î#
y
2È t 1
yÈ y 4yÈt 1
2È y (t 1) 2tÈt 1
; yÈt 1 2tÈy œ 4
2Èy Š Èt y ‹
; thus
; t œ 0 Ê x 2x$Î# œ 0 Ê x ˆ1 2x"Î# ‰ œ 0 Ê x œ 0; t œ 0
t sin t 2t œ y Ê sin t t cos t 2 œ
Ê xœ
4
œ 2t 1 Ê
œ0 Ê
dx
dt
dy
dt ;
sin 1 1 cos 1 2
–
1
Š 1# ‹ cos 1
sin 1 2
dy
dx ¹ tœ0
œ
œ 1 Ê (sin t 2)
thus
œ
dy
dx
œ
4 1 8
21
È4
È0 1
Œ 2È4(0 1) 2(0)È0 1
4
dx
dt
" #yÈt
È È
œ #t t "2 t œ
dy
dt
dx
dt
œ
dy
dx
10È3
9
œ
Ê yÈ0 1 2(0)Èy œ 4 Ê y œ 4; therefore
18. x sin t 2x œ t Ê
" "Î#
; y(t 1) œ Èt Ê y (t 1) dy
dt œ # t
t œ 4 Ê x œ É5 È4 œ È3; t œ 4 Ê y † 3 œ È4 Ê y œ
cyÈy c 4yÈt b 1
dy/dt
dx/dt
3
œ 16
4È t É 5 È t
; thus
Èt 1 y ˆ " ‰ (t 1)"Î# 2Èy 2t ˆ " y"Î# ‰
#
#
Ê ŠÈt 1
dy
dx
3a"b2
4a#b#
œ
ˆ "# t"Î# ‰ œ
2Š" 2a 23 bÈ4‹É& È4
17. x 2x$Î# œ t# t Ê
Ê
ˆ5 Èt‰
y Ê
#ˆ" #yÈt‰É& Èt
;
"t
œ
"
#
œ
dx
dt
dy
dx ¹ tœ2
4(4)
2(0) 1
Œ 1 3(0)"Î#
dx
dt
œ 6
œ 1 x cos t Ê
sin t t cos t 2
c x cos t ‰
ˆ 1sin
tb2
dx
dt
œ
1 x cos t
sin t2
; t œ 1 Ê x sin 1 2x œ 1
œ 4
—
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
;
dy
dt
œ0
Section 11.2 Calculus With Parametric Curves
19. x œ t3 t, y 2t3 œ 2x t2 Ê
Ê
œ
dy
dx
2t 2
3t2 1
Ê
dy
dx ¹ tœ1
œ
dx
2
dt œ 3t
2 a1 b 2
œ1
3 a1 b2 1
20. t œ lnax tb, y œ t et Ê 1 œ
Ê
œ
dy
dx
21. A œ '0
21
t et et
xt1;
1 ˆ dx
x t dt
1,
6t2 œ 2 dx
dt 2t Ê
dy
dt
1‰ Ê x t œ
t œ 0 Ê 0 œ lnax 0b Ê x œ 1 Ê
dx
dx
dt 1 Ê dt œ
a0be0 e0
dy
dx ¹ tœ0 œ 1 0 1
dy
dt
œ 2a3t2 1b 2t 6t2 œ 2t 2
x t 1,
œ
657
dy
dt
œ t et et ;
1
2
y dx œ '0 aa1 cos tbaa1 cos tbdt œ a2 '0 a1 cos tb2 dt œ a2 '0 a1 2cos t cos2 tbdt
21
œ a2 '0 ˆ1 2cos t
21
21
1 cos 2t ‰
dt
2
21
œ a2 '0 ˆ 23 2cos t 12 cos 2t‰dt œ a2 ” 23 t 2sin t 14 sin 2t•
21
21
0
œ a2 a31 0 0b 0 œ 31 a2
22. A œ '0 x dy œ '0 at t2 baet bdt ”u œ t t2 Ê du œ a1 2tbdt; dv œ aet bdt Ê v œ et •
1
1
1
œ et at t2 bº '0 et a1 2tbdt
1
t
t
”u œ 1 2t Ê du œ 2dt; dv œ e dt Ê v œ e •
0
1
1
0
0
œ et at t2 bº ”et a1 2tbº '0 2et dt• œ ”et at t2 b et a1 2tb 2et •º
1
0
œ ae1 a0b e1 a1b 2e1 b ae0 a0b e0 a1b 2e0 b œ 1 3e1 œ 1
23. A œ 2'1 y dx œ 2'1 ab sin tbaa sin tbdt œ 2ab'0 sin2 t dt œ 2ab'0
0
1
1
0
1
1
3
e
1 cos 2t
2
dt œ ab'0 a1 cos 2tb dt
1
œ ab’t 12 sin 2t“ œ abaa1 0b !b œ 1 ab
0
24. (a) x œ t2 , y œ t6 , 0 Ÿ t Ÿ 1 Ê A œ '0 y dx œ '0 at6 b2t dt œ '0 2t7 dt œ ’ 14 t8 “ œ
1
1
1
1
0
1
4
0œ
(b) x œ t3 , y œ t9 , 0 Ÿ t Ÿ 1 Ê A œ '0 y dx œ '0 at9 b3t2 dt œ '0 3t11 dt œ ’ 14 t12 “ œ
1
1
1
1
0
25.
dx
dt
œ sin t and
dy
dt
1
4
1
4
0œ
1
4
#
#
‰ Š dy
Éasin tb# a1 cos tb# œ È2 2 cos t
œ 1 cos t Ê Êˆ dx
dt
dt ‹ œ
cos t ‰
È2 ' É sin# t dt
Ê Length œ '0 È2 2 cos t dt œ È2 '0 Ɉ 11
cos t (1 cos t) dt œ
1 cos t
0
1
œ È2 '0
1
sin t
È1 cos t
1
dt (since sin t
1
0 on [0ß 1]); [u œ 1 cos t Ê du œ sin t dt; t œ 0 Ê u œ 0,
#
t œ 1 Ê u œ 2] Ä È2 '0 u"Î# du œ È2 2u"Î# ‘ ! œ 4
2
26.
dx
dt
œ 3t# and
dy
dt
È3
Ê Length œ '0
Ä '1
4
27.
dx
dt
3
#
#
#
‰ Š dy
Éa3t# b# (3t)# œ È9t% 9t# œ 3tÈt# 1 Šsince t
œ 3t Ê Êˆ dx
dt
dt ‹ œ
3tÈt# 1 dt; ’u œ t# 1 Ê
3
#
0 on ’0ß È3“‹
du œ 3t dt; t œ 0 Ê u œ 1, t œ È3 Ê u œ 4“
%
u"Î# du œ u$Î# ‘ " œ (8 1) œ 7
œ t and
dy
dt
#
#
Èt# a2t 1b œ Éat 1b# œ kt 1k œ t 1 since 0 Ÿ t Ÿ 4
‰ Š dy
œ (2t 1)"Î# Ê Êˆ dx
dt
dt ‹ œ
Ê Length œ '0 at 1b dt œ ’ t2 t“ œ a8 4b œ 12
4
#
%
!
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
658
28.
Chapter 11 Parametric Equations and Polar Coordinates
dx
dt
œ a2t 3b"Î# and
dy
dt
#
#
‰ Š dy
Éa2t 3b a1 tb# œ Èt# 4t 4 œ kt 2k œ t 2
œ 1 t Ê Êˆ dx
dt
dt ‹ œ
since 0 Ÿ t Ÿ 3 Ê Length œ '0 (t 2) dt œ ’ t2 2t“ œ
3
3
#
!
29.
dx
dt
œ 8t cos t and
dy
dt
dx
dt
#
#
‰ Š dy
Éa8t cos tb# a8t sin tb# œ È64t# cos# t 64t# sin# t
œ 8t sin t Ê Êˆ dx
dt
dt ‹ œ
1
#
œ k8tk œ 8t since 0 Ÿ t Ÿ
30.
21
#
Ê Length œ '0
1Î2
1Î#
8t dt œ c4t# d !
œ ˆ sec t" tan t ‰ asec t tan t sec# tb cos t œ sec t cos t and
œ 1#
œ Éasec t cos tb# asin tb# œ Èsec# t 1 œ Ètan# t œ ktan tk œ tan t since 0 Ÿ t Ÿ
Ê Length œ '0
1Î3
31.
dx
dt
œ sin t and
dy
dt
tan t dt œ '0
1Î3
1Î$
dt œ c ln kcos tkd !
sin t
cos t
#
#
‰ Š dy
œ sin t Ê Êˆ dx
dt
dt ‹
dy
dt
"
#
œ ln
1
3
ln 1 œ ln 2
‰ Š dy
Éasin tb# acos tb# œ 1 Ê Area œ ' 21y ds
œ cos t Ê Êˆ dx
dt
dt ‹ œ
#
#
œ '0 21a2 sin tba1bdt œ 21 c2t cos td #!1 œ 21[a41 1b a0 1b] œ 81#
21
32.
dx
dt
œ t"Î# and
È3
œ '0
dy
dt
#
21 ˆ 23 t$Î# ‰ É t
"
t
È3
'0
#
21 ˆ 23 t$Î# ‰ É t
#
fatb œ 21 ˆ 23 t$Î# ‰ É t
Ê
33.
dx
dt
È3
'0
281
9
Fatb dt œ
œ 1 and
È2
dy
dt
#
"
t
dt œ
41
3
È3
'0
1
t
Ê Area œ ' 21x ds
tÈt# 1 dt; cu œ t# 1 Ê du œ 2t dt; t œ 0 Ê u œ 1,
'14 231 Èu du œ 491 u$Î# ‘ %" œ 2891
’t œ È3 Ê u œ 4“ Ä
Note:
#
#
Èt t" œ É t
‰ Š dy
œ t"Î# Ê Êˆ dx
dt
dt ‹ œ
1
t
dt is an improper integral but limb fatb exists and is equal to 0, where
tÄ!
. Thus the discontinuity is removable: define Fatb œ fatb for t 0 and Fa0b œ 0
.
#
#
#
È2‹ œ Ét# 2È2 t 3 Ê Area œ ' 21x ds
‰# Š dy
œ t È2 Ê Êˆ dx
dt
dt ‹ œ Ê1 Št
œ 'cÈ2 21 Št È2‹ Ét# 2È2 t 3 dt; ’u œ t# 2È2 t 3 Ê du œ Š2t 2È2‹ dt; t œ È2 Ê u œ 1,
’t œ È2 Ê u œ 9“ Ä '1 1Èu du œ 23 1u$Î# ‘ " œ
9
*
21
3
a27 1b œ
521
3
' 21y ds œ '0
‰ Š dy
34. From Exercise 30, ʈ dx
dt
dt ‹ œ tan t Ê Area œ
#
#
1Î$
œ 21 c cos td !
35.
dx
dt
œ 2 and
dy
dt
#
1 Î3
21 cos t tan t dt œ 21 '0
1 Î3
sin t dt
œ 21 "# (1)‘ œ 1
È2# 1# œ È5 Ê Area œ ' 21y ds œ ' 21at 1bÈ5 dt
‰ Š dy
œ 1 Ê Êˆ dx
dt
dt ‹ œ
0
#
#
1
"
œ 21È5 ’ t2 t“ œ 31È5. Check: slant height is È5 Ê Area is 1a1 2bÈ5 œ 31È5 .
!
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.2 Calculus With Parametric Curves
36.
dx
dt
œ h and
659
Èh# r# Ê Area œ ' 21y ds œ ' 21rtÈh# r# dt
‰ Š dy
œ r Ê Êˆ dx
dt
dt ‹ œ
0
#
#
dy
dt
œ 2 1 r È h# r #
1
'01 t dt œ 21rÈh# r# ’ t2 “ " œ 1rÈh# r# .
#
!
Check: slant height is Èh# r# Ê Area is
1rÈh# r# .
37. Let the density be $ œ 1. Then x œ cos t t sin t Ê
dx
dt
œ t cos t, and y œ sin t t cos t Ê
dy
dt
#
#
1
#
È(t cos t)# (t sin t)# œ ktk dt œ t dt since 0 Ÿ t Ÿ
‰ Š dy
Ê dm œ 1 † ds œ ʈ dx
dt
dt ‹ dt œ
M œ ' dm œ '0
1Î2
1Î#
œ csin t t cos td !
yœ
Mx
M
œ
1#
Š3
4
‹
œ
#
Š 18 ‹
xœ
My
M
œ
3‰
#
Š 18 ‹
1Î#
ct# sin t 2 sin t 2t cos td !
24
1#
1Î#
ct# cos t 2 cos t 2t sin td !
œ
12
1
Therefore axß yb œ ˆ 12
1
38. Let the density be $ œ 1. Then x œ et cos t Ê
dx
dt
1 Î2
œ
24
1#
, where we integrated by parts. Therefore,
acos t t sin tb t dt œ '0
1 Î2
24
1# .
1#
4
œ 3
2. Next, My œ ' µ
x dm œ '0
. The curve's mass is
1Î2
1Î2
1Î2
. Also Mx œ ' µ
y dm œ '0 asin t t cos tb t dt œ '0 t sin t dt '0 t# cos t dt
1Î#
œ ccos t t sin td !
ˆ 3#1
1#
8
t dt œ
œ t sin t
31
#
t cos t dt '0
1 Î2
t# sin t dt
3, again integrating by parts. Hence
‰
ß 24
1# 2 .
œ et cos t et sin t, and y œ et sin t Ê
dy
dt
œ et sin t et cos t
#
#
‰ Š dy
Éaet cos t et sin tb# aet sin t et cos tb# dt œ È2e2t dt œ È2 et dt.
Ê dm œ 1 † ds œ ʈ dx
dt
dt ‹ dt œ
1
1
The curve's mass is M œ ' dm œ '0 È2 et dt œ È2 e1 È2 . Also Mx œ ' µ
y dm œ '0 aet sin tb ŠÈ2 et ‹ dt
2t
21
œ '0 È2 e2t sin t dt œ È2 ’ e5 (2 sin t cos t)“ œ È2 Š e5 5" ‹ Ê y œ
1
1
!
Mx
M
œ
È2 Š e21 " ‹
5
5
È 2 e1 È 2
œ
e21 "
5 ae1 1b
.
2t
21
Next My œ ' µ
x dm œ '0 aet cos tb ŠÈ2 et ‹ dt œ '0 È2 e2t cos t dt œ È2 ’ e5 a2 cos t sin tb“ œ È2 Š 2e5 52 ‹
1
My
M
œ
1
!
21
Ê xœ
1
È2 Š 2e5 25 ‹
È 2 e1 È 2
21
21
21
œ 52eae1 12b . Therefore axß yb œ Š 52eae1 12b ß 5 eae1 11b ‹.
39. Let the density be $ œ 1. Then x œ cos t Ê
dx
dt
œ sin t, and y œ t sin t Ê
dy
dt
œ 1 cos t
#
#
‰ Š dy
Éasin tb# a1 cos tb# dt œ È2 2 cos t dt. The curve's mass
Ê dm œ 1 † ds œ ʈ dx
dt
dt ‹ dt œ
is M œ ' dm œ '0 È2 2 cos t dt œ È2'0 È1 cos t dt œ È2 '0 É2 cos# ˆ #t ‰ dt œ 2 '0 ¸cos ˆ #t ‰¸ dt
1
1
œ 2 '0 cos ˆ #t ‰ dt ˆsince 0 Ÿ t Ÿ 1 Ê 0 Ÿ
1
t
#
1
1
1
Ÿ 1# ‰ œ 2 2 sin ˆ 2t ‰‘ ! œ 4. Also Mx œ ' µ
y dm
œ '0 at sin tb ˆ2 cos #t ‰ dt œ '0 2t cos ˆ #t ‰ dt '0 2 sin t cos ˆ #t ‰ dt
1
1
1
1
1
œ 2 4 cos ˆ 2t ‰ 2t sin ˆ #t ‰‘ ! 2 "3 cos ˆ #3 t‰ cos ˆ "# t‰‘ ! œ 41
16
3
Ê yœ
Next My œ ' µ
x dm œ '0 acos tbˆ2 cos #t ‰ dt œ '0 cos t cos ˆ #t ‰ dt œ 2 ’sin ˆ 2t ‰
1
œ
4
3
Ê xœ
My
M
ˆ 43 ‰
4
œ
œ
1
"
3
ˆ41 16
‰
Mx
3
œ1
M œ
4
sin ˆ 3# t‰ 1
“ œ 2 23
3
!
43 .
. Therefore axß yb œ ˆ 3" ß 1 43 ‰.
40. Let the density be $ œ 1. Then x œ t$ Ê
dx
dt
œ 3t# , and y œ
3t#
#
Ê
dy
dt
œ 3t Ê dm œ 1 † ds
#
#
‰ Š dy
Éa3t# b# (3t)# dt œ 3 ktk Èt# 1 dt œ 3tÈt# 1 dt since 0 Ÿ t Ÿ È3. The curve's mass
œ ʈ dx
dt
dt ‹ dt œ
È3
is M œ ' dm œ '0
œ
9
#
È3
'0
$Î#
3tÈt# 1 dt œ ’at# 1b “
t$ Èt# 1 dt œ
87
5
È3
!
È3
œ 7. Also Mx œ ' µ
y dm œ '0
œ 17.4 (by computer) Ê y œ
Mx
M
œ
17.4
7
3t#
#
Š3tÈt# 1‹ dt
¸ 2.49. Next My œ ' µ
x dm
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
660
Chapter 11 Parametric Equations and Polar Coordinates
È3
È3
œ '0 t$ † 3t Èat# 1b dt œ 3 '0 t% Èt# 1 dt ¸ 16.4849 (by computer) Ê x œ
My
M
œ
¸ 2.35.
16.4849
7
Therefore, axß yb ¸ a2.35ß 2.49b.
œ 2 sin 2t and
dx
dt
41. (a)
Ê Length œ '0
1Î2
œ 1 cos 1t and
dx
dt
(b)
#
#
#
#
‰ Š dy
Éa2 sin 2tb# a2 cos 2tb# œ 2
œ 2 cos 2t Ê Êˆ dx
dt
dt ‹ œ
dy
dt
1Î#
2 dt œ c2td !
œ1
‰ Š dy
Éa1 cos 1tb# a1 sin 1tb# œ 1
œ 1 sin 1t Ê Êˆ dx
dt
dt ‹ œ
dy
dt
Ê Length œ '1Î2 1 dt œ c1td "Î# œ 1
1 Î2
"Î#
42. (a) x œ gayb has the parametrization x œ gayb and y œ y for c Ÿ y Ÿ d Ê
dx
dy
œ gw ayb and
œ 1; then
dy
dy
dx
dx
'
' È1 [gw ayb]# dy
Length œ 'c ÊŠ dy
dy ‹ Š dy ‹ dy œ c Ê1 Š dy ‹ dy œ c
#
d
(b) x œ y3Î2 , 0 Ÿ y Ÿ
œ
3 Î2
8
27 a4b
4
3
#
Ê
3 Î2
8
27 a1b
(c) x œ 23 y2Î3 , 0 Ÿ y Ÿ 1 Ê
œ lim
3
2
a Ä0
œ 32 y1Î2 Ê L œ '0
4 Î3
dx
dy
œ
#
d
56
27
dx
dy
d
É1 ˆ 32 y1Î2 ‰# dy œ '
0
1
1
Ê
œ
œ
œ
1
È œ
2
3
2
2È 3 1
È3 2
2 sin )b,
dy
dx º
)œ0
(b) x œ ˆ1 2 sinˆ 1# ‰‰cosˆ 1# ‰ œ 0, y œ ˆ1 2 sinˆ 1# ‰‰sinˆ 1# ‰ œ 3;
È3 1
œ
y œ ˆ1 2 sinˆ 431 ‰‰sinˆ 431 ‰ œ
45.
dy
d)
œ
œ
01
20
2 sinˆ2ˆ 1# ‰‰ cosˆ 1# ‰
2 cosˆ2ˆ 1# ‰‰ sinˆ 1# ‰
3 È3
2
;
dy
dx º
)œ41/3
œ
dx
dt
œ 1,
dy
dt
œ sin t Ê
dy
dx
œ
sin t
1
œ sin t Ê
or t œ
31
2
d2 y
dx2
Ê
dy
dx ,
d dy
dt Š dx ‹
(a) the minimum slope is
dy
dx º
tœ31Î2
œ sinˆ 321 ‰ œ 1, which occurs at x œ
œ 2 cos 2t Ê
dy
dx
Ê 2 cos# t 1 œ 0 Ê cos t œ „
y œ sin 2 ˆ 14 ‰ œ 1 Ê Š
00
2 1
œ0
È2
# ß 1‹
œ
d2 y
dx2
œ
œ cos t. The
cos t
1
d2 y
dx2
œ0
œ ± ±
31Î2
1 Î2
œ sinˆ 12 ‰ œ 1, which occurs at x œ 12 , y œ 1 cosˆ 12 ‰ œ 1
dy
dt
œ
1
2
in other words, points where
dy
dx º
tœ1Î2
œ cos t and
œ
2 sinˆ2ˆ 431 ‰‰ cosˆ 431 ‰
2 cosˆ2ˆ 431 ‰‰ sinˆ 431 ‰
œ cos t Ê
(a) the maximum slope is
dx
dt
2 3
œ 2cos ) sin ) cos )a1 2 sin )b
2 sina2a0bb cosa0b
2 cosa2a0bb sina0b
dy
dx º
)œ1/2
maximum and minimum slope will occur at points that maximize/minimize
Ê cos t œ 0 Ê t œ
2 3
a Ä0
œ Š 4 3È 3 ‹
44. x œ t, y œ 1 cos t, 0 Ÿ t Ÿ 21 Ê
1
2
'a1 É y yÎ Î 1 dy
dy œ lim
a
(a) x œ a1 2 sina0bbcosa0b œ 1, y œ a1 2 sina0bbsina0b œ 0;
È3 1
,
2
0
3 Î2
3 Î2
lim ” 32 † 23 ˆy2Î3 1‰ • œ lim Ša2b3Î2 ˆa2Î3 1‰ ‹ œ 2È2 1
a Ä0
a Ä0
dx
2
d) œ 2cos ) sin )a1
4cos ) sin ) cos )
2 sin 2) cos )
2cos2 ) 2sin2 ) sin ) œ 2 cos 2) sin )
(c) x œ ˆ1 2 sinˆ 431 ‰‰cosˆ 431 ‰ œ
1
y2Î3
4Î3
1
'a1 ˆy2Î3 1‰1Î2 ˆ 23 y1Î3 ‰ dy œ
2cos ) sin ) cos )a1 2 sin )b
2cos2 ) sin )a1 2 sin )b
3Î2
É1 94 y dy œ ” 49 † 23 ˆ1 94 y‰ •
#
œ y1Î3 Ê L œ '0 É1 ay1Î3 b dy œ '0 É1
43. x œ a1 2 sin )bcos ), y œ a1 2 sin )bsin ) Ê
dy
dx
4Î3
dy/dt
dx/dt
"
È2
œ
2 cos 2t
cos t
Ê tœ
1
4
,
œ
31
4
,
2 a2 cos# t 1b
cos t
51
4
,
71
4
; then
31
2 ,
dy
dx
y œ 1 cosˆ 321 ‰ œ 1
œ0 Ê
2 a2 cos# t 1b
cos t
. In the 1st quadrant: t œ
1
4
œ0
Ê x œ sin
1
4
œ
È2
#
is the point where the tangent line is horizontal. At the origin: x œ 0 and y œ 0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
and
Section 11.2 Calculus With Parametric Curves
Ê sin t œ 0 Ê t œ 0 or t œ 1 and sin 2t œ 0 Ê t œ 0,
the origin. Tangents at origin:
46.
dx
dt
œ
dy
dx
œ 2 cos 2t and
dy
dt
œ 3 cos 3t Ê
3 ca2 cos# t 1b (cos t) 2 sin t cos t sin td
2 a2 cos# t 1b
(3 cos t) a4 cos# t3b
2 a2 cos# t 1b
œ0 Ê
œ 2 Ê y œ 2x and
dy
dx ¹ tœ0
œ
dy
dx
œ
œ
dy/dt
dx/dt
and y œ sin 3 ˆ 16 ‰ œ 1 Ê Š
3(cos 2t cos t sin 2t sin t)
2 a2 cos# t1b
(3 cos t) a2 cos# t 1 2 sin# tb
2 a2 cos# t 1b
È3
#
È3
# ß 1‹
Ê tœ
1
6
,
51
6
,
œ
(3 cos t) a4 cos# t 3b
2 a2 cos# t 1b
; then
71
6
,
111
6
. In the 1st quadrant: t œ
1
6
1
#
,
31
#
and
Ê x œ sin 2 ˆ 16 ‰ œ
È3
#
is the point where the graph has a horizontal tangent. At the origin: x œ 0
and y œ 0 Ê sin 2t œ 0 and sin 3t œ 0 Ê t œ 0,
1
#
, 1,
the tangent lines at the origin. Tangents at the origin:
3 cos (31)
2 cos (21)
œ
3 cos 3t
2 cos 2t
1 give the tangent lines at
œ 0 Ê 3 cos t œ 0 or 4 cos# t 3 œ 0: 3 cos t œ 0 Ê t œ
4 cos# t 3 œ 0 Ê cos t œ „
œ
1
31
# , 1, # ; thus t œ 0 and t œ
dy
dx ¹ tœ1 œ 2 Ê y œ 2x
661
31
#
and t œ 0,
dy
dx ¹ tœ0
œ
3 cos 0
2 cos 0
1
3
œ
,
21
3
, 1,
41
3
,
51
3
3
#
x, and
Ê t œ 0 and t œ 1 give
3
#
Ê yœ
dy
dt
œ a sin t Ê Length
dy
dx ¹ tœ1
œ 3# Ê y œ 3# x
47. (a) x œ aat sin tb, y œ aa1 cos tb, 0 Ÿ t Ÿ 21 Ê
dx
dt
œ aa1 cos tb,
œ '0 Éaaa1 cos tbb# aa sin tb# dt œ '0 Èa# 2a# cos t a# cos# t a# sin# t dt
21
21
œ aÈ2'0 È1 cos t dt œ aÈ2'0 É2 sin2 ˆ 2t ‰ dt œ 2a'0 sinˆ 2t ‰ dt œ ’4a cosˆ 2t ‰“
21
21
21
21
0
œ 4a cos 1 4a cosa0b œ 8a
(b) a œ 1 Ê x œ t sin t, y œ 1 cos t, 0 Ÿ t Ÿ 21 Ê
dx
dt
21
œ 1 cos t,
dy
dt
œ sin t Ê Surface area œ
œ '0 21a1 cos tbÉa1 cos tb# asin tb# dt œ '0 21a1 cos tbÈ1 2 cos t cos# t sin# t dt
21
3Î2
œ 21'0 a1 cos tbÈ2 2 cos t dt œ 2È21'0 a1 cos tb3Î2 dt œ 2È21'0 ˆ1 cos ˆ2 † 2t ‰‰ dt
21
21
21
3 Î2
œ 2È21'0 ˆ2 sin2 ˆ 2t ‰‰ dt œ 81'0 sin3 ˆ 2t ‰ dt
21
21
’u œ
t
2
Ê du œ 21 dt Ê dt œ 2 du; t œ 0 Ê u œ 0, t œ 21 Ê u œ 1“
œ 161'0 sin3 u du œ 161'0 sin2 u sin u du œ 161'0 a1 cos2 u bsin u du œ 161'0 sin u du 161'0 cos2 u sin u du
1
œ ’161cos u
1
1
161
3
3 cos u“0
1
œ ˆ161
161 ‰
3
1
ˆ161
161 ‰
3
œ
641
3
48. x œ t sin t, y œ 1 cos t, 0 Ÿ t Ÿ 21; Volume œ '0 1 y2 dx œ '0 1a1 cos tb2 a1 cos tbdt
21
21
2t ‰
œ 1'0 a1 3cos t 3cos2 t cos3 tbdt œ 1'0 ˆ1 3cos t 3ˆ 1 cos
cos2 t cos t‰dt
2
21
21
œ 1'0 ˆ 52 3cos t 32 cos 2t a1 sin2 tb cos t‰dt œ 1'0 ˆ 52 4cos t 32 cos 2t sin2 t cos t‰dt
21
21
21
œ 1’ 52 t 4sin t 34 sin 2t 31 sin3 t “
0
œ 1a51 0 0 0b 0 œ 512
47-50. Example CAS commands:
Maple:
with( plots );
with( student );
x := t -> t^3/3;
y := t -> t^2/2;
a := 0;
b := 1;
N := [2, 4, 8 ];
for n in N do
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1
662
Chapter 11 Parametric Equations and Polar Coordinates
tt := [seq( a+i*(b-a)/n, i=0..n )];
pts := [seq([x(t),y(t)],t=tt)];
L := simplify(add( student[distance](pts[i+1],pts[i]), i=1..n ));
# (b)
T := sprintf("#47(a) (Section 11.2)\nn=%3d L=%8.5f\n", n, L );
P[n] := plot( [[x(t),y(t),t=a..b],pts], title=T ):
# (a)
end do:
display( [seq(P[n],n=N)], insequence=true );
ds := t ->sqrt( simplify(D(x)(t)^2 + D(y)(t)^2) ):
# (c)
L := Int( ds(t), t=a..b ):
L = evalf(L);
11.3 POLAR COORDINATES
1. a, e; b, g; c, h; d, f
2. a, f; b, h; c, g; d, e
3. (a) ˆ2ß 1# 2n1‰ and ˆ2ß 1# (2n 1)1‰ , n an integer
(b) (#ß 2n1) and (#ß (2n 1)1), n an integer
(c) ˆ2ß 3#1 2n1‰ and ˆ2ß 3#1 (2n 1)1‰ , n an integer
(d) (#ß (2n 1)1) and (#ß 2n1), n an integer
4. (a) ˆ3ß 14 2n1‰ and ˆ3ß 541 2n1‰ , n an integer
(b) ˆ3ß 14 2n1‰ and ˆ3ß 541 2n1‰ , n an integer
(c) ˆ3ß 14 2n1‰ and ˆ3ß 341 2n1‰ , n an integer
(d) ˆ3ß 14 2n1‰ and ˆ3ß 341 2n1‰ , n an integer
5. (a) x œ r cos ) œ 3 cos 0 œ 3, y œ r sin ) œ 3 sin 0 œ 0 Ê Cartesian coordinates are ($ß 0)
(b) x œ r cos ) œ 3 cos 0 œ 3, y œ r sin ) œ 3 sin 0 œ 0 Ê Cartesian coordinates are ($ß 0)
(c) x œ r cos ) œ 2 cos 21 œ 1, y œ r sin ) œ 2 sin 21 œ È3 Ê Cartesian coordinates are Š1ß È3‹
3
(d) x œ r cos ) œ 2 cos
71
3
3
œ 1, y œ r sin ) œ 2 sin
71
3
œ È3 Ê Cartesian coordinates are Š1ß È3‹
(e) x œ r cos ) œ 3 cos 1 œ 3, y œ r sin ) œ 3 sin 1 œ 0 Ê Cartesian coordinates are (3ß 0)
(f) x œ r cos ) œ 2 cos 1 œ 1, y œ r sin ) œ 2 sin 1 œ È3 Ê Cartesian coordinates are Š1ß È3‹
3
3
(g) x œ r cos ) œ 3 cos 21 œ 3, y œ r sin ) œ 3 sin 21 œ 0 Ê Cartesian coordinates are (3ß 0)
(h) x œ r cos ) œ 2 cos ˆ 1 ‰ œ 1, y œ r sin ) œ 2 sin ˆ 1 ‰ œ È3 Ê Cartesian coordinates are Š1ß È3‹
3
6. (a) x œ È2 cos
1
4
œ 1, y œ È2 sin
3
1
4
œ 1 Ê Cartesian coordinates are (1ß 1)
(b) x œ 1 cos 0 œ 1, y œ 1 sin 0 œ 0 Ê Cartesian coordinates are (1ß 0)
(c) x œ 0 cos 1# œ 0, y œ 0 sin 1# œ 0 Ê Cartesian coordinates are (!ß 0)
(d) x œ È2 cos ˆ 1 ‰ œ 1, y œ È2 sin ˆ 1 ‰ œ 1 Ê Cartesian coordinates are (1ß 1)
4
(e) x œ 3 cos
51
6
œ
4
3È 3
2
, y œ 3 sin
51
6
È
œ 3# Ê Cartesian coordinates are Š 3 # 3 ß 3# ‹
(f) x œ 5 cos ˆtan" 43 ‰ œ 3, y œ 5 sin ˆtan" 43 ‰ œ 4 Ê Cartesian coordinates are ($ß 4)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.3 Polar Coordinates
663
(g) x œ 1 cos 71 œ 1, y œ 1 sin 71 œ 0 Ê Cartesian coordinates are (1ß 0)
(h) x œ 2È3 cos 231 œ È3, y œ 2È3 sin 231 œ 3 Ê Cartesian coordinates are ŠÈ3ß 3‹
7. (a) a1, 1b Ê r œ È12 12 œ È2, sin ) œ
1
È2
and cos ) œ
1
È2
1
4
Ê)œ
Ê Polar coordinates are ŠÈ2, 14 ‹
(b) a3, 0b Ê r œ Éa3b2 02 œ 3, sin ) œ 0 and cos ) œ 1 Ê ) œ 1 Ê Polar coordinates are a3, 1b
2
(c) ŠÈ3, 1‹ Ê r œ ÊŠÈ3‹ a1b2 œ 2, sin ) œ 12 and cos ) œ
(d) a3, 4b Ê r œ Éa3b2 42 œ 5, sin ) œ
4
5
È3
2
111
6
Ê)œ
Ê Polar coordinates are ˆ2,
111 ‰
6
and cos ) œ 35 Ê ) œ 1 arctanˆ 43 ‰ Ê Polar coordinates are
ˆ5, 1 arctanˆ 43 ‰‰
8. (a) a2, 2b Ê r œ Éa2b2 a2b2 œ 2È2, sin ) œ È12 and cos ) œ È12 Ê ) œ 341 Ê Polar coordinates are
Š2È2, 341 ‹
(b) a0, 3b Ê r œ È02 32 œ 3, sin ) œ 1 and cos ) œ 0 Ê ) œ
2
(c) ŠÈ3, 1‹ Ê r œ ÊŠÈ3‹ 12 œ 2, sin ) œ
1
2
1
2
Ê Polar coordinates are ˆ3, 12 ‰
and cos ) œ
(d) a5, 12b Ê r œ É52 a12b2 œ 13, sin ) œ 12
13 and cos ) œ
5
12
È3
2
Ê)œ
51
6
Ê Polar coordinates are ˆ2,
51 ‰
6
‰
Ê ) œ arctanˆ 12
5 Ê Polar coordinates are
ˆ13, arctanˆ 12
‰‰
5
9. (a) a3, 3b Ê r œ È32 32 œ 3È2, sin ) œ È12 and cos ) œ È12 Ê ) œ
Š3È2,
51
4
Ê Polar coordinates are
51
4 ‹
(b) a1, 0b Ê r œ Éa1b2 02 œ 1, sin ) œ 0 and cos ) œ 1 Ê ) œ 0 Ê Polar coordinates are a1, 0b
2
(c) Š1, È3‹ Ê r œ Êa1b2 ŠÈ3‹ œ 2, sin ) œ
ˆ2,
È3
2
and cos ) œ
1
2
Ê)œ
51
3
Ê Polar coordinates are
51 ‰
3
(d) a4, 3b Ê r œ É42 a3b2 œ 5, sin ) œ
3
5
and cos ) œ 45 Ê ) œ 1 arctanˆ 34 ‰ Ê Polar coordinates are
ˆ5, 1 arctanˆ 43 ‰‰
10. (a) a2, 0b Ê r œ Éa2b2 02 œ 2, sin ) œ 0 and cos ) œ 1 Ê ) œ 0 Ê Polar coordinates are a2, 0b
(b) a1, 0b Ê r œ È12 02 œ 1, sin ) œ 0 and cos ) œ 1 Ê ) œ 1 or ) œ 1 Ê Polar coordinates are a1, 1b or
a1, 1b
(c) a0, 3b Ê r œ É02 a3b2 œ 3, sin ) œ 1 and cos ) œ 0 Ê ) œ
(d) Š
È3 1
2 , 2‹
are ˆ1,
Ê r œ ÊŠ
71 ‰
6
È3 2
2 ‹
2
ˆ 21 ‰ œ 1, sin ) œ 12 and cos ) œ
1
2
Ê Polar coordinates are ˆ3, 12 ‰
È3
2
Ê)œ
71
6
or ) œ 561 Ê Polar coordinates
or ˆ1, 561 ‰
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
664
Chapter 11 Parametric Equations and Polar Coordinates
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.3 Polar Coordinates
665
26.
27. r cos ) œ 2 Ê x œ 2, vertical line through (#ß 0)
28. r sin ) œ 1 Ê y œ 1, horizontal line through (0ß 1)
29. r sin ) œ 0 Ê y œ 0, the x-axis
30. r cos ) œ 0 Ê x œ 0, the y-axis
31. r œ 4 csc ) Ê r œ
4
sin )
32. r œ 3 sec ) Ê r œ
Ê r sin ) œ 4 Ê y œ 4, a horizontal line through (0ß 4)
3
cos )
Ê r cos ) œ 3 Ê x œ 3, a vertical line through (3ß 0)
33. r cos ) r sin ) œ 1 Ê x y œ 1, line with slope m œ 1 and intercept b œ 1
34. r sin ) œ r cos ) Ê y œ x, line with slope m œ 1 and intercept b œ 0
35. r# œ 1 Ê x# y# œ 1, circle with center C œ (!ß 0) and radius 1
36. r# œ 4r sin ) Ê x# y# œ 4y Ê x# y# 4y 4 œ 4 Ê x# (y 2)# œ 4, circle with center C œ (0ß 2) and radius 2
37. r œ
5
sin )2 cos )
Ê r sin ) 2r cos ) œ 5 Ê y 2x œ 5, line with slope m œ 2 and intercept b œ 5
38. r# sin 2) œ 2 Ê 2r# sin ) cos ) œ 2 Ê (r sin ))(r cos )) œ 1 Ê xy œ 1, hyperbola with focal axis y œ x
)‰ˆ " ‰
39. r œ cot ) csc ) œ ˆ cos
Ê r sin# ) œ cos ) Ê r# sin# ) œ r cos ) Ê y# œ x, parabola with vertex (0ß 0)
sin )
sin )
which opens to the right
sin ) ‰
40. r œ 4 tan ) sec ) Ê r œ 4 ˆ cos
Ê r cos# ) œ 4 sin ) Ê r# cos# ) œ 4r sin ) Ê x# œ 4y, parabola with
#)
vertex œ (!ß 0) which opens upward
41. r œ (csc )) er cos ) Ê r sin ) œ er cos ) Ê y œ ex , graph of the natural exponential function
42. r sin ) œ ln r ln cos ) œ ln (r cos )) Ê y œ ln x, graph of the natural logarithm function
43. r# 2r# cos ) sin ) œ 1 Ê x# y# 2xy œ 1 Ê x# 2xy y# œ 1 Ê (x y)# œ 1 Ê x y œ „ 1, two parallel
straight lines of slope 1 and y-intercepts b œ „ 1
44. cos# ) œ sin# ) Ê r# cos# ) œ r# sin# ) Ê x# œ y# Ê kxk œ kyk Ê „ x œ y, two perpendicular
lines through the origin with slopes 1 and 1, respectively.
45. r# œ 4r cos ) Ê x# y# œ 4x Ê x# 4x y# œ 0 Ê x# 4x 4 y# œ 4 Ê (x 2)# y# œ 4, a circle with
center C(2ß 0) and radius 2
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
666
Chapter 11 Parametric Equations and Polar Coordinates
46. r# œ 6r sin ) Ê x# y# œ 6y Ê x# y# 6y œ 0 Ê x# y# 6y 9 œ 9 Ê x# (y 3)# œ 9, a circle with
center C(0ß 3) and radius 3
47. r œ 8 sin ) Ê r# œ 8r sin ) Ê x# y# œ 8y Ê x# y# 8y œ 0 Ê x# y# 8y 16 œ 16 Ê x# (y 4)# œ 16, a
circle with center C(0ß 4) and radius 4
48. r œ 3 cos ) Ê r# œ 3r cos ) Ê x# y# œ 3x Ê x# y# 3x œ 0 Ê x# 3x
#
Ê ˆx 3# ‰ y# œ
9
4
, a circle with center C ˆ 3# ß !‰ and radius
9
4
y# œ
9
4
3
#
49. r œ 2 cos ) 2 sin ) Ê r# œ 2r cos ) 2r sin ) Ê x# y# œ 2x 2y Ê x# 2x y# 2y œ 0
Ê (x 1)# (y 1)# œ 2, a circle with center C(1ß 1) and radius È2
50. r œ 2 cos ) sin ) Ê r# œ 2r cos ) r sin ) Ê x# y# œ 2x y Ê x# 2x y# y œ 0
#
Ê (x 1)# ˆy "# ‰ œ 54 , a circle with center C ˆ1ß "# ‰ and radius
È5
#
È
51. r sin ˆ) 16 ‰ œ 2 Ê r ˆsin ) cos 16 cos ) sin 16 ‰ œ 2 Ê #3 r sin ) "# r cos ) œ 2 Ê
Ê È3 y x œ 4, line with slope m œ " and intercept b œ 4
È3
È3
#
È3
È
52. r sin ˆ 231 )‰ œ 5 Ê r ˆsin 231 cos ) cos 231 sin )‰ œ 5 Ê #3 r cos ) "# r sin ) œ 5 Ê
Ê È3 x y œ 10, line with slope m œ È3 and intercept b œ 10
53. x œ 7 Ê r cos ) œ 7
55. x œ y Ê r cos ) œ r sin ) Ê ) œ
y "# x œ 2
È3
#
x "# y œ 5
54. y œ 1 Ê r sin ) œ 1
1
4
56. x y œ 3 Ê r cos ) r sin ) œ 3
57. x# y# œ 4 Ê r# œ 4 Ê r œ 2 or r œ 2
58. x# y# œ 1 Ê r# cos# ) r# sin# ) œ 1 Ê r# acos# ) sin# )b œ 1 Ê r# cos 2) œ 1
59.
x#
9
y#
4
œ 1 Ê 4x# 9y# œ 36 Ê 4r# cos# ) 9r# sin# ) œ 36
60. xy œ 2 Ê (r cos ))(r sin )) œ 2 Ê r# cos ) sin ) œ 2 Ê 2r# cos ) sin ) œ 4 Ê r# sin 2) œ 4
61. y# œ 4x Ê r# sin# ) œ 4r cos ) Ê r sin# ) œ 4 cos )
62. x# xy y# œ 1 Ê x# y# xy œ 1 Ê r# r# sin ) cos ) œ 1 Ê r# (1 sin ) cos )) œ 1
63. x# (y 2)# œ 4 Ê x# y# 4y 4 œ 4 Ê x# y# œ 4y Ê r# œ 4r sin ) Ê r œ 4 sin )
64. (x 5)# y# œ 25 Ê x# 10x 25 y# œ 25 Ê x# y# œ 10x Ê r# œ 10r cos ) Ê r œ 10 cos )
65. (x 3)# (y 1)# œ 4 Ê x# 6x 9 y# 2y 1 œ 4 Ê x# y# œ 6x 2y 6 Ê r# œ 6r cos ) 2r sin ) 6
66. (x 2)# (y 5)# œ 16 Ê x# 4x 4 y# 10y 25 œ 16 Ê x# y# œ 4x 10y 13
Ê r# œ 4r cos ) 10r sin ) 13
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.4 Graphing in Polar Coordinates
67. (!ß )) where ) is any angle
68. (a) x œ a Ê r cos ) œ a Ê r œ
(b) y œ b Ê r sin ) œ b Ê r œ
a
cos )
b
sin )
Ê r œ a sec )
Ê r œ b csc )
11.4 GRAPHING IN POLAR COORDINATES
1. 1 cos ()) œ 1 cos ) œ r Ê symmetric about the
x-axis; 1 cos ()) Á r and 1 cos (1 ))
œ 1 cos ) Á r Ê not symmetric about the y-axis;
therefore not symmetric about the origin
2. 2 2 cos ()) œ 2 2 cos ) œ r Ê symmetric about the
x-axis; 2 # cos ()) Á r and 2 2 cos (1 ))
œ 2 2 cos ) Á r Ê not symmetric about the y-axis;
therefore not symmetric about the origin
3. 1 sin ()) œ 1 sin ) Á r and 1 sin (1 ))
œ 1 sin ) Á r Ê not symmetric about the x-axis;
1 sin (1 )) œ 1 sin ) œ r Ê symmetric about
the y-axis; therefore not symmetric about the origin
4. 1 sin ()) œ 1 sin ) Á r and 1 sin (1 ))
œ 1 sin ) Á r Ê not symmetric about the x-axis;
1 sin (1 )) œ 1 sin ) œ r Ê symmetric about the
y-axis; therefore not symmetric about the origin
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
667
668
Chapter 11 Parametric Equatins and Polar Coordinates
5. 2 sin ()) œ 2 sin ) Á r and 2 sin (1 ))
œ 2 sin ) Á r Ê not symmetric about the x-axis;
2 sin (1 )) œ 2 sin ) œ r Ê symmetric about the
y-axis; therefore not symmetric about the origin
6. 1 2 sin ()) œ 1 2 sin ) Á r and 1 2 sin (1 ))
œ 1 2 sin ) Á r Ê not symmetric about the x-axis;
1 2 sin (1 )) œ 1 2 sin ) œ r Ê symmetric about the
y-axis; therefore not symmetric about the origin
7. sin ˆ #) ‰ œ sin ˆ #) ‰ œ r Ê symmetric about the y-axis;
sin ˆ 21#) ‰ œ sin ˆ 2) ‰ , so the graph is symmetric about the
x-axis, and hence the origin.
8. cos ˆ #) ‰ œ cos ˆ #) ‰ œ r Ê symmetric about the x-axis;
cos ˆ 21#) ‰ œ cos ˆ 2) ‰ , so the graph is symmetric about the
y-axis, and hence the origin.
9. cos ()) œ cos ) œ r# Ê (rß )) and (rß )) are on the
graph when (rß )) is on the graph Ê symmetric about the
x-axis and the y-axis; therefore symmetric about the origin
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.4 Graphing in Polar Coordinates
10. sin (1 )) œ sin ) œ r# Ê (rß 1 )) and (rß 1 )) are on
the graph when (rß )) is on the graph Ê symmetric about
the y-axis and the x-axis; therefore symmetric about the
origin
11. sin (1 )) œ sin ) œ r# Ê (rß 1 )) and (rß 1 ))
are on the graph when (rß )) is on the graph Ê symmetric
about the y-axis and the x-axis; therefore symmetric about
the origin
12. cos ()) œ cos ) œ r# Ê (rß )) and (rß )) are on
the graph when (rß )) is on the graph Ê symmetric about
the x-axis and the y-axis; therefore symmetric about the
origin
13. Since a „ rß )b are on the graph when (rß )) is on the graph
ˆa „ rb# œ 4 cos 2( )) Ê r# œ 4 cos 2)‰ , the graph is
symmetric about the x-axis and the y-axis Ê the graph is
symmetric about the origin
14. Since (rß )) on the graph Ê (rß )) is on the graph
ˆa „ rb# œ 4 sin 2) Ê r# œ 4 sin 2)‰ , the graph is
symmetric about the origin. But 4 sin 2()) œ 4 sin 2)
Á r# and 4 sin 2(1 )) œ 4 sin (21 2)) œ 4 sin (2))
œ 4 sin 2) Á r# Ê the graph is not symmetric about
the x-axis; therefore the graph is not symmetric about
the y-axis
15. Since (rß )) on the graph Ê (rß )) is on the graph
ˆa „ rb# œ sin 2) Ê r# œ sin 2)‰ , the graph is
symmetric about the origin. But sin 2()) œ ( sin 2))
sin 2) Á r# and sin 2(1 )) œ sin (21 2))
œ sin (2)) œ ( sin 2)) œ sin 2) Á r# Ê the graph
is not symmetric about the x-axis; therefore the graph is
not symmetric about the y-axis
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
669
670
Chapter 11 Parametric Equatins and Polar Coordinates
16. Sincea „ rß )b are on the graph when (rß )) is on the
graph ˆa „ rb# œ cos 2()) Ê r# œ cos 2)‰, the
graph is symmetric about the x-axis and the y-axis Ê the
graph is symmetric about the origin.
Ê r œ 1 Ê ˆ1ß 1# ‰ , and ) œ 1# Ê r œ 1
w
)r cos )
Ê ˆ1ß 1# ‰ ; rw œ ddr) œ sin ); Slope œ rrw sin
cos )r sin )
17. ) œ
1
#
sin# )r cos )
sin ) cos )r sin )
sin# ˆ 1# ‰(1) cos 1#
sin 1# cos 1# (1) sin 1#
œ
Ê Slope at ˆ1ß 1# ‰ is
œ 1; Slope at ˆ1ß 1# ‰ is
sin# ˆ 1# ‰(1) cos ˆ 1# ‰
sin ˆ 1# ‰ cos ˆ 1# ‰(1) sin ˆ 1# ‰
œ1
18. ) œ 0 Ê r œ 1 Ê ("ß 0), and ) œ 1 Ê r œ 1
dr
Ê ("ß 1); rw œ d)
œ cos );
rw sin )r cos )
cos ) sin )r cos )
rw cos )r sin ) œ cos ) cos )r sin )
0 sin 0(1) cos 0
cos ) sin )r cos )
Ê Slope at ("ß 0) is coscos
# 0(1) sin 0
cos# )r sin )
cos 1 sin 1(1) cos 1
1; Slope at ("ß 1) is cos# 1(1) sin 1 œ 1
Slope œ
œ
œ
Ê r œ 1 Ê ˆ"ß 14 ‰ ; ) œ 14 Ê r œ 1
Ê ˆ1ß 14 ‰ ; ) œ 341 Ê r œ 1 Ê ˆ"ß 341 ‰ ;
) œ 341 Ê r œ 1 Ê ˆ1ß 341 ‰ ;
19. ) œ
rw œ
1
4
dr
d)
œ 2 cos 2);
Slope œ
r sin )r cos )
r cos )r sin )
w
w
Ê Slope at ˆ1ß 14 ‰ is
Slope at ˆ1ß 14 ‰ is
Slope at ˆ1ß 341 ‰ is
Slope at ˆ1ß 341 ‰ is
2 cos 2) sin )r cos )
2 cos 2) cos )r sin )
2 cos ˆ 1# ‰ sin ˆ 14 ‰(1) cos ˆ 14 ‰
2 cos ˆ 1 ‰ cos ˆ 1 ‰(1) sin ˆ 1 ‰
œ
#
4
4
œ 1;
2 cos ˆ 1# ‰ sin ˆ 14 ‰(1) cos ˆ 14 ‰
2 cos ˆ 1# ‰ cos ˆ 14 ‰(1) sin ˆ 14 ‰
2 cos Š 3#1 ‹ sin Š 341 ‹(1) cos Š 341 ‹
2 cos Š 3#1 ‹ cos Š 341 ‹(1) sin Š 341 ‹
œ 1;
œ 1;
2 cos Š 3#1 ‹ sin Š 341 ‹(1) cos Š 341 ‹
2 cos Š 3#1 ‹ cos Š 341 ‹(1) sin Š 341 ‹
œ 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.4 Graphing in Polar Coordinates
20. ) œ 0 Ê r œ 1 Ê (1ß 0); ) œ 12 Ê r œ 1 Ê ˆ1ß 12 ‰ ;
) œ 1# Ê r œ 1 Ê ˆ"ß 12 ‰ ; ) œ 1 Ê r œ 1
Ê (1ß 1); rw œ
dr
d) œ 2 sin 2);
)r cos )
2 sin 2) sin )r cos )
Slope œ rr sin
cos )r sin ) œ 2 sin 2) cos )r sin )
2 sin 0 sin 0cos 0
Ê Slope at (1ß 0) is
2 sin 0 cos 0sin 0 , which is undefined;
2 sin 2 ˆ 1 ‰ sin ˆ 1 ‰(1) cos ˆ 1 ‰
Slope at ˆ1ß 12 ‰ is 2 sin 2 ˆ 12 ‰ cos ˆ21 ‰(1) sin ˆ 21 ‰ œ 0;
w
w
2
Slope at ˆ1ß 12 ‰ is
Slope at ("ß 1) is
2
2
2 sin 2 ˆ 1# ‰ sin ˆ 1# ‰(1) cos ˆ 1# ‰
2 sin 2 ˆ 1 ‰ cos ˆ 1 ‰(1) sin ˆ 1 ‰
#
2 sin 21 sin 1cos 1
2 sin 21 cos 1sin 1
#
#
œ 0;
, which is undefined
21. (a)
(b)
22. (a)
(b)
23. (a)
(b)
24. (a)
(b)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
671
672
Chapter 11 Parametric Equatins and Polar Coordinates
25.
26. r œ 2 sec ) Ê r œ
27.
2
cos )
Ê r cos ) œ 2 Ê x œ 2
28.
29. Note that (rß )) and (rß ) 1) describe the same point in the plane. Then r œ 1 cos ) Í 1 cos () 1)
œ 1 (cos ) cos 1 sin ) sin 1) œ 1 cos ) œ (1 cos )) œ r; therefore (rß )) is on the graph of
r œ 1 cos ) Í (rß ) 1) is on the graph of r œ 1 cos ) Ê the answer is (a).
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.4 Graphing in Polar Coordinates
30. Note that (rß )) and (rß ) 1) describe the same point in the plane. Then r œ cos 2) Í sin ˆ2() 1)) 1# ‰
œ sin ˆ2) 5#1 ‰ œ sin (2)) cos ˆ 5#1 ‰ cos (2)) sin ˆ 5#1 ‰ œ cos 2) œ r; therefore (rß )) is on the graph of
r œ sin ˆ2) 1# ‰ Ê the answer is (a).
31.
33. (a)
34. (a)
32.
(b)
(c)
(b)
(d)
(c)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
673
674
Chapter 11 Parametric Equatins and Polar Coordinates
(d)
(e)
11.5 AREA AND LENGTHS IN POLAR COORDINATES
1. A œ '0 "# )# d) œ 16 )3 ‘ ! œ
1
1
13
6
2. A œ '1Î4 "# a2 sin )b# d) œ 2'1Î4 sin2 ) d) œ 2'1Î4
1Î2
1Î2
œ ˆ 12 0‰ ˆ 14 12 ‰ œ
3. A œ '0
21
"
#
1
4
1Î2
4. A œ '0
21
(4 2 cos ))# d) œ '0
21
œ
"
#
a#
œ '0
"
#
21
"
#
# [a(1 cos ))] d)
21
ˆ #3 2 cos ) #"
0
'
5. A œ 2 '0
1Î4
6. A œ '1Î6
1 Î6
œ "4 )
7. A œ '0
1Î2
"
#
"
#
1Î4
1
‘ 1 Î6
6 sin 6) 1Î6
"
#
1Î2
1Î6
"
#
"
#
#1
"
2
sin 2)‘ ! œ 181
a# a1 2 cos ) cos# )b d) œ
"
#
"
4
a# #3 ) 2 sin )
d) œ
"
#
cos2 3) d) œ
)
"
#
sin 4) ‘ 1Î%
4
!
'11ÎÎ66
œ "4 ˆ 16 0‰ "4 ˆ 16 0‰ œ
(4 sin 2)) d) œ '0
8. A œ (6)(2)'0
1Î2
21
1 cos 4)
#
' 11ÎÎ66
"
#
1Î2
2 ) ‰‘
a16 16 cos ) 4 cos# )b d) œ '0 8 8 cos ) 2 ˆ 1 cos
d)
#
cos 2)‰ d) œ
cos# 2) d) œ '0
acos 3)b2 d) œ
d) œ '1Î4 a1 cos 2)bd) œ ) 12 sin 2)‘1Î4
1
2
œ '0 (9 8 cos ) cos 2)) d) œ 9) 8 sin )
21
1 cos 2)
2
œ
1 cos 6)
2
a#
2) ‰
'021 ˆ1 2 cos ) 1 cos
d)
#
#1
sin 2)‘ ! œ
3
#
1a#
1
8
d) œ
"
4
'11ÎÎ66
a1 cos 6)b d)
1
12
1Î#
2 sin 2) d) œ c cos 2)d !
œ2
(2 sin 3)) d) œ 12 '0 sin 3) d) œ 12 cos3 3) ‘ !
1Î6
"
#
1Î'
œ4
9. r œ 2 cos ) and r œ 2 sin ) Ê 2 cos ) œ 2 sin )
Ê cos ) œ sin ) Ê ) œ 14 ; therefore
A œ 2 '0
1Î4
œ '0
1Î4
"
#
(2 sin ))# d) œ '0
1Î4
2) ‰
4 ˆ 1 cos
d) œ '0
#
œ c2) sin
1Î4
1Î%
2) d !
œ
1
#
4 sin# ) d)
(2 2 cos 2)) d)
1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.5 Area and Lengths in Polar Coordinates
10. r œ 1 and r œ 2 sin ) Ê 2 sin ) œ 1 Ê sin ) œ
1
6
Ê )œ
or
51
6
"
#
; therefore
A œ 1(1)# '1Î6
51Î6
"
#
c(2 sin ))# 1# d d)
œ 1 '1Î6 ˆ2 sin# ) "# ‰ d)
51Î6
œ 1 '1Î6 ˆ1 cos 2) "# ‰ d)
51Î6
œ 1 '1Î6 ˆ "# cos 2)‰ d) œ 1 "2 )
sin 2) ‘ &1Î'
#
1Î'
œ 1 ˆ 511#
41 3È 3
6
51Î6
"
#
sin
51 ‰
3
1
ˆ 12
"
#
sin 13 ‰ œ
11. r œ 2 and r œ 2(1 cos )) Ê 2 œ 2(1 cos ))
Ê cos ) œ 0 Ê ) œ „ 1# ; therefore
A œ 2 '0
1Î2
œ '0
1Î2
œ '0
1Î2
œ '0
1Î2
"
#
[2(1 cos ))]# d) "# area of the circle
4 a1 2 cos ) cos# )b d) ˆ "# 1‰ (2)#
4 ˆ1 2 cos )
1 cos 2) ‰
#
d) 21
(4 8 cos ) 2 2 cos 2)) d) 21
1Î#
œ c6) 8 sin ) sin 2)d !
2 1 œ 51 8
12. r œ 2(1 cos )) and r œ 2(1 cos )) Ê 1 cos )
œ 1 cos ) Ê cos ) œ 0 Ê ) œ 1# or 3#1 ; the graph also
gives the point of intersection (0ß 0); therefore
A œ 2 '0
1Î2
"
#
[2(1 cos ))]# d) 2 '1Î2 "# [2(1 cos ))]# d)
1
œ '0 4a1 2cos ) cos# )bd)
1Î2
'1Î2 4 a1 2 cos ) cos# )bd)
1
œ '0
4 ˆ1 2 cos )
œ '0
(6 8 cos ) 2 cos 2)) d) '1Î2 (6 8 cos ) 2 cos 2)) d)
1Î2
1Î2
1 cos 2) ‰
#
d) '1Î2 4 ˆ1 2 cos )
1
1 cos 2) ‰
#
d)
1
1Î#
œ c6) 8 sin ) sin 2)d !
c6) 8 sin ) sin 2)d 11Î# œ 61 16
13. r œ È3 and r# œ 6 cos 2) Ê 3 œ 6 cos 2) Ê cos 2) œ
1
6
Ê )œ
"
#
(in the 1st quadrant); we use symmetry of the
graph to find the area, so
A œ 4 '0 ” "# (6 cos 2)) "# ŠÈ3‹ • d)
1Î6
#
œ 2 '0 (6 cos 2) 3) d) œ 2 c3 sin 2) 3)d !
1Î6
1Î'
œ 3È 3 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
675
676
Chapter 11 Parametric Equatins and Polar Coordinates
14. r œ 3a cos ) and r œ a(1 cos )) Ê 3a cos ) œ a(1 cos ))
Ê 3 cos ) œ 1 cos ) Ê cos ) œ "# Ê ) œ 13 or 13 ;
the graph also gives the point of intersection (0ß 0); therefore
A œ 2 '0
1Î3
"
#
c(3a cos ))# a# (1 cos ))# d d)
œ '0 a9a# cos# ) a# 2a# cos ) a# cos# )b d)
1Î3
œ '0
1Î3
a8a# cos# ) 2a# cos ) a# b d)
œ '0 c4a# (1 cos 2)) 2a# cos ) a# d d)
1Î3
œ '0 a3a# 4a# cos 2) 2a# cos )b d)
1Î3
1Î$
œ c3a# ) 2a# sin 2) 2a# sin )d !
œ 1a# 2a# ˆ "# ‰ 2a# Š
È3
# ‹
œ a# Š1 1 È3‹
15. r œ 1 and r œ 2 cos ) Ê 1 œ 2 cos ) Ê cos ) œ "#
Ê )œ
A œ 2'
1
21
3
in quadrant II; therefore
"
21Î3 #
c(2 cos ))# 1# d d) œ '21Î3 a4 cos# ) 1b d)
1
œ '21Î3 [2(1 cos 2)) 1] d) œ '21Î3 (1 2 cos 2)) d)
1
1
œ c) sin 2)d 1#1Î$ œ
1
3
È3
#
16. r œ 6 and r œ 3 csc ) Ê 6 sin ) œ 3 Ê sin ) œ
Ê )œ
1
6
or
51
6
œ '1Î6 ˆ18
51Î6
9
#
; therefore A œ '1Î6
51Î6
csc# )‰ d) œ 18)
"
#
"
#
a6# 9 csc# )b d)
9
#
cot )‘ 1Î'
&1Î'
œ Š151 9# È3‹ Š31 9# È3‹ œ 121 9È3
17. r œ sec ) and r œ 4 cos ) Ê 4 cos ) œ sec ) Ê cos2 ) œ
Ê ) œ 13 , 231 , 431 , or 531 ; therefore
1Î3
A œ 2 0 "# a16 cos# ) sec# )b d)
1Î3
œ 0 a8 8 cos 2) sec# )b d)
1Î3
œ c8) 4 sin 2) tan )d0
1
4
'
'
œ Š 831 2È3 È3‹ a0 0 0b œ
81
3
È3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.5 Area and Lengths in Polar Coordinates
18. r œ 3 csc ) and r œ 4 sin ) Ê 4 sin ) œ 3 csc ) Ê sin2 ) œ
Ê ) œ 13 ,
21 41
51
3 , 3 , or 3 ; therefore
1 Î2
"
a16 sin# ) 9 csc#
1 Î3 #
A œ 41 2'
677
3
4
)b d)
œ 41 '1Î3 a8 8 cos 2) 9 csc# )b d)
1Î2
1Î2
œ 41 c8) 4 sin 2) 9 cot )d1Î3
œ 41 ’a41 0 0b Š 831 2È3 3È3‹“
œ
81
3
È3
19. (a) r œ tan ) and r œ Š
Ê sin# ) œ Š
Ê cos# ) Š
È2
#
È2
# ‹
È2
# ‹
csc ) Ê tan ) œ Š
È2
# ‹
cos ) Ê 1 cos# ) œ Š
È2
# ‹cos
csc )
È2
# ‹
cos )
) 1 œ 0 Ê cos ) œ È2 or
(use the quadratic formula) Ê ) œ
1
4
(the solution
in the first quadrant); therefore the area of R" is
A" œ '0
1Î4
È2
#
œ
"
#
and OB œ Š
'01Î4 asec# ) 1b d) œ "# ctan ) )d 1! Î% œ "# ˆtan 14 14 ‰ œ "# 18 ; AO œ Š È#2 ‹ csc 1#
"
#
tan# ) d) œ
È2
# ‹
1
4
csc
œ 1 Ê AB œ Ê1# Š
È2 #
# ‹
therefore the area of the region shaded in the text is 2 ˆ "#
1
8
œ
È2
#
Ê the area of R# is A# œ
"4 ‰ œ
3
#
1
4
but does not generate the segment AB of the liner œ
_ on the line r œ
(b)
lim
) Ä 1Î2
œ
lim
È2
#
sin )
ˆ cos
)
r œ sec ) as ) Ä
œ
"
4
;
1
4
generates the arc OB of r œ tan )
csc ).
" ‰
cos )
1c
#
È2
È2
# ‹Š # ‹
csc ). Instead the interval generates the half-line from B to
tan ) œ _ and the line x œ 1 is r œ sec ) in polar coordinates; then
) Ä 1Î2c
Š
. Note: The area must be found this way
since no common interval generates the region. For example, the interval 0 Ÿ ) Ÿ
È2
#
"
#
œ
lim
) Ä 1 Î2 c
ˆ sincos) ) 1 ‰ œ
lim
) Ä 1Î2c
(tan ) sec ))
) ‰
ˆ cos
sin ) œ 0 Ê r œ tan ) approaches
lim
) Ä 1 Î2 c
Ê r œ sec ) (or x œ 1) is a vertical asymptote of r œ tan ). Similarly, r œ sec ) (or x œ 1)
is a vertical asymptote of r œ tan ).
20. It is not because the circle is generated twice from ) œ 0 to 21. The area of the cardioid is
A œ 2 '0
1
œ 32)
"
#
2)
(cos ) 1)# d) œ '0 acos# ) 2 cos ) 1b d) œ '0 ˆ 1 cos
2 cos ) 1‰ d)
#
sin 2)
4
1
1
2 sin )‘ ! œ
21. r œ )# , 0 Ÿ ) Ÿ È5 Ê
#
. The area of the circle is A œ 1 ˆ "# ‰ œ
È5
œ 2); therefore Length œ '0
dr
d)
È5
31
#
œ '0 k)k È)# 4 d) œ (since )
) œ È5 Ê u œ 9“ Ä '4
9
22. r œ
e)
È2
,0Ÿ)Ÿ1 Ê
dr
d)
1
"
#
œ
Èu du œ
e)
È2
È5
0) '0
1
4
Ê the area requested is actually 3#1
È5
Éa)# b# (2))# d) œ ' È)% 4)# d)
0
) È ) # 4 d ) ; u œ ) # 4 Ê
" 2 $Î# ‘ *
# 3 u
%
œ
"
#
du œ ) d); ) œ 0 Ê u œ 4,
19
3
; therefore Length œ '0 ÊŠ Èe 2 ‹ Š Èe 2 ‹ d) œ '0 Ê2 Š e# ‹ d)
1
)
#
)
#
1
œ '0 e) d) œ e) ‘ ! œ e1 1
1
1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
2)
1
4
œ
51
4
678
Chapter 11 Parametric Equatins and Polar Coordinates
23. r œ 1 cos ) Ê
dr
d)
œ sin ); therefore Length œ '0 È(1 cos ))# ( sin ))# d)
21
1
œ 2 '0 È2 2 cos ) d) œ 2'0 É 4(1 #cos )) d) œ 4 '0 É 1 #cos ) d) œ 4 '0 cos ˆ #) ‰ d) œ 4 2 sin 2) ‘ ! œ 8
1
1
24. r œ a sin#
)
#
1
, 0 Ÿ ) Ÿ 1, a 0 Ê
œ '0 Éa# sin%
1
)
#
)
#
a# sin#
dr
d)
)
#
cos#
œ a sin
)
#
cos
)
#
1
#
; therefore Length œ '0 Ɉa sin# #) ‰ ˆa sin
1
d) œ '0 a ¸sin #) ¸ Ésin#
1
)
#
)
#
cos#
1
6
1 cos )
œ '0
1Î2
,0Ÿ)Ÿ
1
#
"
1 cos )
dr
d)
œ
; therefore Length œ '0
1Î2
6 sin )
(1 cos ))#
d) œ 6 '0
1Î2
36 sin# )
a1 cos )b%
É (1 36
cos ))#
œ ˆsince
Ê
"
¸ 1cos
¸
) É1
0
1Î2
)
#
sin# )
(1 cos ))#
d)
cos# ) sin# )
0 on 0 Ÿ ) Ÿ 1# ‰ 6 '0 ˆ 1 "cos ) ‰ É 1 2 cos(1)cos
d)
) )#
1Î2
1Î2
#
#
6 sin )
ʈ 1 6cos ) ‰ Š (1
cos ))# ‹ d)
1Î2
cos )
È '
œ 6 '0 ˆ 1 "cos ) ‰ É (12 2cos
) ) # d) œ 6 2 0
œ 3'0 sec$
#
cos #) ‰ d)
d) œ (since 0 Ÿ ) Ÿ 1) a ' sin ˆ #) ‰ d)
1
œ 2a cos 2) ‘ ! œ 2a
25. r œ
)
#
d) œ 6'0
1Î4
d)
(1 cos ))$Î#
œ 6È2 '0
1Î2
1Î%
sec$ u du œ (use tables) 6 Œ sec u2tan u ‘ !
d)
ˆ2 cos# #) ‰$Î#
"
#
'01Î4
œ 3'0
1Î2
¸sec$ #) ¸ d)
sec u du
1Î%
œ 6 Š È"2 2" ln ksec u tan uk‘ ! ‹ œ 3 ’È2 ln Š1 È2‹“
26. r œ
2
1 cos )
1
#
,
Ÿ)Ÿ1 Ê
4
œ '1Î2 Ê (1 cos
) ) # Š1
1
œ ˆsince 1 cos )
œ
dr
d)
2 sin )
(1 cos ))#
sin# )
‹
a1 cos )b#
0 on
1
#
sin )
; therefore Length œ '1Î2 ʈ 1 2cos ) ‰ Š (12cos
) )# ‹ d )
1
) sin
d) œ '1Î2 ¸ 1 2cos ) ¸ É (1 (1cos )cos
) )#
1
#
1
œ '1Î2 csc$ ˆ #) ‰ d) œ ˆsince csc
1
1Î#
"
#
'11ÎÎ42
)
d)
#
1
2Œ csc u2cot u ‘ 1Î%
#
cos ) sin
Ÿ ) Ÿ 1‰ 2 '1Î2 ˆ 1 "cos ) ‰ É 1 2 cos(1)cos
))#
cos )
d)
È '
È '
œ 2 '1Î2 ˆ 1 "cos ) ‰ É (12 2cos
))# d) œ 2 2 1Î2 (1 cos ))$Î# œ 2 2 1Î2
1
)
#
#
d)
ˆ2 sin# )# ‰$Î#
)
d)
œ '1Î2 ¸csc$ #) ¸ d)
1
Ÿ ) Ÿ 1‰ 2 '1Î4 csc$ u du œ (use tables)
1Î2
1
#
0 on
1
#
#
1Î#
csc u du œ 2 Š È" 2" ln kcsc u cot uk‘ 1Î% ‹ œ 2 ’ È"
2
2
"
#
ln ŠÈ2 1‹“
œ È2 ln Š1 È2‹
27. r œ cos$
œ '0
1Î4
œ '0
)
3
Ê
dr
d)
œ sin
)
3
cos#
)
3
; therefore Length œ '0
Écos' ˆ 3) ‰ sin# ˆ 3) ‰ cos% ˆ 3) ‰ d) œ '0
1Î4
1Î4 1cos ˆ 2) ‰
3
#
d) œ
"
#
)
3
2
sin
28. r œ È1 sin 2) , 0 Ÿ ) Ÿ 1È2 Ê
Length œ '0
È
1 2
œ '0
È
1 2
1Î4
É(1 sin 2))
sin 2)
'
É 212sin
2 ) d) œ 0
È
1 2
2) ‘ 1Î%
3 !
dr
d)
œ
cos# 2)
(1 sin 2))
œ
"
#
1
8
Ɉcos$ 3) ‰# ˆ sin
)
3
#
cos# 3) ‰ d)
ˆcos# 3) ‰ Écos# ˆ 3) ‰ sin# ˆ 3) ‰ d) œ '
1Î4
0
cos# ˆ 3) ‰ d)
3
8
(1 sin 2))"Î# (2 cos 2)) œ (cos 2))(1 sin 2))"Î# ; therefore
d) œ '0
È2 d) œ ’È2 )“
È
1 2
1È#
!
#
sin 2) cos
É 1 2 sin 2)1
sin 2)
#
2)
d)
œ 21
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.6 Conic Sections
29. Let r œ f()). Then x œ f()) cos ) Ê
dx
d)
679
‰# œ cf w ()) cos ) f()) sin )d#
œ f w ()) cos ) f()) sin ) Ê ˆ dx
d)
œ cf w ())d# cos# ) 2f w ()) f()) sin ) cos ) [f())]# sin# ); y œ f()) sin ) Ê
#
dy
d)
œ f w ()) sin ) f()) cos )
#
#
w
w
#
w
#
#
Ê Š dy
d) ‹ œ cf ()) sin ) f()) cos )d œ cf ())d sin ) 2f ())f()) sin ) cos ) [f())] cos ). Therefore
#
#
#
w
#
#
#
#
#
w
#
#
ˆ dx
‰# Š dy
ˆ dr ‰#
d)
d) ‹ œ cf ())d acos ) sin )b [f())] acos ) sin )b œ cf ())d [f())] œ r d) .
' Ér# ˆ ddr) ‰# d).
‰# Š dy
Thus, L œ '! ʈ dx
d)
d) ‹ d) œ !
"
30. (a) r œ a Ê
"
#
œ 0; Length œ '0 Èa# 0# d) œ '0 kak d) œ ca)d #!1 œ 21a
21
dr
d)
(b) r œ a cos ) Ê
dr
d)
œ a sin ); Length œ '0 È(a cos ))# (a sin ))# d) œ '0 Èa# acos# ) sin# )b d)
dr
d)
œ a cos ); Length œ '0 È(a cos ))# (a sin ))# d) œ '0 Èa# acos# ) sin# )b d)
1
œ '0 kak d) œ ca)d 1! œ 1a
1
(c) r œ a sin ) Ê
21
1
1
1
œ '0 kak d) œ ca)d 1! œ 1a
1
'021 a(1 cos )) d) œ 2a1 c) sin )d #!1 œ a
21
rav œ 21"0 '0 a d) œ #"1 ca)d #!1 œ a
1Î2
1Î#
rav œ ˆ 1 ‰"ˆ 1 ‰ 'c1Î2 a cos ) d) œ 1" ca sin )d 1Î# œ 2a
1
31. (a) rav œ
(b)
(c)
"
210
#
#
32. r œ 2f()), ! Ÿ ) Ÿ " Ê
dr
d)
œ 2f w ()) Ê r# ˆ ddr) ‰ œ [2f())]# c2f w ())d# Ê Length œ '! É4[f())]# 4 cf w ())d# d)
"
#
œ 2 '! É[f())]# cf w ())d# d) which is twice the length of the curve r œ f()) for ! Ÿ ) Ÿ " .
"
11.6 CONIC SECTIONS
1. x œ
y#
8
Ê 4p œ 8 Ê p œ 2; focus is (2ß 0), directrix is x œ 2
#
2. x œ y4 Ê 4p œ 4 Ê p œ 1; focus is (1ß 0), directrix is x œ 1
#
3. y œ x6 Ê 4p œ 6 Ê p œ
4. y œ
x#
2
Ê 4p œ 2 Ê p œ
1
#
3
#
; focus is ˆ!ß 3# ‰ , directrix is y œ
3
#
; focus is ˆ!ß #1 ‰ , directrix is y œ 1#
5.
x#
4
y#
9
œ 1 Ê c œ È4 9 œ È13 Ê foci are Š „ È13ß !‹ ; vertices are a „ 2ß 0b ; asymptotes are y œ „ 3# x
6.
x#
4
y#
9
œ 1 Ê c œ È9 4 œ È5 Ê foci are Š0ß „ È5‹ ; vertices are a0ß „ 3b
7.
x#
2
y# œ 1 Ê c œ È2 1 œ 1 Ê foci are a „ 1ß 0b ; vertices are Š „ È2ß !‹
8.
y#
4
x# œ 1 Ê c œ È4 1 œ È5 Ê foci are Š0ß „ È5‹ ; vertices are a!ß „ 2b ; asymptotes are y œ „ 2x
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
680
Chapter 11 Parametric Equatins and Polar Coordinates
9. y# œ 12x Ê x œ
y#
1#
#
10. x# œ 6y Ê y œ x6 Ê 4p œ 6 Ê p œ
focus is ˆ!ß 3# ‰ , directrix is y œ 3#
Ê 4p œ 12 Ê p œ 3;
focus is ($ß !), directrix is x œ 3
11. x# œ 8y Ê y œ
x#
8
focus is ˆ!ß
" ‰
16 ,
x#
ˆ 4" ‰
Ê 4p œ
"
4
Ê pœ
directrix is y œ
#
#
focus is ˆ
"
‰
1# ß ! ,
directrix is x œ
#
14. y œ 8x# Ê y œ ˆx" ‰ Ê 4p œ
;
8
"
16
15. x œ 3y# Ê x œ ˆy" ‰ Ê 4p œ
3
"
16
"
3
"
1#
focus is ˆ!ß
Ê pœ
;
y
12. y# œ 2x Ê x œ #
Ê 4p œ 2 Ê p œ
"
ˆ
‰
focus is # ß ! , directrix is x œ "#
Ê 4p œ 8 Ê p œ 2;
focus is (!ß 2), directrix is y œ 2
13. y œ 4x# Ê y œ
3
#
"
1#
;
" ‰
32 ,
16. x œ 2y# Ê x œ
focus is
ˆ "8 ß !‰ ,
Ê 4p œ
"
#
directrix is x œ
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Ê pœ
"
8
;
Ê pœ
"
3#
directrix is y œ
y#
ˆ "# ‰
"
8
"
#
"
8
;
"
32
;
Section 11.6 Conic Sections
#
#
y
17. 16x# 25y# œ 400 Ê #x5 16
œ1
Ê c œ Èa# b# œ È25 16 œ 3
#
19. 2x# y# œ 2 Ê x# y# œ 1
Ê c œ Èa# b# œ È2 1 œ 1
#
#
21. 3x# 2y# œ 6 Ê x# y3 œ 1
Ê c œ Èa# b# œ È3 2 œ 1
#
#
x
18. 7x# 16y# œ 112 Ê 16
y7 œ 1
Ê c œ Èa# b# œ È16 7 œ 3
#
#
20. 2x# y# œ 4 Ê x# y4 œ 1
Ê c œ Èa# b# œ È4 2 œ È2
#
#
x
22. 9x# 10y# œ 90 Ê 10
y9 œ 1
Ê c œ Èa# b# œ È10 9 œ 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
681
682
Chapter 11 Parametric Equations and Polar Coordinates
#
#
23. 6x# 9y# œ 54 Ê x9 y6 œ 1
Ê c œ Èa# b# œ È9 6 œ È3
#
#
y
x
24. 169x# 25y# œ 4225 Ê 25
169
œ1
Ê c œ Èa# b# œ È169 25 œ 12
#
25. Foci: Š „ È2ß !‹ , Vertices: a „ 2ß 0b Ê a œ 2, c œ È2 Ê b# œ a# c# œ 4 ŠÈ2‹ œ 2 Ê
26. Foci: a!ß „ 4b , Vertices: a0ß „ 5b Ê a œ 5, c œ 4 Ê b# œ 25 16 œ 9 Ê
27. x# y# œ 1 Ê c œ Èa# b# œ È1 1 œ È2 ;
asymptotes are y œ „ x
#
#
29. y# x# œ 8 Ê y8 x8 œ 1 Ê c œ Èa# b#
œ È8 8 œ 4; asymptotes are y œ „ x
x#
9
#
y#
#5
x#
4
y#
#
œ1
#
x
28. 9x# 16y# œ 144 Ê 16
y9 œ 1
Ê c œ Èa# b# œ È16 9 œ 5;
asymptotes are y œ „ 34 x
#
#
30. y# x# œ 4 Ê y4 x4 œ 1 Ê c œ Èa# b#
œ È4 4 œ 2È2; asymptotes are y œ „ x
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
œ1
Section 11.6 Conic Sections
683
31. 8x# 2y# œ 16 Ê x# y8 œ 1 Ê c œ Èa# b#
œ È2 8 œ È10 ; asymptotes are y œ „ 2x
32. y# 3x# œ 3 Ê y3 x# œ 1 Ê c œ Èa# b#
œ È3 1 œ 2; asymptotes are y œ „ È3x
#
#
33. 8y# 2x# œ 16 Ê y# x8 œ 1 Ê c œ Èa# b#
œ È2 8 œ È10 ; asymptotes are y œ „ x
y
x
34. 64x# 36y# œ 2304 Ê 36
64
œ 1 Ê c œ È a# b #
œ È36 64 œ 10; asymptotes are y œ „ 4
#
#
#
#
#
#
3
35. Foci: Š!ß „ È2‹ , Asymptotes: y œ „ x Ê c œ È2 and
a
b
œ 1 Ê a œ b Ê c# œ a# b# œ 2a# Ê 2 œ 2a#
Ê a œ 1 Ê b œ 1 Ê y# x# œ 1
36. Foci: a „ 2ß !b , Asymptotes: y œ „
Ê 4œ
4a#
3
"
È3
x Ê c œ 2 and
Ê a# œ 3 Ê a œ È3 Ê b œ 1 Ê
x#
3
b
a
œ
"
È3
Ê bœ
a
È3
4
3
Ê c# œ a# b# œ a#
y# œ 1
37. Vertices: a „ 3ß 0b , Asymptotes: y œ „ 43 x Ê a œ 3 and
b
a
œ
4
3
Ê bœ
(3) œ 4 Ê
38. Vertices: a!ß „ 2b , Asymptotes: y œ „ 12 x Ê a œ 2 and
a
b
œ
1
2
Ê b œ 2(2) œ 4 Ê
x#
9
y#
4
39. (a) y# œ 8x Ê 4p œ 8 Ê p œ 2 Ê directrix is x œ 2,
focus is (#ß !), and vertex is (!ß 0); therefore the new
directrix is x œ 1, the new focus is (3ß 2), and the
new vertex is (1ß 2)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
y#
16
x#
16
œ1
œ1
a#
3
œ
4a#
3
684
Chapter 11 Parametric Equations and Polar Coordinates
40. (a) x# œ 4y Ê 4p œ 4 Ê p œ 1 Ê directrix is y œ 1,
focus is (!ß 1), and vertex is (!ß 0); therefore the new
directrix is y œ 4, the new focus is (1ß 2), and the
new vertex is (1ß 3)
41. (a)
x#
16
y#
9
œ 1 Ê center is (!ß 0), vertices are (4ß 0)
(b)
(b)
and (%ß !); c œ Èa# b# œ È7 Ê foci are ŠÈ7ß 0‹
and ŠÈ7ß !‹ ; therefore the new center is (%ß $), the
new vertices are (!ß 3) and (8ß 3), and the new foci are
Š4 „ È7ß $‹
42. (a)
x#
9
y#
25
œ 1 Ê center is (!ß 0), vertices are (0ß 5)
and (0ß 5); c œ Èa# b# œ È16 œ 4 Ê foci are
(b)
(!ß 4) and (!ß 4) ; therefore the new center is (3ß 2),
the new vertices are (3ß 3) and (3ß 7), and the new
foci are (3ß 2) and (3ß 6)
43. (a)
x#
16
y#
9
œ 1 Ê center is (!ß 0), vertices are (4ß 0)
(b)
and (4ß 0), and the asymptotes are œ „ or
Èa# b# œ È25 œ 5 Ê foci are
y œ „ 3x
4 ;cœ
x
4
y
3
(5ß 0) and (5ß 0) ; therefore the new center is (2ß 0), the
new vertices are (2ß 0) and (6ß 0), the new foci
are (3ß 0) and (7ß 0), and the new asymptotes are
yœ „
3(x 2)
4
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.6 Conic Sections
y#
4
44. (a)
x#
5
œ 1 Ê center is (!ß 0), vertices are (0ß 2)
and (0ß 2), and the asymptotes are
yœ „
2x
È5
y
2
œ „
x
È5
685
(b)
or
; c œ Èa# b# œ È9 œ 3 Ê foci are
(0ß 3) and (0ß 3) ; therefore the new center is (0ß 2),
the new vertices are (0ß 4) and (0ß 0), the new foci
are (0ß 1) and (0ß 5), and the new asymptotes are
2x
y2œ „ È
5
45. y# œ 4x Ê 4p œ 4 Ê p œ 1 Ê focus is ("ß 0), directrix is x œ 1, and vertex is (0ß 0); therefore the new
vertex is (2ß 3), the new focus is (1ß 3), and the new directrix is x œ 3; the new equation is
(y 3)# œ 4(x 2)
46. y# œ 12x Ê 4p œ 12 Ê p œ 3 Ê focus is (3ß 0), directrix is x œ 3, and vertex is (0ß 0); therefore the new
vertex is (4ß 3), the new focus is (1ß 3), and the new directrix is x œ 7; the new equation is (y 3)# œ 12(x 4)
47. x# œ 8y Ê 4p œ 8 Ê p œ 2 Ê focus is (0ß 2), directrix is y œ 2, and vertex is (0ß 0); therefore the new
vertex is (1ß 7), the new focus is (1ß 5), and the new directrix is y œ 9; the new equation is
(x 1)# œ 8(y 7)
Ê focus is ˆ!ß #3 ‰ , directrix is y œ #3 , and vertex is (0ß 0); therefore the new
vertex is (3ß 2), the new focus is ˆ3ß "# ‰ , and the new directrix is y œ 7# ; the new equation is
48. x# œ 6y Ê 4p œ 6 Ê p œ
3
#
(x 3)# œ 6(y 2)
49.
x#
6
y#
9
œ 1 Ê center is (!ß 0), vertices are (0ß 3) and (!ß 3); c œ Èa# b# œ È9 6 œ È3 Ê foci are Š!ß È3‹
and Š!ß È3‹ ; therefore the new center is (#ß 1), the new vertices are (2ß 2) and (#ß 4), and the new foci
are Š#ß 1 „ È3‹ ; the new equation is
50.
x#
2
(x 2)#
6
(y 1)#
9
œ1
y# œ 1 Ê center is (!ß 0), vertices are ŠÈ2ß !‹ and ŠÈ2ß !‹ ; c œ Èa# b# œ È2 1 œ 1 Ê foci are
(1ß 0) and ("ß !); therefore the new center is (3ß 4), the new vertices are Š3 „ È2ß 4‹ , and the new foci are (2ß 4)
and (4ß 4); the new equation is
51.
x#
3
y#
#
(x 3)#
#
(y 4)# œ 1
œ 1 Ê center is (!ß 0), vertices are ŠÈ3ß !‹ and ŠÈ3ß !‹ ; c œ Èa# b# œ È3 2 œ 1 Ê foci are
(1ß 0) and ("ß !); therefore the new center is (2ß 3), the new vertices are Š2 „ È3ß 3‹ , and the new foci are (1ß 3)
and (3ß 3); the new equation is
52.
x#
16
y#
#5
(x 2)#
3
(y 3)#
#
œ1
œ 1 Ê center is (!ß 0), vertices are (!ß &) and (!ß 5); c œ Èa# b# œ È25 16 œ 3 Ê foci are
(0ß 3) and (0ß 3); therefore the new center is (4ß 5), the new vertices are (4ß 0) and (4ß 10), and the new
foci are (4ß 2) and (4ß 8); the new equation is
53.
x#
4
y#
5
(x 4)#
16
(y 5)#
#5
œ1
œ 1 Ê center is (!ß 0), vertices are (2ß 0) and (2ß 0); c œ Èa# b# œ È4 5 œ 3 Ê foci are ($ß !) and
(3ß 0); the asymptotes are „
x
#
œ
y
È5
Ê yœ „
È5x
#
; therefore the new center is (2ß 2), the new vertices are
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
686
Chapter 11 Parametric Equations and Polar Coordinates
(4ß 2) and (0ß 2), and the new foci are (5ß 2) and (1ß 2); the new asymptotes are y 2 œ „
equation is
54.
x#
16
y#
9
(x 2)#
4
(y 2)#
5
È5 (x 2)
#
; the new
œ1
œ 1 Ê center is (!ß 0), vertices are (4ß 0) and (4ß 0); c œ Èa# b# œ È16 9 œ 5 Ê foci are (5ß !)
and (5ß 0); the asymptotes are „
œ
x
4
Ê yœ „
y
3
3x
4
; therefore the new center is (5ß 1), the new vertices are
(1ß 1) and (9ß 1), and the new foci are (10ß 1) and (0ß 1); the new asymptotes are y 1 œ „
the new equation is
(x 5)
16
#
#
(y 1)
9
3(x 5)
4
;
œ1
55. y# x# œ 1 Ê center is (!ß 0), vertices are (0ß 1) and (0ß 1); c œ Èa# b# œ È1 1 œ È2 Ê foci are
Š!ß „ È2‹ ; the asymptotes are y œ „ x; therefore the new center is (1ß 1), the new vertices are (1ß 0) and
(1ß 2), and the new foci are Š1ß 1 „ È2‹ ; the new asymptotes are y 1 œ „ (x 1); the new equation is
(y 1)# (x 1)# œ 1
56.
y#
3
x# œ 1 Ê center is (!ß 0), vertices are Š0ß È3‹ and Š!ß È3‹ ; c œ Èa# b# œ È3 1 œ 2 Ê foci are (!ß #)
and (!ß 2); the asymptotes are „ x œ
y
È3
Ê y œ „ È3x; therefore the new center is (1ß 3), the new vertices are
Š"ß $ „ È3‹ , and the new foci are ("ß &) and (1ß 1); the new asymptotes are y 3 œ „ È3 (x 1); the new equation is
(y 3)#
3
(x 1)# œ 1
57. x# 4x y# œ 12 Ê x# 4x 4 y# œ 12 4 Ê (x 2)# y# œ 16; this is a circle: center at C(2ß 0), a œ 4
58. 2x# 2y# 28x 12y 114 œ 0 Ê x# 14x 49 y# 6y 9 œ 57 49 9 Ê (x 7)# (y 3)# œ 1;
this is a circle: center at C(7ß 3), a œ 1
59. x# 2x 4y 3 œ 0 Ê x# 2x 1 œ 4y 3 1 Ê (x 1)# œ 4(y 1); this is a parabola: V(1ß 1), F(1ß 0)
60. y# 4y 8x 12 œ 0 Ê y# 4y 4 œ 8x 12 4 Ê (y 2)# œ 8(x 2); this is a parabola: V(#ß 2), F(!ß #)
61. x# 5y# 4x œ 1 Ê x# 4x 4 5y# œ 5 Ê (x 2)# 5y# œ 5 Ê
(x 2)#
5
y# œ 1; this is an ellipse: the
center is (2ß 0), the vertices are Š2 „ È5ß 0‹ ; c œ Èa# b# œ È5 1 œ 2 Ê the foci are (4ß 0) and (!ß 0)
62. 9x# 6y# 36y œ 0 Ê 9x# 6 ay# 6y 9b œ 54 Ê 9x# 6(y 3)# œ 54 Ê
x#
6
(y 3)#
9
œ 1; this is an ellipse:
the center is (0ß 3), the vertices are (!ß 0) and (!ß 6); c œ Èa# b# œ È9 6 œ È3 Ê the foci are Š0ß 3 „ È3‹
63. x# 2y# 2x 4y œ 1 Ê x# 2x 1 2 ay# 2y 1b œ 2 Ê (x 1)# 2(y 1)# œ 2
#
Ê (x1) (y 1)# œ 1; this is an ellipse: the center is (1ß 1), the vertices are Š" „ È2ß "‹ ;
2
c œ Èa# b# œ È2 1 œ 1 Ê the foci are (2ß 1) and (0ß 1)
64. 4x# y# 8x 2y œ 1 Ê 4 ax# 2x 1b y# 2y 1 œ 4 Ê 4(x 1)# (y 1)# œ 4
Ê (x 1)#
(y1)#
4
œ 1; this is an ellipse: the center is (1ß 1), the vertices are (1ß 3) and
(1ß 1); c œ Èa# b# œ È4 1 œ È3 Ê the foci are Š1ß " „ È3‹
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.6 Conic Sections
65. x# y# 2x 4y œ 4 Ê x# 2x 1 ay# 4y 4b œ 1 Ê (x 1)# (y 2)# œ 1; this is a hyperbola:
the center is (1ß 2), the vertices are (2ß 2) and (!ß 2); c œ Èa# b# œ È1 1 œ È2 Ê the foci are Š1 „ È2ß #‹ ;
the asymptotes are y 2 œ „ (x 1)
66. x# y# 4x 6y œ 6 Ê x# 4x 4 ay# 6y 9b œ 1 Ê (x 2)# (y 3)# œ 1; this is a hyperbola:
the center is (2ß 3), the vertices are (1ß 3) and (3ß 3); c œ Èa# b# œ È1 1 œ È2 Ê the foci are
Š2 „ È2ß 3‹ ; the asymptotes are y 3 œ „ (x 2)
(y 3)#
6
67. 2x# y# 6y œ 3 Ê 2x# ay# 6y 9b œ 6 Ê
x#
3
œ 1; this is a hyperbola: the center is (!ß $),
the vertices are Š!ß 3 „ È6‹ ; c œ Èa# b# œ È6 3 œ 3 Ê the foci are (0ß 6) and (!ß 0); the asymptotes are
y 3
È6
œ „
x
È3
Ê y œ „ È2x 3
68. y# 4x# 16x œ 24 Ê y# 4 ax# 4x 4b œ 8 Ê
y#
8
(x 2)#
2
œ 1; this is a hyperbola: the center is (2ß 0),
the vertices are Š2ß „ È8‹ ; c œ Èa# b# œ È8 2 œ È10 Ê the foci are Š2ß „ È10‹ ; the asymptotes are
y
È8
x 2
È2
œ „
Ê y œ „ 2(x 2)
y#
k
69. (a) y# œ kx Ê x œ
; the volume of the solid formed by
Èkx
revolving R" about the y-axis is V" œ '0
œ
1
k#
Èkx
'0
y% dy œ
1x# Èkx
5
#
#
1 Š yk ‹ dy
; the volume of the right
circular cylinder formed by revolving PQ about the
y-axis is V# œ 1x# Èkx Ê the volume of the solid
formed by revolving R# about the y-axis is
V$ œ V# V" œ
41x# Èkx
5
. Therefore we can see the
ratio of V$ to V" is 4:1.
(b) The volume of the solid formed by revolving R# about the x-axis is V" œ '0 1 ŠÈkt‹ dt œ 1k'0 t dt
x
œ
1kx#
#
#
x
. The volume of the right circular cylinder formed by revolving PS about the x-axis is
#
V# œ 1 ŠÈkx‹ x œ 1kx# Ê the volume of the solid formed by revolving R" about the x-axis is
V$ œ V# V" œ 1kx#
70. y œ '
w
H
x dx œ
w
H
#
1kx#
#
Š x# ‹ C œ
wx#
2H
œ
1kx#
#
. Therefore the ratio of V$ to V" is 1:1.
C; y œ 0 when x œ 0 Ê 0 œ
w(0)#
2H
C Ê C œ 0; therefore y œ
wx#
2H
is the
equation of the cable's curve
71. x# œ 4py and y œ p Ê x# œ 4p# Ê x œ „ 2p. Therefore the line y œ p cuts the parabola at points (2pß p) and
(2pß p), and these points are È[2p (2p)]# (p p)# œ 4p units apart.
72. x lim
Š b x ba Èx# a# ‹ œ
Ä_ a
œ
b
a x lim
Ä_
ax# a# b
“
x È x# a#
’x
#
œ
b
a x lim
Ä_
b
a x lim
Ä_
’
Šx È x # a# ‹ œ
a#
“
x È x# a#
b
a x lim
Ä_
–
Šx Èx# a# ‹ Šx Èx# a# ‹
x È x # a#
œ0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
—
687
688
Chapter 11 Parametric Equations and Polar Coordinates
73. Let y œ É1
x#
4
on the interval 0 Ÿ x Ÿ 2. The area of the inscribed rectangle is given by
x#
4‹
A(x) œ 2x Š2É1
Ê Aw (x) œ 4É1
x#
4
œ 4xÉ1
x#
É1 x4#
x#
4
(since the length is 2x and the height is 2y)
x#
4
. Thus Aw (x) œ 0 Ê 4É1
x#
É1 x4#
œ 0 Ê 4 Š1
x#
4‹
x# œ 0 Ê x# œ 2
Ê x œ È2 (only the positive square root lies in the interval). Since A(0) œ A(2) œ 0 we have that A ŠÈ2‹ œ 4
is the maximum area when the length is 2È2 and the height is È2.
74. (a) Around the x-axis: 9x# 4y# œ 36 Ê y# œ 9 94 x# Ê y œ „ É9 94 x# and we use the positive root
#
Ê V œ 2 '0 1 ŠÉ9 94 x# ‹ dx œ 2 '0 1 ˆ9 94 x# ‰ dx œ 21 9x 34 x$ ‘ ! œ 241
2
2
#
(b) Around the y-axis: 9x# 4y# œ 36 Ê x# œ 4 49 y# Ê x œ „ É4 49 y# and we use the positive root
#
Ê V œ 2'0 1 ŠÉ4 49 y# ‹ dy œ 2 '0 1 ˆ4 49 y# ‰ dy œ 21 4y
3
75. 9x# 4y# œ 36 Ê y# œ
œ
91
4
9x# 36
4
'24 ax# 4b dx œ 941 ’ x3
$
3
4
27
$
y$ ‘ ! œ 161
Ê y œ „ 3# Èx# 4 on the interval 2 Ÿ x Ÿ 4 Ê V œ '2 1 Š #3 Èx# 4‹ dx
#
4
%
4x“ œ
#
91
4
ˆ 64
‰ ˆ8
‰‘ œ
3 16 3 8
91
4
ˆ 56
‰
3 8 œ
31
4
(56 24) œ 241
76. Let P" (pß y" ) be any point on x œ p, and let P(xß y) be a point where a tangent intersects y# œ 4px. Now
y# œ 4px Ê 2y
dy
dx
œ 4p Ê
dy
dx
œ
2p
y
Ê y# yy" œ 2px 2p# . Since x œ
Ê
"
#
y# yy" 2p# œ 0 Ê y œ
tangents from P" are m" œ
; then the slope of a tangent line from P" is
y#
4p
œ
dy
dx
#
œ
y
, we have y# yy" œ 2p Š 4p
‹ 2p# Ê y# yy" œ
2y" „ È4y#" 16p#
#
2p
y" Èy#" 4p#
y y"
x (p)
and m# œ
"
#
2p
y
y# 2p#
œ y" „ Èy"# 4p# . Therefore the slopes of the two
Ê m" m# œ
2p
y" Èy#" 4p#
4p#
y#" ay#" 4p# b
œ 1
Ê the lines are perpendicular
77. (x 2)# (y 1)# œ 5 Ê 2(x 2) 2(y 1)
dy
dx
œ0 Ê
dy
dx
2
#
#
œ xy
1 ; y œ 0 Ê (x 2) (0 1) œ 5
Ê (x 2)# œ 4 Ê x œ 4 or x œ 0 Ê the circle crosses the x-axis at (4ß 0) and (!ß 0); x œ 0
Ê (0 2)# (y 1)# œ 5 Ê (y 1)# œ 1 Ê y œ 2 or y œ 0 Ê the circle crosses the y-axis at (!ß 2) and (!ß !).
At (4ß 0):
At (!ß !):
At (!ß #):
2
œ 40
1 œ 2 Ê the tangent line is y œ 2(x 4) or y œ 2x 8
dy
dx
dy
dx
dy
dx
2
œ 00
1 œ 2 Ê the tangent line is y œ 2x
2
œ 02
1 œ 2 Ê the tangent line is y 2 œ 2x or y œ 2x 2
78. x# y# œ 1 Ê x œ „ È1 y# on the interval 3 Ÿ y Ÿ 3 Ê V œ 'c3 1 ˆÈ1 y# ‰ dy œ 2'0 1 ˆÈ1 y# ‰ dy
3
œ 21'0 a1 y# b dy œ 21 ’y
3
79. Let y œ É16
vertical strip:
16
9
$
y$
3 “!
É16
aµ
x ßµ
y b œ xß
#
#
É16 16
9 x
#
#
3
œ 241
x# on the interval 3 Ÿ x Ÿ 3. Since the plate is symmetric about the y-axis, x œ 0. For a
Ê mass œ dm œ $ dA œ $É16
µ
y dm œ
#
Š$ É16
16
9
16
9
16
9
x#
, length œ É16
16
9
x# , width œ dx Ê area œ dA œ É16
x# dx. Moment of the strip about the x-axis:
x# ‹ dx œ $ ˆ8 98 x# ‰ dx so the moment of the plate about the x-axis is
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
16
9
x# dx
Section 11.7 Conics in Polar Coordinates
Mx œ ' µ
y dm œ 'c3 $ ˆ8 89 x# ‰ dx œ $ 8x
3
M œ 'c3 $ É16
3
16
9
8
27
$
x$ ‘ $ œ 32$ ; also the mass of the plate is
x# dx œ 'c3 4$ É1 ˆ "3 x‰ dx œ 4$ 'c1 3È1 u# du where u œ
3
689
1
#
x
3
Ê 3 du œ dx; x œ 3
Ê u œ 1 and x œ 3 Ê u œ 1. Hence, 4$ 'c1 3È1 u# du œ 12$ 'c1 È1 u# du
1
œ 12$ ’ "2 ŠuÈ1 u# sin" u‹“
80. y œ Èx# 1 Ê
dy
dx
"
#
œ
È2
1
'
œ É 2x
x# 1 Ê S œ 0
#
–
u œ È2x
— Ä
du œ È2 dx
21
È2
ax# 1b
"
"
1
œ 61$ Ê y œ
"Î#
(2x) œ
x
È x# 1
Mx
M
œ
32$
61$
#
œ
Ê Š dy
dx ‹ œ
È2
16
31
. Therefore the center of mass is ˆ!ß 3161 ‰ .
x#
x # 1
#
É1
Ê Ê1 Š dy
dx ‹ œ
È2
1
È #
È #
'
'
É 2x
21yÊ1 Š dy
dx ‹ dx œ 0 21 x 1
x# 1 dx œ 0 21 2x 1 dx ;
#
'02 Èu# 1 du œ È21
2
#
#
’ 2" ŠuÈu# 1 ln Šu Èu# 1‹‹“ œ
!
1
È2
81. (a) tan " œ mL Ê tan " œ f w (x! ) where f(x) œ È4px ;
f w (x) œ
œ
2p
y!
"
#
(4px)"Î# (4p) œ
(c) tan ! œ
œ
2p
È4px
Ê f w (x! ) œ
2p
È4px!
Ê tan " œ
(b) tan 9 œ mFP œ
2p
y! .
y! 0
y!
x! p œ x! p
tan 9 tan "
1 tan 9 tan "
y#! 2p(x! p)
y! (x! p 2p)
œ
œ
y
Š x ! p c y2p ‹
!
!
y
1 b Š x ! p ‹ Š y2p ‹
!
!
4px! 2px! 2p#
y! (x! p)
œ
2p(x! p)
y! (x! p)
œ
2p
y!
11.7 CONICS IN POLAR COORDINATES
#
y#
1. 16x# 25y# œ 400 Ê #x5 16
œ 1 Ê c œ È a# b #
œ È25 16 œ 3 Ê e œ ca œ 35 ; F a „ 3ß 0b ;
directrices are x œ 0 „
a
e
œ „
5
ˆ 35 ‰
œ „
25
3
#
x#
2. 7x# 16y# œ 112 Ê 16
y7 œ 1 Ê c œ Èa# b#
œ È16 7 œ 3 Ê e œ ca œ 34 ; F a „ 3ß 0b ;
directrices are x œ 0 „
a
e
x#
x# 1
œ „
4
ˆ 34 ‰
œ „
16
3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
’2È5 ln Š2 È5‹“
690
Chapter 11 Parametric Equations and Polar Coordinates
3. 2x# y# œ 2 Ê x# y2 œ 1 Ê c œ Èa# b#
œ È2 1 œ 1 Ê e œ ca œ È1 ; F a0ß „ 1b ;
#
directrices are y œ 0 „
4. 2x# y# œ 4 Ê
x#
#
a
e
œ „
a
e
œ „2
Š È12 ‹
œ 1 Ê c œ Èa# b#
y#
4
œ È4 2 œ È2 Ê e œ
directrices are y œ 0 „
2
È2
c
a
œ
È2
2
; F Š0ß „ È2‹ ;
œ „ È22 œ „ 2È2
Š ‹
2
#
#
5. 3x# 2y# œ 6 Ê x# y3 œ 1 Ê c œ Èa# b#
œ È3 2 œ 1 Ê e œ ca œ È13 ; F a0ß „ 1b ;
directrices are y œ 0 „
a
e
œ „
È3
œ „3
Š È13 ‹
#
x#
6. 9x# 10y# œ 90 Ê 10
y9 œ 1 Ê c œ Èa# b#
œ È10 9 œ 1 Ê e œ ca œ È110 ; F a „ 1ß 0b ;
directrices are x œ 0 „
7. 6x# 9y# œ 54 Ê
x#
9
a
e
œ „
y#
6
œ È9 6 œ È3 Ê e œ
directrices are x œ 0 „
a
e
È10
Š È110 ‹
œ „ 10
œ 1 Ê c œ Èa# b#
c
a
œ
È3
3
; F Š „ È3ß 0‹ ;
œ „ È33 œ „ 3È3
Š ‹
3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.7 Conics in Polar Coordinates
691
y#
x#
8. 169x# 25y# œ 4225 Ê 25
169
œ 1 Ê c œ Èa# b#
œ È169 25 œ 12 Ê e œ c œ 12 ; F a0ß „ 12b ;
a
directrices are y œ 0 „
a
e
œ „
13
13
ˆ 12
‰
13
œ „
169
12
x#
#7
y#
36
9. Foci: a0ß „ 3b , e œ 0.5 Ê c œ 3 and a œ
c
e
œ
3
0.5
œ 6 Ê b# œ 36 9 œ 27 Ê
10. Foci: a „ 8ß 0b , e œ 0.2 Ê c œ 8 and a œ
c
e
œ
8
0.#
œ 40 Ê b# œ 1600 64 œ 1536 Ê
œ1
x#
1600
y#
1536
11. Vertices: a0ß „ 70b , e œ 0.1 Ê a œ 70 and c œ ae œ 70(0.1) œ 7 Ê b# œ 4900 49 œ 4851 Ê
œ1
x#
4851
12. Vertices: a „ 10ß 0b , e œ 0.24 Ê a œ 10 and c œ ae œ 10(0.24) œ 2.4 Ê b# œ 100 5.76 œ 94.24 Ê
13. Focus: ŠÈ5ß !‹ , Directrix: x œ
Ê eœ
È5
3
. Then PF œ
È5
3
Ê x# 2È5 x 5 y# œ
14. Focus: (%ß 0), Directrix: x œ
PF
œ
È
œ #3 PD
3 ˆ #
32
4 x 3
9
È5
Ê c œ ae œ È5 and
256 ‰
9
Ê
"
4
œ
#
5
9
Šx#
16
3
18
È5
x
Ê
81
5 ‹
Ê c œ ae œ 4 and
x# y# œ
16
3
È3
#
¸x
Ê
#
x
ˆ 64
‰
3
4
9
a
e
È5
3
16
3
Ê
œ
ae
e#
16
3
œ
ae
e#
¹x
x# y# œ 4 Ê
œ
16 ¸
3
Ê
9
È5
PD Ê ÊŠx È5‹ (y 0)# œ
Ê È(x 4)# (y 0)# œ
x
a
e
9
È5 ¹
x#
9
Ê
Ê (x 4)# y# œ
#
y
ˆ 16
‰
3
9
È5
y#
4
4
e#
3
4
Ê
È5
e#
œ
x#
100
Ê e# œ
9
È5
#
Ê Šx È5‹ y# œ
5
9
y#
4900
y#
94.24
#
1
4
#
ax# 32x 256b Ê
3
4
x# y# œ 48 Ê
Šx
œ
"
#
1
È2
. Then PF œ
#
1
È2
#
œ1
œ
16
3
ˆx
Ê e# œ
16 ‰#
3
Ê eœ
3
4
È3
#
. Then
Ê x# 8x 16 y#
1
#
. Then
4
x#
64
y#
48
œ1
#
PD Ê ÊŠx È2‹ (y 0)# œ
Šx 2È2‹ Ê x# 2È2 x 2 y# œ
9
È5 ‹
œ1
16. Focus: ŠÈ2ß !‹ , Directrix: x œ 2È2 Ê c œ ae œ È2 and
Ê eœ
œ1
5
9
"
4
#
15. Focus: (%ß 0), Directrix: x œ 16 Ê c œ ae œ 4 and ae œ 16 Ê ae
e# œ 16 Ê e# œ 16 Ê e œ 4 Ê e œ
PF œ 1 PD Ê È(x 4)# (y 0)# œ 1 kx 16k Ê (x 4)# y# œ 1 (x 16)# Ê x# 8x 16 y#
œ
œ1
"
#
a
e
œ 2È 2 Ê
1
È2
ae
e#
œ 2È 2 Ê
È2
e#
œ 2 È 2 Ê e# œ
#
¹x 2È2¹ Ê Šx È2‹ y#
Šx# 4È2 x 8‹ Ê
"
#
x# y# œ 2 Ê
x#
4
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
y#
#
œ1
"
#
692
Chapter 11 Parametric Equations and Polar Coordinates
17. x# y# œ 1 Ê c œ Èa# b# œ È1 1 œ È2 Ê e œ
œ
È2
1
c
a
œ È2 ; asymptotes are y œ „ x; F Š „ È2 ß !‹ ;
directrices are x œ 0 „
a
e
œ „
"
È2
#
x#
18. 9x# 16y# œ 144 Ê 16
y9 œ 1 Ê c œ Èa# b#
œ È16 9 œ 5 Ê e œ ca œ 54 ; asymptotes are
y œ „ 34 x; F a „ 5ß !b ; directrices are x œ 0 „
œ „
a
e
"6
5
#
#
19. y# x# œ 8 Ê y8 x8 œ 1 Ê c œ Èa# b#
œ È8 8 œ 4 Ê e œ ca œ È48 œ È2 ; asymptotes are
y œ „ x; F a0ß „ 4b ; directrices are y œ 0 „
œ „
È8
È2
a
e
œ „2
20. y# x# œ 4 Ê
y#
4
x#
4
œ 1 Ê c œ Èa# b#
œ È 4 4 œ 2È 2 Ê e œ
c
a
œ
2È 2
2
œ È2 ; asymptotes
are y œ „ x; F Š0ß „ 2È2‹ ; directrices are y œ 0 „
œ „
2
È2
a
e
œ „ È2
21. 8x# 2y# œ 16 Ê
x#
2
y#
8
œ È2 8 œ È10 Ê e œ
œ 1 Ê c œ Èa# b#
c
a
œ
È10
È2
œ È5 ; asymptotes
are y œ „ 2x; F Š „ È10ß !‹ ; directrices are x œ 0 „
œ „
È2
È5
œ „
a
e
2
È10
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.7 Conics in Polar Coordinates
693
#
22. y# 3x# œ 3 Ê y3 x# œ 1 Ê c œ Èa# b#
œ È3 1 œ 2 Ê e œ ca œ È23 ; asymptotes are
y œ „ È3 x; F a0ß „ 2b ; directrices are y œ 0 „
œ „
È3
Š È23 ‹
œ „
a
e
3
#
23. 8y# 2x# œ 16 Ê
y#
2
x#
8
œ È2 8 œ È10 Ê e œ
œ 1 Ê c œ Èa# b#
c
a
È10
È2
œ
œ È5 ; asymptotes
are y œ „ x# ; F Š0ß „ È10‹ ; directrices are y œ 0 „
œ „
È2
È5
œ „
a
e
2
È10
y#
x#
24. 64x# 36y# œ 2304 Ê 36
64
œ 1 Ê c œ È a# b #
5
œ È36 64 œ 10 Ê e œ ca œ 10
6 œ 3 ; asymptotes are
y œ „ 43 x; F a „ 10ß !b ; directrices are x œ 0 „
œ „
6
ˆ 53 ‰
œ „
a
e
18
5
25. Vertices a!ß „ 1b and e œ 3 Ê a œ 1 and e œ
c
a
œ 3 Ê c œ 3a œ 3 Ê b# œ c# a# œ 9 1 œ 8 Ê y#
x#
8
œ1
26. Vertices a „ 2ß !b and e œ 2 Ê a œ 2 and e œ
c
a
œ 2 Ê c œ 2a œ 4 Ê b# œ c# a# œ 16 4 œ 12 Ê
x#
4
y#
1#
œ1
œ 3 Ê c œ 3a Ê a œ 1 Ê b# œ c# a# œ 9 1 œ 8 Ê x#
y#
8
œ1
27. Foci a „ 3ß !b and e œ 3 Ê c œ 3 and e œ
c
a
28. Foci a!ß „ 5b and e œ 1.25 Ê c œ 5 and e œ
œ 25 16 œ 9 Ê
#
y
16
#
x
9
c
a
œ 1.25 œ
5
4
Ê cœ
5
4
a Ê 5œ
5
4
a Ê a œ 4 Ê b# œ c# a#
œ1
29. e œ 1, x œ 2 Ê k œ 2 Ê r œ
2(1)
1 (1) cos )
œ
2
1cos )
30. e œ 1, y œ 2 Ê k œ 2 Ê r œ
2(1)
1 (1) sin )
œ
2
1sin )
31. e œ 5, y œ 6 Ê k œ 6 Ê r œ
6(5)
1 5 sin )
32. e œ 2, x œ 4 Ê k œ 4 Ê r œ
4(2)
1 2 cos )
33. e œ "# , x œ 1 Ê k œ 1 Ê r œ
ˆ "# ‰ (1)
1 ˆ "# ‰ cos )
œ
œ
30
15 sin )
8
12 cos )
œ
1
2cos )
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
694
Chapter 11 Parametric Equations and Polar Coordinates
ˆ "4 ‰ (2)
1 ˆ "4 ‰ cos )
34. e œ "4 , x œ 2 Ê k œ 2 Ê r œ
35. e œ 5" , y œ 10 Ê k œ 10 Ê r œ
36. e œ "3 , y œ 6 Ê k œ 6 Ê r œ
37. r œ
"
1 cos )
38. r œ
6
2 cos )
œ
ˆ "5 ‰ (10)
1 ˆ "5 ‰ sin )
ˆ "3 ‰ (6)
1 ˆ 3" ‰ sin )
œ
2
4cos )
œ
10
5sin )
6
3sin )
Ê e œ 1, k œ 1 Ê x œ 1
œ
3
1 ˆ "# ‰ cos )
Ê eœ
"
#
, k œ 6 Ê x œ 6;
#
a a1 e# b œ ke Ê a ’1 ˆ "# ‰ “ œ 3 Ê
3
4
aœ3
Ê a œ 4 Ê ea œ 2
39. r œ
25
10 5 cos )
Ê eœ
"
#
Ê rœ
#
40. r œ
4
22 cos )
41. r œ
400
16 8 sin )
"
#
œ
ˆ 5# ‰
1 ˆ "# ‰ cos )
, k œ 5 Ê x œ 5; a a1 e# b œ ke
Ê a ’1 ˆ "# ‰ “ œ
eœ
ˆ 25
‰
10
5 ‰
1 ˆ 10
cos )
Ê rœ
5
#
Ê
2
1cos )
Ê rœ
3
4
aœ
5
#
Ê aœ
10
3
Ê ea œ
5
3
Ê e œ 1, k œ 2 Ê x œ 2
ˆ 400
‰
16
8 ‰
1 ˆ 16
sin )
Ê rœ
25
1 ˆ "# ‰ sin )
, k œ 50 Ê y œ 50; a a1 e# b œ ke
#
Ê a ’1 ˆ "# ‰ “ œ 25 Ê
Ê ea œ
3
4
a œ 25 Ê a œ
100
3
50
3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 11.7 Conics in Polar Coordinates
42. r œ
12
3 3 sin )
Ê rœ
4
1 sin )
Ê e œ 1,
43. r œ
kœ4 Ê yœ4
44. r œ
4
2 sin )
8
2 2 sin )
Ê rœ
4
1 sin )
Ê e œ 1,
k œ 4 Ê y œ 4
Ê rœ
2
1 ˆ "# ‰ sin )
Ê eœ
"
#
,kœ4
#
Ê y œ 4; a a1 e# b œ ke Ê a ’1 ˆ "# ‰ “ œ 2
Ê
3
4
aœ2 Ê aœ
8
3
Ê ea œ
4
3
45. r cos ˆ) 14 ‰ œ È2 Ê r ˆcos ) cos 14 sin ) sin 14 ‰
œ È2 Ê " r cos ) " r sin ) œ È2 Ê " x
È2
È2
È2
"
È2
y
œ È2 Ê x y œ 2 Ê y œ 2 x
46. r cos ˆ)
Ê
31 ‰
4
œ 1 Ê r ˆcos ) cos
È2
2
r cos )
Ê y œ x È 2
47. r cos ˆ)
21 ‰
3
È3
2
sin ) sin
31 ‰
4
œ1
r sin ) œ 1 Ê x y œ È2
œ 3 Ê r ˆcos ) cos
Ê r cos )
1
2
È2
2
31
4
21
3
sin ) sin
"
#
r sin ) œ 3 Ê x
Ê x È 3 y œ 6 Ê y œ
È3
3
È3
#
21 ‰
3
œ3
yœ3
x 2È 3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
695
696
Chapter 11 Parametric Equations and Polar Coordinates
48. r cos ˆ) 13 ‰ œ 2 Ê r ˆcos ) cos
Ê
1
2
r cos )
È3
2
r sin ) œ 2 Ê
Ê x È3 y œ 4 Ê y œ
È3
3
1
3
sin ) sin 13 ‰ œ 2
"
#
x
x
4È 3
3
È3
#
yœ2
È
49. È2 x È2 y œ 6 Ê È2 r cos ) È2 r sin ) œ 6 Ê r Š #2 cos )
È2
#
sin )‹ œ 3 Ê r ˆcos
1
4
cos ) sin
œ 3 Ê r cos ˆ) 14 ‰ œ 3
È
50. È3 x y œ 1 Ê È3 r cos ) r sin ) œ 1 Ê r Š #3 cos )
œ
"
#
Ê r cos ˆ) 16 ‰ œ
1
#
sin )‹ œ
"
#
Ê r ˆcos
1
6
cos ) sin
1
6
sin )‰
"
#
51. y œ 5 Ê r sin ) œ 5 Ê r sin ) œ 5 Ê r sin ()) œ 5 Ê r cos ˆ 1# ())‰ œ 5 Ê r cos ˆ) 1# ‰ œ 5
52. x œ 4 Ê r cos ) œ 4 Ê r cos ) œ 4 Ê r cos () 1) œ 4
53.
54.
55.
56.
57. (x 6)# y# œ 36 Ê C œ (6ß 0), a œ 6
Ê r œ 12 cos ) is the polar equation
58. (x 2)# y# œ 4 Ê C œ (2ß 0), a œ 2
Ê r œ 4 cos ) is the polar equation
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1
4
sin )‰
Section 11.7 Conics in Polar Coordinates
59. x# (y 5)# œ 25 Ê C œ (!ß 5), a œ 5
Ê r œ 10 sin ) is the polar equation
60. x# (y 7)# œ 49 Ê C œ (!ß 7), a œ 7
Ê r œ 14 sin ) is the polar equation
61. x# 2x y# œ 0 Ê (x 1)# y# œ 1
Ê C œ (1ß 0), a œ 1 Ê r œ 2 cos ) is
the polar equation
62. x# 16x y# œ 0 Ê (x 8)# y# œ 64
Ê C œ (8ß 0), a œ 8 Ê r œ 16 cos ) is the
polar equation
#
63. x# y# y œ 0 Ê x# ˆy "# ‰ œ 4"
Ê C œ ˆ!ß "# ‰ , a œ "# Ê r œ sin ) is the
#
64. x# y# 34 y œ 0 Ê x# ˆy 32 ‰ œ 49
Ê C œ ˆ0ß 23 ‰ , a œ 23 Ê r œ 43 sin ) is the
polar equation
65.
polar equation
66.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
697
698
Chapter 11 Parametric Equations and Polar Coordinates
67.
68.
69.
70.
71.
72.
73.
74.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 11 Practice Exercises
75. (a) Perihelion œ a ae œ a(1 e), Aphelion œ ea a œ a(1 e)
(b)
Planet
Perihelion
Aphelion
Mercury
0.3075 AU
0.4667 AU
Venus
0.7184 AU
0.7282 AU
Earth
0.9833 AU
1.0167 AU
Mars
1.3817 AU
1.6663 AU
Jupiter
4.9512 AU
5.4548 AU
Saturn
9.0210 AU
10.0570 AU
Uranus
18.2977 AU
20.0623 AU
Neptune
29.8135 AU
30.3065 AU
(0.3871) a1 0.2056# b
0.3707
œ 1 0.2056
1 0.2056 cos )
cos )
(0.7233) a1 0.0068# b
0.7233
Venus: r œ 1 0.0068 cos ) œ 1 0.0068 cos )
0.0167# b
0.9997
Earth: r œ 11a10.0167
cos ) œ 1 0.0617 cos )
a1 0.0934# b
1.511
Mars: r œ (1.524)
œ 1 0.0934
1 0.0934 cos )
cos )
#
a1 0.0484 b
5.191
Jupiter: r œ (5.203)
œ
1 0.0484 cos )
1 0.0484 cos )
(9.539) a1 0.0543# b
9.511
Saturn: r œ 1 0.0543 cos ) œ 1 0.0543
cos )
a1 0.0460# b
19.14
Uranus: r œ (19.18)
œ
1 0.0460 cos )
1 0.0460 cos )
(30.06) a1 0.0082# b
30.06
Neptune: r œ 1 0.0082 cos ) œ 1 0.0082
cos )
76. Mercury: r œ
CHAPTER 11 PRACTICE EXERCISES
1. x œ
t
#
and y œ t 1 Ê 2x œ t Ê y œ 2x 1
"
# tan t
and y# œ "4
#
#
3. x œ
and y œ
"
#
sec t Ê x# œ
"
4
tan# t
sec# t Ê 4x# œ tan# t and
4y œ sec t Ê 4x# 1 œ 4y# Ê 4y# 4x# œ 1
2. x œ Èt and y œ 1 Èt Ê y œ 1 x
4. x œ 2 cos t and y œ 2 sin t Ê x# œ 4 cos# t and
y# œ 4 sin# t Ê x# y# œ 4
699
700
Chapter 11 Parametric Equations and Polar Coordinates
5. x œ cos t and y œ cos# t Ê y œ (x)# œ x#
6. x œ 4 cos t and y œ 9 sin t Ê x# œ 6 cos# t and
x#
16
y# œ 81 sin# t Ê
x#
9
7. 16x# 9y# œ 144 Ê
y#
16
y#
81
œ1
œ 1 Ê a œ 3 and b œ 4 Ê x œ 3 cos t and y œ 4 sin t, 0 Ÿ t Ÿ 21
8. x# y# œ 4 Ê x œ 2 cos t and y œ 2 sin t, 0 Ÿ t Ÿ 61
9. x œ
"
#
"
#
tan t, y œ
Ê xœ
"
#
tan
1
3
sec t Ê
œ
œ 2 cos3 ˆ 13 ‰ œ
È3
#
dy
dx
œ
dy/dt
dx/dt
œ
"
#
sec
1
3
and y œ
"
#
sec t tan t
"
#
# sec t
œ
tan t
sec t
œ sin t Ê
œ1 Ê yœ
È3
#
x 4" ;
d# y
dx#
dy
dx ¹ tœ1Î3
œ
dyw /dt
dx/dt
œ sin
œ
"
#
1
3
cos t
sec# t
œ
È3
#
;tœ
1
3
œ 2 cos3 t Ê
d# y
dx# ¹ tœ1Î3
"
4
10. x œ "
"
t#
,yœ"
yœ1
3
#
œ #" Ê y œ 3x
Ê
3
t
dy
dx
œ
11. (a) x œ 4t2 , y œ t3 1 Ê t œ „
(b) x œ cos t, y œ tan t Ê sec t œ
"3
4
;
Èx
2
1
x
Š t3# ‹
œ
dy/dt
dx/dt
Š t2$ ‹
d# y
dx#
œ
œ 32 t Ê
dyw /dt
dx/dt
œ
ÊyœŠ„
ˆ 3# ‰
Š t2$ ‹
Èx 3
2 ‹
dy
dx ¹ tœ2
œ
œ 3# (2) œ 3; t œ 2 Ê x œ 1
3 $
4 t
Ê
1œ „
d# y
dx# ¹ tœ2
x3Î2
8
1
1
x2
1œ
Ê tan2 t 1 œ sec2 t Ê y2 œ
œ
3
4
1 x2
x2
"
##
œ
5
4
and
(2)$ œ 6
È 1 x2
x
Êyœ „
12. (a) The line through a1, 2b with slope 3 is y œ 3x 5 Ê x œ t, y œ 3t 5, _ t _
(b) ax 1b2 ay 2b2 œ 9 Ê x 1 œ 3 cos t, y 2 œ 3 sin t Ê x œ 1 3 cos t, y œ 2 3 sin t, 0 Ÿ t Ÿ 21
(c) y œ 4x2 x Ê x œ t, y œ 4t2 t, _ t _
(d) 9x2 4y2 œ 36 Ê
13. y œ x"Î#
x$Î#
3
Ê
dy
dx
x2
4
œ
"
#
y2
9
œ 1 Ê x œ 2 cos t, y œ 3 sin t, 0 Ÿ t Ÿ 21
#
x"Î# #" x"Î# Ê Š dy
dx ‹ œ
"
4
ˆ x" 2 x‰ Ê L œ ' É1 4" ˆ x" 2 x‰ dx
1
4
#
Ê L œ '1 É 4" ˆ x" 2 x‰ dx œ '1 É 4" ax"Î# x"Î# b dx œ '1
4
œ
"
#
ˆ4
2
3
14. x œ y#Î$ Ê
œ '1
8
4
† 8‰ ˆ2 32 ‰‘ œ
dx
dy
È9x#Î$ 4
3x"Î$
œ
5
12
ˆ2
#
14 ‰
3
x"Î$ Ê Š dx
dy ‹ œ
œ
4x #Î$
9
"
#
ˆx"Î# x"Î# ‰ dx œ
"
#
2x"Î# 23 x$Î# ‘ %
"
10
3
' É1
Ê L œ '1 Ê1 Š dx
dy ‹ dy œ 1
#
8
8
4
9x#Î$
dy
'18 È9x#Î$ 4 ˆx"Î$ ‰ dx; u œ 9x#Î$ 4 Ê du œ 6y"Î$ dy; x œ 1
40
" '
" 2 $Î# ‘ %!
Ä L œ 18
u"Î# du œ 18
œ #"7 40$Î# 13$Î# ‘ ¸ 7.634
3 u
"$
13
dx œ
x œ 8 Ê u œ 40d
15. y œ
2
3
"
#
4
"
3
x'Î& 58 x%Î& Ê
dy
dx
œ
"
#
#
x"Î& "# x"Î& Ê Š dy
dx ‹ œ
"
4
Ê u œ 13,
ˆx#Î& 2 x#Î& ‰
#
Ê L œ '1 É1 "4 ax#Î& 2 x#Î& b dx Ê L œ '1 É 4" ax#Î& 2 x#Î& b dx œ ' É "4 ax"Î& x"Î& b dx
32
32
32
1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 11 Practice Exercises
œ '1
32
œ
"
48
16. x œ
"
#
ˆx"Î& x"Î& ‰ dx œ
(1260 450) œ
"
1#
y$
"
y
Ê
" %
œ '1 É 16
y
2
"
#
œ
1710
48
dx
dt
$#
x'Î& 54 x%Î& ‘ " œ
#
dx
dy
œ
"
4
"
y#
"
y%
dy œ '1 ÊŠ 4" y#
y#
8
œ ˆ 12
"# ‰ ˆ 1"# 1‰ œ
17.
" 5
# 6
285
8
Ê Š dx
dy ‹ œ
œ 5 sin t 5 sin 5t and
"
#
œ
ˆ 56 † 2'
y%
"
#
5
4
† 2% ‰ ˆ 56 54 ‰‘ œ
"
#
ˆ 315
6
" %
Ê L œ '1 Ê1 Š 16
y
2
"
y%
dy œ '1 Š 4" y#
#
"
y# ‹
2
7
1#
"
16
"
#
2
"
y# ‹
dy œ ’ 1"# y$ y" “
"
#
75 ‰
4
"
y% ‹
dy
#
"
13
12
#
#
‰ Š dy
œ 5 cos t 5 cos 5t Ê Êˆ dx
dt
dt ‹
dy
dt
œ Éa5 sin t 5 sin 5tb# a5 cos t 5 cos 5tb#
œ 5Èsin# 5t #sin t sin 5t sin# t cos# t #cos t cos 5t cos# 5t œ &È# #asin t sin 5t cos t cos 5 tb
œ 5È#a" cos %tb œ 5É%ˆ "# ‰a" cos %tb œ "!Èsin# #t œ "!lsin #tl œ "!sin #t (since ! Ÿ t Ÿ 1# )
Ê Length œ '!
1 Î2
18.
dx
dt
1Î#
"!sin #t dt œ c5 cos #td !
œ 3t2 12t and
dy
dt
œ a&ba"b a&ba"b œ "!
#
#
‰ Š dy
Éa3t2 12tb# a3t2 12tb# œ È288t# "8t4
œ 3t2 12t Ê Êˆ dx
dt
dt ‹ œ
œ 3È2 ktkÈ16 t2 Ê Length œ '! 3È2 ktkÈ16 t2 dt œ 3È2'! t È16 t2 dt; ’u œ 16 t2 Ê du œ 2t dt
"
"
Ê "# du œ t dt; t œ 0 Ê u œ 16; t œ 1 Ê u œ 17“;
œ
19.
dx
d)
3È 2
2
œ $ sin ) and
#
20. x œ t and y œ
t$
3
$ d) œ $'!
$1Î2
d) œ $ˆ $#1 !‰ œ
t, È3 Ÿ t Ÿ È3 Ê
È3
Èt% #t# " dt œ
È3
#
#
‰ Š dy
Éa$ sin )b# a$ cos )b# œ È$asin# ) cos# )b œ $
œ $ cos ) Ê Êˆ dx
d)
d) ‹ œ
dy
d)
$1Î2
'
'16"7 Èu du œ 3È2 2 23 u3/2 ‘1617 œ 3È2 2 Š 23 a17b3/2 23 a16b3/2 ‹
† 23 Ša17b3/2 64‹ œ È2Ša17b3/2 64‹ ¸ 8.617.
Ê Length œ '!
œ
3È 2
2
'
dx
dt
*1
#
œ 2t and
È3
Èt% 2t# " dt œ
È 3
œt
dy
dt
'
#
" Ê Length œ '
È3
È 3
È3
È 3
Éat# "b# dt œ
Éa2tb# at# "b# dt
È
'È33 at# "b dt œ ’ t3 t“
3
È3
È3
œ 4È 3
21. x œ
t#
#
and y œ 2t, 0 Ÿ t Ÿ È5 Ê
*
œ 21 23 u$Î# ‘ % œ
22. x œ t#
"
2t
761
3
dx
dt
"
È2
ŸtŸ1 Ê
Ê Surface Area œ '1ÎÈ2 21 ˆt#
1
1
œ 21 Š2
dy
dt
È5
œ 2 Ê Surface Area œ '0
21(2t)Èt# 4 dt œ '4 21u"Î# du
9
, where u œ t# 4 Ê du œ 2t dt; t œ 0 Ê u œ 4, t œ È5 Ê u œ 9
and y œ 4Èt ,
œ 21 '1ÎÈ2 ˆt#
œ t and
" ‰ˆ
2t
2t
" ‰
2t#
"‰
#t
dx
dt
œ 2t
ʈ2t
" ‰#
2t#
"
2t#
dy
dt
œ
2
Èt
Š È2 t ‹ dt œ 21 '1ÎÈ2 ˆt#
dt œ 21 '1ÎÈ2 ˆ2t$
1
and
#
3
#
1
" ‰ Ɉ
2t
#t
"
" ‰#
#t#
"4 t$ ‰ dt œ 21 "2 t% 3# t 8" t# ‘ "ÎÈ#
3È 2
4 ‹
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
dt
701
702
Chapter 11 Parametric Equations and Polar Coordinates
23. r cos ˆ) 13 ‰ œ 2È3 Ê r ˆcos ) cos
1
3
sin ) sin 13 ‰
È
r cos ) #3 r sin ) œ 2È3
Ê r cos ) È3 r sin ) œ 4È3 Ê x È3 y œ 4È3
"
#
œ 2È 3 Ê
Ê yœ
È3
3
24. r cos ˆ)
œ
È2
#
x4
31 ‰
4
Ê
œ
È2
#
È2
#
Ê r ˆcos ) cos
r cos )
Ê yœx1
25. r œ 2 sec ) Ê r œ
2
cos )
È2
#
31
4
r sin ) œ
sin ) sin
È2
#
31 ‰
4
Ê x y œ 1
Ê r cos ) œ 2 Ê x œ 2
26. r œ È2 sec ) Ê r cos ) œ È2 Ê x œ È2
27. r œ 3# csc ) Ê r sin ) œ 3# Ê y œ 3#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 11 Practice Exercises
28. r œ 3È3 csc ) Ê r sin ) œ 3È3 Ê y œ 3È3
29. r œ 4 sin ) Ê r# œ 4r sin ) Ê x# y# 4y œ 0
Ê x# (y 2)# œ 4; circle with center (!ß 2) and
radius 2.
30. r œ 3È3 sin ) Ê r# œ 3È3 r sin )
Ê x# y# 3È3 y œ 0 Ê x# Šy
circle with center Š!ß
3È 3
# ‹
and radius
3È 3
# ‹
#
œ
27
4
;
3È 3
#
31. r œ 2È2 cos ) Ê r# œ 2È2 r cos )
#
Ê x# y# 2È2 x œ 0 Ê Šx È2‹ y# œ 2;
circle with center ŠÈ2ß 0‹ and radius È2
32. r œ 6 cos ) Ê r# œ 6r cos ) Ê x# y# 6x œ 0
Ê (x 3)# y# œ 9; circle with center (3ß 0) and
radius 3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
703
704
Chapter 11 Parametric Equations and Polar Coordinates
#
33. x# y# 5y œ 0 Ê x# ˆy #5 ‰ œ
and a œ
5
#
25
4
Ê C œ ˆ!ß #5 ‰
; r# 5r sin ) œ 0 Ê r œ 5 sin )
34. x# y# 2y œ 0 Ê x# (y 1)# œ 1 Ê C œ (!ß 1) and
a œ 1; r# 2r sin ) œ 0 Ê r œ 2 sin )
#
35. x# y# 3x œ 0 Ê ˆx 3# ‰ y# œ
and a œ
3
#
9
4
Ê C œ ˆ 3# ß !‰
; r# 3r cos ) œ 0 Ê r œ 3 cos )
36. x# y# 4x œ 0 Ê (x 2)# y# œ 4 Ê C œ (2ß 0)
and a œ 2; r# 4r cos ) œ 0 Ê r œ 4 cos )
37.
38.
39. d
40.
e
41.
l
42.
f
43. k
44.
h
45.
i
46.
j
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 11 Practice Exercises
47. A œ 2'0
1
" #
# r
d) œ '0 (2 cos ))# d) œ '0 a4 4 cos ) cos# )b d) œ '0 ˆ4 4 cos )
1
œ '0 ˆ 9# 4 cos )
1
48. A œ '0
1Î3
"
#
1
cos 2) ‰
#
1
sin 2) ‘ 1
4
!
d) œ 92 ) 4 sin )
asin# 3)b d) œ '0
1Î3
6) ‰
ˆ 1 cos
d) œ
#
"
4
)
"
6
œ
9
#
A œ 4'0
"
#
1Î$
sin 6)‘ !
œ
1
12
1
#
1
4
Ê )œ
c(1 cos 2))# 1# d d) œ 2 '0 a1 2 cos 2) cos# 2) 1b d)
1Î4
œ 2'0 ˆ2 cos 2)
1Î4
"
#
cos 4) ‰
#
d) œ 2 sin 2)
"
2
)
d)
1
49. r œ 1 cos 2) and r œ 1 Ê 1 œ 1 cos 2) Ê 0 œ cos 2) Ê 2) œ
1Î4
1 cos 2) ‰
#
sin 4) ‘ 1Î%
8
!
œ 2 ˆ1
1
8
; therefore
0‰ œ 2
1
4
50. The circle lies interior to the cardioid. Thus,
1Î2
A œ 2 ' 1Î2
1Î2
"
#
[2(1 sin ))]# d) 1 (the integral is the area of the cardioid minus the area of the circle)
1Î2
œ ' 1Î2 4 a1 2 sin ) sin# )b d) 1 œ ' 1Î2 (6 8 sin ) 2 cos 2)) d) 1 œ c6) 8 cos ) sin 2)d 1Î# 1
œ c31 (31)d 1 œ 51
51. r œ 1 cos ) Ê
dr
d)
œ sin ); Length œ '0 È(1 cos ))# ( sin ))# d) œ '0 È2 2 cos ) d)
21
œ '0 É 4(1 #cos )) d) œ '0 2 sin
21
1Î#
21
52. r œ 2 sin ) 2 cos ), 0 Ÿ ) Ÿ
1
#
53. r œ 8 sin$ ˆ 3) ‰ , 0 Ÿ ) Ÿ
1
4
Ê
dr
d)
#1
d) œ 4 cos 2) ‘ ! œ (4)(1) (4)(1) œ 8
)
#
Ê
œ 8 asin# ) cos# )b œ 8 Ê L œ
21
dr
d)
#
œ 2 cos ) 2 sin ); r# ˆ ddr) ‰ œ (2 sin ) 2 cos ))# (2 cos ) 2 sin ))#
'01Î2 È8 d) œ ’2È2 )“ 1Î# œ 2È2 ˆ 1# ‰ œ 1È2
!
#
œ 64 sin% ˆ 3) ‰ Ê L œ '0 É64 sin% ˆ 3) ‰ d) œ '0
1Î4
#
œ 8 sin# ˆ 3) ‰ cos ˆ 3) ‰ ; r# ˆ ddr) ‰ œ 8 sin$ ˆ 3) ‰‘ 8 sin# ˆ 3) ‰ cos ˆ 3) ‰‘
1Î4
8 sin# ˆ 3) ‰ d) œ '0 8 ’
1Î4
1cos ˆ 23) ‰
“
#
d)
1Î%
œ '0 4 4 cos ˆ 23) ‰‘ d) œ 4) 6 sin ˆ 23) ‰‘ ! œ 4 ˆ 14 ‰ 6 sin ˆ 16 ‰ 0 œ 1 3
1Î4
54. r œ È1 cos 2) Ê
dr
d)
œ
"
#
(1 cos 2))"Î# (2 sin 2)) œ
#
Ê r# ˆ ddr) ‰ œ 1 cos 2)
œ
2 2 cos 2)
1 cos 2)
1Î2
sin# 2)
1 cos 2)
œ
(1 cos 2))# sin# 2)
1 cos 2)
sin 2)
È1 cos 2)
œ
#
Ê ˆ ddr) ‰ œ
sin# 2)
1 cos 2)
1 2 cos 2) cos# 2) sin# 2)
1cos 2)
œ 2 Ê L œ ' 1Î2 È2 d) œ È2 1# ˆ 1# ‰‘ œ È2 1
#
55. x# œ 4y Ê y œ x4 Ê 4p œ 4 Ê p œ 1;
therefore Focus is (0ß 1), Directrix is y œ 1
x#
#
œ y Ê 4p œ 2 Ê p œ "# ;
therefore Focus is ˆ!ß "# ‰; Directrix is y œ "#
56. x# œ 2y Ê
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
#
705
706
Chapter 11 Parametric Equations and Polar Coordinates
57. y# œ 3x Ê x œ
y#
3
Ê 4p œ 3 Ê p œ
3
4
#
58. y# œ 38 x Ê x œ ˆy8 ‰ Ê 4p œ
;
3
therefore Focus is ˆ 34 ß 0‰ , Directrix is x œ 34
x#
7
59. 16x# 7y# œ 112 Ê
y#
16
61. 3x# y# œ 3 Ê x#
Ê c œ 2; e œ
c
a
œ
2
1
y#
3
x#
4
c
a
œ
62. 5y# 4x# œ 20 Ê
y#
4
3
# ;
60. x# 2y# œ 4 Ê
c
a
œ
Ê pœ
therefore Focus is ˆ 23 ß !‰ , Directrix is x œ
œ1
Ê c# œ 16 7 œ 9 Ê c œ 3; e œ
8
3
Ê c œ È2 ; e œ
3
4
œ 1 Ê c# œ 1 3 œ 4
œ 2; the asymptotes are
Ê c œ 3, e œ
c
a
œ
y#
# œ
È2
#
x#
5
2
3
;
2
3
1 Ê c# œ 4 2 œ 2
œ 1 Ê c# œ 4 5 œ 9
the asymptotes are y œ „
2
È5
x
y œ „ È3 x
#
63. x# œ 12y Ê 1x# œ y Ê 4p œ 12 Ê p œ 3 Ê focus is (!ß 3), directrix is y œ 3, vertex is (0ß 0); therefore new
vertex is (2ß 3), new focus is (2ß 0), new directrix is y œ 6, and the new equation is (x 2)# œ 12(y 3)
#
y
64. y# œ 10x Ê 10
œ x Ê 4p œ 10 Ê p œ 5# Ê focus is ˆ 5# ß 0‰ , directrix is x œ 5# , vertex is (0ß 0); therefore new
vertex is ˆ "# ß 1‰ , new focus is (2ß 1), new directrix is x œ 3, and the new equation is (y 1)# œ 10 ˆx "# ‰
65.
x#
9
y#
#5
œ 1 Ê a œ 5 and b œ 3 Ê c œ È25 9 œ 4 Ê foci are a!ß „ 4b , vertices are a!ß „ 5b , center is
(0ß 0); therefore the new center is ($ß 5), new foci are (3ß 1) and (3ß 9), new vertices are ($ß 10) and
($ß 0), and the new equation is
(x 3)#
9
(y 5)#
#5
œ1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 11 Practice Exercises
66.
x#
169
y#
144
707
œ 1 Ê a œ 13 and b œ 12 Ê c œ È169 144 œ 5 Ê foci are a „ 5ß 0b , vertices are a „ 13ß 0b , center
is (0ß 0); therefore the new center is (5ß 12), new foci are (10ß 12) and (0ß 12), new vertices are (18ß 12) and
(8ß 12), and the new equation is
67.
y#
8
x#
2
(x 5)#
169
(y 12)#
144
œ1
œ 1 Ê a œ 2È2 and b œ È2 Ê c œ È8 2 œ È10 Ê foci are Š0ß „ È10‹ , vertices are
Š0ß „ 2È2‹ , center is (0ß 0), and the asymptotes are y œ „ 2x; therefore the new center is Š2ß 2È2‹, new foci are
Š2ß 2È2 „ È10‹ , new vertices are Š2ß 4È2‹ and (#ß 0), the new asymptotes are y œ 2x 4 2È2 and
#
y œ 2x 4 2È2; the new equation is
68.
x#
36
y#
64
Šy 2È2‹
8
(x 2)#
#
œ1
œ 1 Ê a œ 6 and b œ 8 Ê c œ È36 64 œ 10 Ê foci are a „ 10ß 0b , vertices are a „ 6ß 0b , the center
is (0ß 0) and the asymptotes are
y
8
œ „
x
6
or y œ „ 43 x; therefore the new center is (10ß 3), the new foci are
(20ß 3) and (0ß 3), the new vertices are (16ß 3) and (4ß 3), the new asymptotes are y œ
y œ 43 x
49
3
; the new equation is
(x 10)#
36
(y 3)#
64
4
3
x
31
3
and
œ1
69. x# 4x 4y# œ 0 Ê x# 4x 4 4y# œ 4 Ê (x 2)# 4y# œ 4 Ê
(x 2)#
4
y# œ 1, a hyperbola; a œ 2 and
b œ 1 Ê c œ È1 4 œ È5 ; the center is (2ß 0), the vertices are (!ß 0) and (4ß 0); the foci are Š2 „ È5 ß 0‹ and
the asymptotes are y œ „
x 2
#
70. 4x# y# 4y œ 8 Ê 4x# y# 4y 4 œ 4 Ê 4x# (y 2)# œ 4 Ê x#
(y 2)#
4
œ 1, a hyperbola; a œ 1 and
b œ 2 Ê c œ È1 4 œ È5 ; the center is (!ß 2), the vertices are (1ß 2) and ("ß 2), the foci are Š „ È5ß 2‹ and
the asymptotes are y œ „ 2x 2
71. y# 2y 16x œ 49 Ê y# 2y 1 œ 16x 48 Ê (y 1)# œ 16(x 3), a parabola; the vertex is ($ß 1);
4p œ 16 Ê p œ 4 Ê the focus is (7ß 1) and the directrix is x œ 1
72. x# 2x 8y œ 17 Ê x# 2x 1 œ 8y 16 Ê (x 1)# œ 8(y 2), a parabola; the vertex is (1ß 2);
4p œ 8 Ê p œ 2 Ê the focus is (1ß 4) and the directrix is y œ 0
73. 9x# 16y# 54x 64y œ 1 Ê 9 ax# 6xb 16 ay# 4yb œ 1 Ê 9 ax# 6x 9b 16 ay# 4y 4b œ 144
Ê 9(x 3)# 16(y 2)# œ 144 Ê
(x 3)#
16
(y 2)#
9
œ 1, an ellipse; the center is (3ß 2); a œ 4 and b œ 3
Ê c œ È16 9 œ È7 ; the foci are Š$ „ È7ß 2‹ ; the vertices are (1ß 2) and (7ß 2)
74. 25x# 9y# 100x 54y œ 44 Ê 25 ax# 4xb 9 ay# 6yb œ 44 Ê 25 ax# 4x 4b 9 ay# 6y 9b œ 225
#
#
Ê (x 2) (y 3) œ 1, an ellipse; the center is (2ß 3); a œ 5 and b œ 3 Ê c œ È25 9 œ 4; the foci are
9
25
(2ß 1) and (2ß 7); the vertices are (2ß 2) and (2ß 8)
75. x# y# 2x 2y œ 0 Ê x# 2x 1 y# 2y 1 œ 2 Ê (x 1)# (y 1)# œ 2, a circle with center (1ß 1) and
radius œ È2
76. x# y# 4x 2y œ 1 Ê x# 4x 4 y# 2y 1 œ 6 Ê (x 2)# (y 1)# œ 6, a circle with center (2ß 1)
and radius œ È6
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
708
Chapter 11 Parametric Equations and Polar Coordinates
77. r œ
2
1 cos )
Ê e œ 1 Ê parabola with vertex at (1ß 0)
78. r œ
8
2 cos )
Ê rœ
ke œ 4 Ê
"
#
4
1 ˆ "# ‰ cos )
Ê eœ
k œ 4 Ê k œ 8; k œ
a
e
"
#
Ê ellipse;
ea Ê 8 œ
a
ˆ "# ‰
"# a
ˆ " ‰ ˆ 16
‰ 8
Ê a œ 16
3 Ê ea œ #
3 œ 3 ; therefore the center is
ˆ 83 ß 1‰ ; vertices are ()ß 1) and ˆ 83 ß 0‰
79. r œ
6
1 2 cos )
Ê e œ 2 Ê hyperbola; ke œ 6 Ê 2k œ 6
Ê k œ 3 Ê vertices are (2ß 1) and (6ß 1)
80. r œ
Ê
12
3 sin )
"
3
Ê rœ
4
1 ˆ "3 ‰ sin )
Ê eœ
"
3
; ke œ 4
#
k œ 4 Ê k œ 12; a a1 e# b œ 4 Ê a ’1 ˆ 3" ‰ “
œ 4 Ê a œ 9# Ê ea œ ˆ "3 ‰ ˆ 9# ‰ œ 3# ; therefore the
center is ˆ 3# ß 3#1 ‰ ; vertices are ˆ3ß 1# ‰ and ˆ6ß 3#1 ‰
81. e œ 2 and r cos ) œ 2 Ê x œ 2 is directrix Ê k œ 2; the conic is a hyperbola; r œ
Ê rœ
ke
1 e cos )
83. e œ
"
#
84. e œ
"
3
85.
ke
1 e sin )
Ê rœ
(4)(1)
1 cos )
Ê rœ
(2) ˆ "# ‰
1 ˆ "# ‰ sin )
2
2 sin )
and r sin ) œ 6 Ê y œ 6 is directrix Ê k œ 6; the conic is an ellipse; r œ
Ê rœ
ke
1 e cos )
4
1 cos )
and r sin ) œ 2 Ê y œ 2 is directrix Ê k œ 2; the conic is an ellipse; r œ
Ê rœ
(2)(2)
1 # cos )
4
1 # cos )
82. e œ 1 and r cos ) œ 4 Ê x œ 4 is directrix Ê k œ 4; the conic is a parabola; r œ
Ê rœ
Ê rœ
ke
1 e sin )
Ê rœ
(6) ˆ "3 ‰
1 ˆ 3" ‰ sin )
6
3 sin )
(a) Around the x-axis: 9x# 4y# œ 36 Ê y# œ 9 94 x# Ê y œ „ É9 94 x# and we use the positive root:
#
V œ 2 '0 1 ŠÉ9 94 x# ‹ dx œ 2 '0 1 ˆ9 94 x# ‰ dx œ 21 9x 34 x$ ‘ ! œ 241
2
2
#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 11 Additional and Advanced Exercises
709
(b) Around the y-axis: 9x# 4y# œ 36 Ê x# œ 4 49 y# Ê x œ „ É4 49 y# and we use the positive root:
#
V œ 2 '0 1 ŠÉ4 49 y# ‹ dy œ 2'0 1 ˆ4 49 y# ‰ dy œ 21 4y
3
86.
9x# 4y# œ 36, x œ 4 Ê y# œ
œ
87.
91
4
%
$
’ x3 4x“ œ
#
(a) r œ
k
1 e cos )
#
91
4
3
9x# 36
4
Ê yœ
ˆ 64
‰ ˆ8
‰‘ œ
3 16 3 8
3
#
91
4
4
27
$
y$ ‘ ! œ 161
Èx# 4 ; V œ ' 1 Š 3 Èx# 4‹ dx œ
#
2
4
ˆ 56
3
24 ‰
3
œ
31
4
#
91
4
'24 ax# 4b dx
(32) œ 241
Ê r er cos ) œ k Ê Èx# y# ex œ k Ê Èx# y# œ k ex Ê x# y#
œ k 2kex e# x# Ê x# e# x# y# 2kex k# œ 0 Ê a1 e# b x# y# 2kex k# œ 0
(b) e œ 0 Ê x# y# k# œ 0 Ê x# y# œ k# Ê circle;
0 e 1 Ê e# 1 Ê e# 1 0 Ê B# 4AC œ 0# 4 a1 e# b (1) œ 4 ae# 1b 0 Ê ellipse;
e œ 1 Ê B# 4AC œ 0# 4(0)(1) œ 0 Ê parabola;
e 1 Ê e# 1 Ê B# 4AC œ 0# 4 a1 e# b (1) œ 4e# 4 0 Ê hyperbola
88.
Let (r" ß )" ) be a point on the graph where r" œ a)" . Let (r# ß )# ) be on the graph where r# œ a)# and
)# œ )" 21. Then r" and r# lie on the same ray on consecutive turns of the spiral and the distance between
the two points is r# r" œ a)# a)" œ a()# )" ) œ 21a, which is constant.
CHAPTER 11 ADDITIONAL AND ADVANCED EXERCISES
1. Directrix x œ 3 and focus (4ß 0) Ê vertex is ˆ 7# ß !‰
Ê pœ
"
#
Ê the equation is x
7
#
œ
y#
#
2. x# 6x 12y 9 œ 0 Ê x# 6x 9 œ 12y Ê
(x3)#
12
œ y Ê vertex is (3ß 0) and p œ 3 Ê focus is (3ß 3) and the
directrix is y œ 3
3. x# œ 4y Ê vertex is (!ß 0) and p œ 1 Ê focus is (!ß 1); thus the distance from P(xß y) to the vertex is Èx# y#
and the distance from P to the focus is Èx# (y 1)# Ê Èx# y# œ 2Èx# (y 1)#
Ê x# y# œ 4 cx# (y 1)# d Ê x# y# œ 4x# 4y# 8y 4 Ê 3x# 3y# 8y 4 œ 0, which is a circle
4. Let the segment a b intersect the y-axis in point A and
intersect the x-axis in point B so that PB œ b and PA œ a
(see figure). Draw the horizontal line through P and let it
intersect the y-axis in point C. Let nPBO œ )
Ê nAPC œ ). Then sin ) œ yb and cos ) œ xa
Ê
x#
a#
y#
b#
œ cos# ) sin# ) œ 1.
5. Vertices are a!ß „ 2b Ê a œ 2; e œ
c
a
Ê 0.5 œ
c
#
Ê c œ 1 Ê foci are a0ß „ 1b
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
710
Chapter 11 Parametric Equations and Polar Coordinates
6. Let the center of the ellipse be (xß 0); directrix x œ 2, focus (4ß 0), and e œ 23 Ê ae c œ 2 Ê ae œ 2 c
a
Ê a œ 32 (2 c). Also c œ ae œ 32 a Ê a œ 32 ˆ2 32 a‰ Ê a œ 34 94 a Ê 95 a œ 34 Ê a œ 12
5 ;x2œ e
28
28
8
#
#
#
‰ ˆ 3# ‰ œ 18
ˆ 28 ‰
Ê x 2 œ ˆ 12
5
5 Ê x œ 5 Ê the center is 5 ß 0 ; x 4 œ c Ê c œ 5 4 œ 5 so that c œ a b
#
#
‰ ˆ 58 ‰ œ
œ ˆ 12
5
80
25
; therefore the equation is
ˆx 28
‰#
5
ˆ 144
‰
25
y#
ˆ 80
‰
25
œ 1 or
7. Let the center of the hyperbola be (0ß y).
(a) Directrix y œ 1, focus (0ß 7) and e œ 2 Ê c ae œ 6 Ê
Ê a œ 2(2a) 12 Ê a œ
Ê b# œ c# a# œ 64 16
(b) e œ 5 Ê c
Ê cœ
œ
625
16
25
4
25
16
a
e
œ6 Ê
; y (1) œ
œ
75
#
a
e
a
e
a
e
‰
25 ˆx 28
5
144
#
Ê
144
a4 a # b
#
5y#
16
œ1
œ c 6 Ê a œ 2c 12. Also c œ ae œ 2a
4 Ê c œ 8; y (1) œ œ #4 œ 2
#
œ 48; therefore the equation is (y161)
a
e
Ê y œ 1 Ê the center is (0ß 1); c# œ a# b#
x#
48
œ1
œ c 6 Ê a œ 5c 30. Also, c œ ae œ 5a Ê a œ 5(5a) 30 Ê 24a œ 30 Ê a œ
ˆ 54 ‰
5
œ
œ
"
4
Ê y œ 43 Ê the center is ˆ!ß 43 ‰ ; c# œ a# b# Ê b# œ c# a#
; therefore the equation is
ˆy 34 ‰#
ˆ 25
‰
16
x#
ˆ 75
‰
#
œ 1 or
16 ˆy 34 ‰
25
#
#
#
#
#
2x#
75
8. The center is (0ß 0) and c œ 2 Ê 4 œ a# b# Ê b# œ 4 a# . The equation is
49
a#
y#
a#
#
œ1
x#
b#
#
œ1 Ê
#
%
49
a#
144
b#
œ1
%
œ 1 Ê 49 a4 a b 144a œ a a4 a b Ê 196 49a 144a œ 4a a Ê a 197a# 196
œ 0 Ê aa 196b aa# 1b œ 0 Ê a œ 14 or a œ 1; a œ 14 Ê b# œ 4 (14)# 0 which is impossible; a œ 1
Ê b# œ 4 1 œ 3; therefore the equation is y#
9. b# x# a# y# œ a# b# Ê
dy
dx
x#
3
œ1
#
#
œ ba# yx ; at (x" ß y" ) the tangent line is y y" œ Š ba# yx"" ‹ (x x" )
Ê a# yy" b# xx" œ b# x"# a# y"# œ a# b# Ê b# xx" a# yy" a# b# œ 0
10. b# x# a# y# œ a# b# Ê
dy
dx
œ
b# x
a# y
#
; at (x" ß y" ) the tangent line is y y" œ Š ba# yx"" ‹ (x x" )
Ê b# xx" a# yy" œ b# x"# a# y"# œ a# b# Ê b# xx" a# yy" a# b# œ 0
11.
12.
13.
14.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
5
4
Chapter 11 Additional and Advanced Exercises
15. a9x# 4y# 36b a4x# 9y# 16b Ÿ 0
Ê 9x# 4y# 36 Ÿ 0 and 4x# 9y# 16 0
or 9x# 4y# 36 0 and 4x# 9y# 16 Ÿ 0
16. a9x# 4y# 36b a4x# 9y# 16b 0, which is the
complement of the set in Exercise 15
sin t
17. (a) x œ e2t cos t and y œ e2t sin t Ê x# y# œ e4t cos# t e4t sin# t œ e4t . Also yx œ ee2t cos
t œ tan t
" ˆ y ‰
#
#
% tan " ayÎxb
#
#
Ê t œ tan
Ê x y œe
is the Cartesian equation. Since r œ x y# and
x
2t
) œ tan" ˆ yx ‰ , the polar equation is r# œ e4) or r œ e2) for r 0
(b) ds# œ r# d)# dr# ; r œ e2) Ê dr œ 2e2) d)
#
#
Ê ds# œ r# d)# ˆ2e2) d)‰ œ ˆe2) ‰ d)# 4e4) d)#
œ 5e4) d)# Ê ds œ È5 e2) d) Ê L œ '0 È5 e2) d)
21
œ’
È5 e2) #1
2 “!
œ
È5
#
ae41 1b
#
#
18. r œ 2 sin$ ˆ 3) ‰ Ê dr œ 2 sin# ˆ 3) ‰ cos ˆ 3) ‰ d) Ê ds# œ r# d)# dr# œ 2 sin$ ˆ 3) ‰‘ d)# 2 sin# ˆ 3) ‰ cos ˆ 3) ‰ d)‘
œ 4 sin' ˆ 3) ‰ d)# 4 sin% ˆ 3) ‰ cos# ˆ 3) ‰ d)# œ 4 sin% ˆ 3) ‰‘ sin# ˆ 3) ‰ cos# ˆ 3) ‰‘ d)# œ 4 sin% ˆ 3) ‰ d)#
Ê ds œ 2 sin# ˆ 3) ‰ d). Then L œ '0 2 sin# ˆ 3) ‰ d) œ '0 1 cos ˆ 23) ‰‘ d) œ )
31
31
3
2
$1
sin ˆ 23) ‰‘ ! œ 31
19. e œ 2 and r cos ) œ 2 Ê x œ 2 is the directrix Ê k œ 2; the conic is a hyperbola with r œ
Ê rœ
(2)(2)
1 2 cos )
œ
ke
1 e cos )
4
1 2 cos )
20. e œ 1 and r cos ) œ 4 Ê x œ 4 is the directrix Ê k œ 4; the conic is a parabola with r œ
Ê rœ
21. e œ
"
#
"
3
œ
4
1 cos )
and r sin ) œ 2 Ê y œ 2 is the directrix Ê k œ 2; the conic is an ellipse with r œ
Ê rœ
22. e œ
(4)(1)
1 cos )
2 ˆ "# ‰
1 ˆ "# ‰ sin
)
œ
ke
1 e sin )
2
2 sin )
and r sin ) œ 6 Ê y œ 6 is the directrix Ê k œ 6; the conic is an ellipse with r œ
Ê rœ
6 ˆ "3 ‰
1 ˆ 3" ‰ sin
)
ke
1 e cos )
œ
ke
1 e sin )
6
3 sin )
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
711
712
Chapter 11 Parametric Equations and Polar Coordinates
23. Arc PF œ Arc AF since each is the distance rolled;
nPCF œ Arcb PF Ê Arc PF œ b(nPCF); ) œ ArcaAF
Ê Arc AF œ a) Ê a) œ b(nPCF) Ê nPCF œ ˆ ba ‰ );
nOCB œ
1
#
) and nOCB œ nPCF nPCE
œ nPCF ˆ 1# !‰ œ ˆ ba ‰ ) ˆ 1# !‰ Ê 1# )
œ ˆ ba ‰ ) ˆ 1# !‰ Ê 1# ) œ ˆ ba ‰ ) 1# !
Ê ! œ 1 ) ˆ ba ‰ ) Ê ! œ 1 ˆ ab b ‰ ).
Now x œ OB BD œ OB EP œ (a b) cos ) b cos ! œ (a b) cos ) b cos ˆ1 ˆ a b b ‰ )‰
œ (a b) cos ) b cos 1 cos ˆˆ a b b ‰ )‰ b sin 1 sin ˆˆ a b b ‰ )‰ œ (a b) cos ) b cos ˆˆ a b b ‰ )‰ and
y œ PD œ CB CE œ (a b) sin ) b sin ! œ (a b) sin ) b sin ˆˆ a b b ‰ )‰
œ (a b) sin ) b sin 1 cos ˆˆ a b b ‰ )‰ b cos 1 sin ˆˆ a b b ‰ )‰ œ (a b) sin ) b sin ˆˆ a b b ‰ )‰ ;
therefore x œ (a b) cos ) b cos ˆˆ a b b ‰ )‰ and y œ (a b) sin ) b sin ˆˆ a b b ‰ )‰
24. x œ a(t sin t) Ê
‰
œ a(1 cos t) and let $ œ 1 Ê dm œ dA œ y dx œ y ˆ dx
dt dt
dx
dt
œ a(1 cos t) a (1 cos t) dt œ a# (1 cos t)# dt; then A œ '0 a# (1 cos t)# dt
21
'021 a1 2 cos t cos# tb dt œ a# '021 ˆ1 2 cos t "# "# cos 2t‰ dt œ a# 32 t 2 sin t sin4 2t ‘ #!1
œ 31a# ; µ
x = x œ a(t sin t) and µ
y = "# y œ "# a(1 cos t) Ê Mx œ ' µ
y dm œ ' µ
y $ dA
21
21
21
œ '0 "# a(1 cos t) a# (1 cos t)# dt œ "# a$ '0 (1 cos t)$ dt œ a# '0 a1 3 cos t 3 cos# t cos$ tb dt
œ a#
$
œ
œ
a$
#
'021 1 3 cos t #3 3 cos# 2t a1 sin# tb (cos t)‘ dt œ a# ’ 25 t 3 sin t 3 sin4 2t sin t sin3 t “ #1
$
51 a
#
$
!
$
. Therefore y œ
Mx
M
œ
$
Š 51#a ‹
œ
31 a#
5
6
a. Also, My œ ' µ
x dm œ ' µ
x $ dA
œ '0 a(t sin t) a# (1 cos t)# dt œ a$ '0 at 2t cos t t cos# t sin t 2 sin t cos t sin t cos# tb dt
21
21
#
œ a$ ’ t2 2 cos t 2t sin t 4" t#
xœ
25.
My
M
œ
31 # a$
31 a#
œ 1a Ê ˆ1aß
5
6
"
8
cos 2t
t
4
sin 2t cos t sin# t
#1
cos$ t
3 “!
œ 31# a$ . Thus
a‰ is the center of mass.
" œ <# <" Ê tan " œ tan (<# <" ) œ
tan <# tan <"
1 tan <# tan <"
;
the curves will be orthogonal when tan " is undefined, or
when tan <# œ tan"<" Ê g (r)) œ "
r
w
’ f ()) “
w
#
w
w
Ê r œ f ( ) ) g ( ) )
26.
r œ sin% ˆ 4) ‰ Ê
27.
r œ 2a sin 3) Ê
œ sin$ ˆ 4) ‰ cos ˆ 4) ‰ Ê tan < œ
dr
d)
dr
d)
œ 6a cos 3) Ê tan < œ
r
ˆ ddr) ‰
œ
sin% ˆ )4 ‰
sin$ ˆ 4) ‰ cos ˆ )4 ‰
2a sin 3)
6a cos 3)
œ
"
3
œ tan ˆ 4) ‰
tan 3); when ) œ
1
6
, tan < œ
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
3
tan
1
#
Ê<œ
1
#
Chapter 11 Additional and Advanced Exercises
28.
(b) r) œ 1 Ê r œ )" Ê
(a)
œ
) "
) #
œ ) Ê
Ê < Ä
1
#
dr
d)
œ )# Ê tan with( plots );
r := t -> [sin(t)-t*cos(t),cos(t)+t*sin(t),t^2];
t0 := 3*Pi/2;
lo := 0;
hi := 6*Pi;
P1 := spacecurve( r(t), t=lo..hi, axes=boxed, thickness=3 ):
display( P1, title="#35(a) (Section 13.1)" );
Dr := unapply( diff(r(t),t), t );
# (b)
Dr(t0);
# (c)
q1 := expand( r(t0) + Dr(t0)*(t-t0) );
T := unapply( q1, t );
P2 := spacecurve( T(t), t=lo..hi, axes=boxed, thickness=3, color=black ):
display( [P1,P2], title="#35(d) (Section 13.1)" );
39-40. Example CAS commands:
Maple:
a := 'a'; b := 'b';
r := (a,b,t) -> [cos(a*t),sin(a*t),b*t];
Dr := unapply( diff(r(a,b,t),t), (a,b,t) );
t0 := 3*Pi/2;
q1 := expand( r(a,b,t0) + Dr(a,b,t0)*(t-t0) );
T := unapply( q1, (a,b,t) );
lo := 0;
hi := 4*Pi;
P := NULL:
for a in [ 1, 2, 4, 6 ] do
P1 := spacecurve( r(a,1,t), t=lo..hi, thickness=3 ):
P2 := spacecurve( T(a,1,t), t=lo..hi, thickness=3, color=black ):
P := P, display( [P1,P2], axes=boxed, title=sprintf("#39 (Section 13.1)\n a=%a",a) );
end do:
display( [P], insequence=true );
35-40. Example CAS commands:
Mathematica: (assigned functions, parameters, and intervals will vary)
The x-y-z components for the curve are entered as a list of functions of t. The unit vectors i, j, k are not inserted.
If a graph is too small, highlight it and drag out a corner or side to make it larger.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
763
764
Chapter 13 Vector-Valued Functions and Motion in Space
Only the components of r[t] and values for t0, tmin, and tmax require alteration for each problem.
Clear[r, v, t, x, y, z]
r[t_]={ Sin[t] t Cos[t], Cos[t] t Sin[t], t^2}
t0= 31 / 2; tmin= 0; tmax= 61;
ParametricPlot3D[Evaluate[r[t]], {t, tmin, tmax}, AxesLabel Ä {x, y, z}];
v[t_]= r'[t]
tanline[t_]= v[t0] t r[t0]
ParametricPlot3D[Evaluate[{r[t], tanline[t]}], {t, tmin, tmax}, AxesLabel Ä {x, y, z}];
For 39 and 40, the curve can be defined as a function of t, a, and b. Leave a space between a and t and b and t.
Clear[r, v, t, x, y, z, a, b]
r[t_,a_,b_]:={Cos[a t], Sin[a t], b t}
t0= 31 / 2; tmin= 0; tmax= 41;
v[t_,a_,b_]= D[r[t, a, b], t]
tanline[t_,a_,b_]=v[t0, a, b] t r[t0, a, b]
pa1=ParametricPlot3D[Evaluate[{r[t, 1, 1], tanline[t, 1, 1]}], {t,tmin, tmax}, AxesLabel Ä {x, y, z}];
pa2=ParametricPlot3D[Evaluate[{r[t, 2, 1], tanline[t, 2, 1]}], {t,tmin, tmax}, AxesLabel Ä {x, y, z}];
pa4=ParametricPlot3D[Evaluate[{r[t, 4, 1], tanline[t, 4, 1]}], {t,tmin, tmax}, AxesLabel Ä {x, y, z}];
pa6=ParametricPlot3D[Evaluate[{r[t, 6, 1], tanline[t, 6, 1]}], {t,tmin, tmax}, AxesLabel Ä {x, y, z}];
Show[GraphicsRow[{pa1, pa2, pa4, pa6}]]
13.2 INTEGRALS OF VECTOR FUNCTIONS; PROJECTILE MOTION
1.
'01 ct$ i 7j (t 1)kd dt œ ’ t4 “ " i [7t] !" j ’ t2 t“ " k œ 4" i 7j 3# k
2.
'12 (6 6t)i 3Èt j ˆ t4 ‰ k‘ dt œ c6t 3t# d #" i 2t$Î# ‘ #" j c4t" d #" k œ 3i Š4È2 2‹ j 2k
3.
' 11Î%Î% c(sin t)i (1 cos t)j asec# tb kd dt œ c cos td 1Î%1Î% i ct sin td 1Î%1Î% j ctan td 1Î%1Î% k œ Š 1 #2È2 ‹ j 2k
4.
'01Î3 casec t tan tbi atan tbj a2 sin t cos tb kd dt œ '01Î3 [asec t tan tbi atan tbj asin 2tbk] dt
%
#
!
!
#
1Î$
1Î$
1Î$
œ csec td ! i c ln acos tbd ! j "# cos 2t‘ ! k œ i (ln 2)j 34 k
5.
'14 ˆ "t i 5 " t j #"t k‰ dt œ
6.
'01 Š È 2
7.
'01 Štet
8.
'1ln 3 atet i et j ln t kb dt œ ctet et d ln1 3 i cet d ln1 3 j ct ln t td 1ln 3 k
1 t#
2
i
È3
1 t#
%
œ cln td %" i c ln (5 t)d %" j "# ln t‘ " k œ (ln 4)i (ln 4)j (ln 2)k
"
"
k‹ dt œ c2 sin" td ! i ’È3 tan" t“ k œ 1i
!
"
"
i et j k‹ dt œ ’ 12 et “ i cet d ! j ctd !" k œ
2
!
e1
2 i
1È3
4
e1
e i
k
k
œ 3a ln 3 1bi a3 ebi aln 3alnaln 3b 1b 1b k
9.
'01Î2 cacos tbi asin 2tbj asin2 tb kd dt œ '01Î2 acos tbi asin 2tbj ˆ 12 12 cos 2t‰ k‘ dt œ
1 Î2
1 Î2
1 Î2
œ csin td ! i 12 cos t‘ ! j 12 t 14 sin 2t‘ ! k œ i j
1
4
k
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 13.2 Integrals of Vector Functions; Projectile Motion
10.
765
'01/4 casec tbi atan2 tbj at sin tb kd dt œ '01/4 casec tbi asec2 t 1bj at sin tb kd dt
œ clnasec t tan tbd 1!/4 i ctan t td 1!/4 j ct cos t sin td !1/4 k œ lnŠ1 È2‹i ˆ1 14 ‰j Š
11. r œ ' (ti tj tk) dt œ t# i
#
Ê r œ Š
t#
#
1‹ i Š
t#
#
t#
#
j
t#
#
1
4È 2
1
È2 ‹k
k C ; r(0) œ 0i 0j 0k C œ i 2j 3k Ê C œ i 2j 3k
#
2‹ j Š t# 3‹ k
$‰
#
#
12. r œ ' c(180t)i a180t 16t# b jd dt œ 90t# i ˆ90t# 16
3 t j C ; r(0) œ 90(0) i 90(0)
$
‰
œ 100j Ê C œ 100j Ê r œ 90t# i ˆ90t# 16
3 t 100 j
(0)$ ‘ j C
16
3
13. r œ ' ˆ 3# (t 1)"Î# ‰ i e t j ˆ t " 1 ‰ k‘ dt œ (t 1)$Î# i e t j ln (t 1)k C ;
r(0) œ (0 1)$Î# i e ! j ln (0 1)k C œ k Ê C œ i j k
Ê r œ (t 1)$Î# 1‘ i a1 e t b j [1 ln (t 1)]k
14. r œ ' cat$ 4tb i tj 2t# kd dt œ Š t4 2t# ‹ i
%
t#
2
j
%
2t$
3
%
k C ; r(0) œ ’ 04 2(0)# “ i
#
œ i j Ê C œ i j Ê r œ Š t4 2t# 1‹ i Š t# 1‹ j
15.
0#
2
j
2(0)$
3
kC
2t$
3 k
œ ' (32k) dt œ 32tk C" ; ddtr (0) œ 8i 8j Ê 32(0)k C" œ 8i 8j Ê C" œ 8i 8j
Ê dr œ 8i 8j 32tk ; r œ ' (8i 8j 32tk) dt œ 8ti 8tj 16t# k C# ; r(0) œ 100k
dr
dt
dt
Ê 8(0)i 8(0)j 16(0)# k C# œ 100k Ê C# œ 100k Ê r œ 8ti 8tj a100 16t# b k
16.
dr
dt
œ ' (i j k) dt œ (ti tj tk) C" ;
Ê
dr
dt
dr
dt
(0) œ 0 Ê (0i 0j 0k) C" œ 0 Ê C" œ 0
#
t#
#
œ (ti tj tk) ; r œ ' (ti tj tk) dt œ Š t# i
#
Ê Š 0# i
0#
#
j
0#
#
#
j
t#
#
k‹ C# ; r(0) œ 10i 10j 10k
k‹ C# œ 10i 10j 10k Ê C# œ 10i 10j 10k
#
#
Ê r œ Š t# 10‹ i Š t# 10‹ j Š t# 10‹ k
17.
dv
dt
œ a œ 3i j k Ê v(t) œ 3ti tj tk C" ; the particle travels in the direction of the vector
(4 1)i (1 2)j (4 3)k œ 3i j k (since it travels in a straight line), and at time t œ 0 it has speed
2 Ê v(0) œ
(3i j k) œ C" Ê
Ê r(t) œ Š 3# t#
6
È11
t‹ i Š #" t#
Ê r(t) œ Š 3# t#
6
È11
t 1‹ i Š "# t#
œ Š "# t#
18.
2
È9 1 1
dv
dt
2
È11
2
È11
dr
dt
œ v(t) œ Š3t
t‹ j Š #" t#
2
È11
2
È11
6
È11 ‹ i
Št
2
È11 ‹ j
Št
2
È11 ‹ k
t‹ k C# ; r(0) œ i 2j 3k œ C#
t 2‹ j Š #" t#
2
È11
t 3‹ k
t‹ (3i j k) (i 2j 3k)
œ a œ 2i j k Ê v(t) œ 2ti tj tk C" ; the particle travels in the direction of the vector
(3 1)i (0 (1))j (3 2)k œ 2i j k (since it travels in a straight line), and at time t œ 0 it has speed 2
Ê v(0) œ
2
È4 1 1
(2i j k) œ C" Ê
Ê r(t) œ Št#
4
È6
t‹ i Š "# t#
Ê r(t) œ Št#
4
È6
t 1‹ i Š "# t#
2
È6
dr
dt
œ v(t) œ Š2t
t‹ j Š "# t#
2
È6
2
È6
4
È6 ‹ i
Št
2
È6 ‹ j
Št
2
È6 ‹ k
t‹ k C# ; r(0) œ i j 2k œ C#
t 1‹ j Š "# t#
2
È6
t 2‹ k œ Š "# t#
2
È6
t‹ (2i j k) (i j 2k)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
766
Chapter 13 Vector-Valued Functions and Motion in Space
m‰
19. x œ (v! cos !)t Ê (21 km)ˆ 1000
œ (840 m/s)(cos 60°)t Ê t œ
1 km
20. R œ
v#!
g
21,000 m
(840 m/s)(cos 60°)
œ 50 seconds
v#
!
sin 2! and maximum R occurs when ! œ 45° Ê 24.5 km œ Š 9.8 m/s
# ‹ (sin 90°)
Ê v! œ È(9.8)(24,500) m# /s# œ 490 m/s
(b)
(c)
v#!
g
(500 m/s)#
9.8 m/s# (sin 90°) ¸ 25,510.2 m
5000 m
x œ (v! cos !)t Ê 5000 m œ (500 m/s)(cos 45°)t Ê t œ (500 m/s)(cos 45°) ¸ 14.14 s; thus,
y œ (v! sin !)t "# gt# Ê y ¸ (500 m/s)(sin 45°)(14.14 s) "# a9.8 m/s# b (14.14 s)# ¸ 4020 m
!)#
45°))#
ymax œ (v! sin
œ ((5002 m/s)(sin
¸ 6378 m
2g
a9.8 m/s# b
2v! sin !
g
21. (a) t œ
œ
2(500 m/s)(sin 45°)
9.8 m/s#
¸ 72.2 seconds; R œ
sin 2! œ
22. y œ y! (v! sin !)t "# gt# Ê y œ 32 ft (32 ft/sec)(sin 30°)t "# a32 ft/sec# b t# Ê y œ 32 16t 16t# ;
the ball hits the ground when y œ 0 Ê 0 œ 32 16t 16t# Ê t œ 1 or t œ 2 Ê t œ 2 sec since t 0; thus,
x œ (v! cos !) t Ê x œ (32 ft/sec)(cos 30°)t œ 32 Š
23. (a) R œ
v#!
g
(b) 6m ¸
È3
# ‹ (2)
¸ 55.4 ft
v#
#
# #
!
sin 2! Ê 10 m œ Š 9.8 m/s
Ê v! ¸ 9.9 m/s;
# ‹ (sin 90°) Ê v! œ 98 m s
(9.9 m/s)#
9.8 m/s#
(sin 2!) Ê sin 2! ¸ 0.59999 Ê 2! ¸ 36.87° or 143.12° Ê ! ¸ 18.4° or 71.6°
24. v! œ 5 ‚ 10' m/s and x œ 40 cm œ 0.4 m; thus x œ (v! cos !)t Ê 0.4m œ a5 ‚ 10' m/sb (cos 0°)t
Ê t œ 0.08 ‚ 10' s œ 8 ‚ 10) s; also, y œ y! (v! sin !)t "# gt#
#
Ê y œ a5 ‚ 10' m/sb (sin 0°) a8 ‚ 10) sb "# a9.8 m/s# b a8 ‚ 10) sb œ 3.136 ‚ 10"% m or
3.136 ‚ 10"# cm. Therefore, it drops 3.136 ‚ 10"# cm.
25. R œ
v#!
g
sin 2! Ê 16,000 m œ
(400 m/s)#
9.8 m/s#
sin 2! Ê sin 2! œ 0.98 Ê 2! ¸ 78.5° or 2! ¸ 101.5° Ê ! ¸ 39.3°
or 50.7°
26. (a) R œ
(2v! )#
g
sin 2! œ
4v#!
g
v#
sin 2! œ 4 Š g! sin !‹ or 4 times the original range.
(b) Now, let the initial range be R œ
Ê
(pv! )
g
#
v#!
g
sin 2!. Then we want the factor p so that pv! will double the range
v#!
sin 2! œ 2 Š g sin 2!‹ Ê p# œ 2 Ê p œ È2 or about 141%. The same percentage will approximately
double the height:
apv0 sin !b2
2g
œ
2av0 sin !b2
2g
Ê p# œ 2 Ê p œ È2.
27. The projectile reaches its maximum height when its vertical component of velocity is zero Ê
Êtœ
v0 sin !
g
2
!
v0 sin !
"
Ê ymax œ av0 sin !bŠ v0 sin
g ‹ # gŠ g ‹ œ
av0 sin !b
g
2
av0 sin !b
2g
2
œ
av0 sin !b
2g
dy
dt
œ v0 sin ! gt œ 0
2
. To find the flight time
we find the time when the projectile lands: av0 sin !bt "# g t2 œ 0 Ê tˆv0 sin ! "# g t‰ œ 0 Ê t œ 0 or t œ
t œ 0 is the time when the projectile is fired, so t œ
the value of the horizontal component when t œ
2v0 sin !
g
2v0 sin !
g
2v0 sin !
.
g
is the time when the projectile strikes the ground. The range is
!
Ê R œ x œ av0 cos !bŠ 2v0 sin
‹œ
g
v02
g a2 sin !cos !b
œ
v02
g sin 2!.
The range is largest when sin 2! œ 1 Ê ! œ 45‰ .
28. When marble A is located R units downrange, we have x œ (v! cos !)t Ê R œ (v! cos !)t Ê t œ
R
v! cos !
#
R
R
"
that time the height of marble A is y œ y! (v! sin !)t "# gt# œ (v! sin !) Š v! cos
! ‹ # g Š v! cos ! ‹
#
R
Ê y œ R tan ! "# g Š v# cos
# ! ‹ . The height of marble B at the same time t œ
!
R
v! cos !
seconds is
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
. At
Section 13.2 Integrals of Vector Functions; Projectile Motion
767
#
R
h œ R tan ! "# gt# œ R tan ! "# g Š v# cos
# ! ‹ . Since the heights are the same, the marbles collide regardless
!
of the initial velocity v! .
29.
dr
dt
œ ' (gj) dt œ gtj C" and
dr
dt
(0) œ (v! cos !)i (v! sin !)j Ê g(0)j C" œ (v! cos !)i (v! sin !)j
Ê C" œ (v! cos !)i (v! sin !)j Ê ddtr œ (v! cos !)i (v! sin ! gt)j ; r œ ' [(v! cos !)i (v! sin ! gt)j] dt
œ (v! t cos !)i ˆv! t sin ! "# gt# ‰ j C# and r(0) œ x! i y! j Ê [v! (0) cos !]i v! (0) sin ! "# g(0)# ‘ j C#
œ x! i y! j Ê C# œ x! i y! j Ê r œ (x! v! t cos !)i ˆy! v! t sin ! "# gt# ‰ j Ê x œ x! v! t cos ! and
y œ y! v! t sin ! "# gt#
30. The maximum height is y œ
(v! sin !)#
#g
and this occurs for x œ
v#!
#g
sin 2! œ
v#! sin ! cos !
g
. These equations describe
parametrically the points on a curve in the xy-plane associated with the maximum heights on the parabolic trajectories in
terms of the parameter (launch angle) !. Eliminating the parameter !, we have x# œ
œ
v%! sin# !
g#
v%! sin% !
g#
Ê x# 4 Šy
œ
v#!
4g ‹
#
v#!
g
œ
(2y) (2y)# Ê x# 4y# Š
v!%
4g#
, where x
2v#!
g ‹y
v%! sin# ! cos# !
g#
v#
œ 0 Ê x# 4 ’y# Š 2g! ‹ y
œ
ˆv%! sin# !‰ a1 sin# !b
g#
v%!
16g# “
œ
v%!
4g#
0.
31. (a) At the time t when the projectile hits the line OR we
have tan " œ yx ; x œ [v! cos (! " )]t and
y œ [v! sin (! " )]t "# gt# 0 since R is
below level ground. Therefore let
kyk œ "# gt# [v! sin (! " )]t 0
"# gt# (v! sin (! " ))t‘
" gt v sin (! " )‘
œ # v! cos! (! ")
[v! cos (! " )]t
v! cos (! " ) tan " œ "# gt v! sin (! " )
t œ 2v! sin (! ") 2vg ! cos (! ") tan " , which is the time
so that tan " œ
Ê
Ê
when the projectile hits the downhill slope. Therefore,
x œ [v! cos (! " )] ’ 2v! sin (! ") 2vg ! cos (! ") tan " “ œ
maximized, then OR is maximized:
dx
d!
œ
2v#!
g
2v#!
g
ccos# (! " ) tan " sin (! " ) cos (! " )d . If x is
[ sin 2(! " ) tan " cos 2(! " )] œ 0
Ê sin 2(! " ) tan " cos 2(! " ) œ 0 Ê tan " œ cot 2(! " ) Ê 2(! " ) œ 90° "
Ê ! " œ "# (90° " ) Ê ! œ "# (90° " ) œ "# of nAOR.
(b) At the time t when the projectile hits OR we have
tan " œ yx ; x œ [v! cos (! " )]t and
y œ [v! sin (! " )]t "# gt#
v! sin (! " ) "# gt‘
v! cos (! " )
v! cos (! " ) tan " œ v! sin (! " ) "# gt
t œ 2v! sin (! ") 2vg ! cos (! ") tan " , which is the
Ê tan " œ
Ê
Ê
cv! sin (! " )d t "# gt#
[v! cos (! " )]t
œ
time
when the projectile hits the uphill slope. Therefore,
x œ [v! cos (! " )] ’ 2v! sin (! ") 2vg ! cos (! ") tan " “ œ
maximized, then OR is maximized: ddx! œ
2v#!
g
2v#!
g
csin (! " ) cos (! " ) cos# (! " ) tan " d . If x is
[cos 2(! " ) sin 2(! " ) tan " ] œ 0
Ê cos 2(! " ) sin 2(! " ) tan " œ 0 Ê cot 2(! " ) tan " œ ! Ê cot 2(! " ) œ tan "
œ tan (" ) Ê 2(! " ) œ 90° (" ) œ 90° " Ê ! œ "# (90° " ) œ "# of nAOR. Therefore v! would bisect
nAOR for maximum range uphill.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
768
Chapter 13 Vector-Valued Functions and Motion in Space
32. v! œ 116 ft/sec, ! œ 45°, and x œ (v! cos !)t
Ê 369 œ (116 cos 45°)t Ê t ¸ 4.50 sec;
also y œ (v! sin !)t "# gt#
Ê y œ (116 sin 45°)(4.50) "# (32)(4.50)#
¸ 45.11 ft. It will take the ball 4.50 sec to travel
369 ft. At that time the ball will be 45.11 ft in
the air and will hit the green past the pin.
33. (a) (Assuming that "x" is zero at the point of impact:)
ratb œ axatbbi ayatbbj; where xatb œ a35 cos 27‰ bt and yatb œ 4 a35 sin 27‰ bt 16t2 .
(b) ymax œ
av0 sin !b2
2g
4 œ
a35sin 27‰ b2
64
4 ¸ 7.945 feet, which is reached at t œ
‰
v0 sin !
g
œ
35sin 27‰
32
¸ 0.497 seconds.
(c) For the time, solve y œ 4 a35 sin 27 bt 16t œ 0 for t, using the quadratic formula
tœ
35 sin 27‰ Éa35 sin 27‰ b2 256
32
2
¸ 1.201 sec. Then the range is about xa1.201b œ a35 cos 27‰ ba1.201b ¸ 37.453 feet.
(d) For the time, solve y œ 4 a35 sin 27‰ bt 16t2 œ 7 for t, using the quadratic formula
tœ
35 sin 27‰ Éa35 sin 27‰ b2 192
32
‰
¸ 0.254 and 0.740 seconds. At those times the ball is about
xa0.254b œ a35 cos 27 ba0.254b ¸ 7.921 feet and xa0.740b œ a35 cos 27‰ ba0.740b ¸ 23.077 feet the impact point,
or about 37.453 7.921 ¸ 29.532 feet and 37.453 23.077 ¸ 14.376 feet from the landing spot.
(e) Yes. It changes things because the ball won't clear the net (ymax ¸ 7.945).
34. x œ x! (v! cos !)t œ 0 (v! cos 40°)t ¸ 0.766 v! t and y œ y! (v! sin !)t "# gt# œ 6.5 (v! sin 40°)t 16t#
¸ 6.5 0.643 v! t 16t# ; now the shot went 73.833 ft Ê 73.833 œ 0.766 v! t Ê t ¸
#
Ê 0 œ 6.5 (0.643)(96.383) 16 Š 96.383
v! ‹ Ê 0 ¸ 68.474
148,635
v#!
96.383
v!
sec; the shot lands when y œ 0
Ê v! ¸ É 148,635
68.474 ¸ 46.6 ft/sec, the shot's initial
speed
35. Flight time œ 1 sec and the measure of the angle of elevation is about 64° (using a protractor) so that t œ
Ê1œ
2v! sin 64°
32
Ê v! ¸ 17.80 ft/sec. Then ymax œ
(17.80 sin 64°)#
2(32)
¸ 4.00 ft and R œ
v#!
g
sin 2! Ê R œ
2v! sin !
g
(17.80)#
32
sin 128°
¸ 7.80 ft Ê the engine traveled about 7.80 ft in 1 sec Ê the engine velocity was about 7.80 ft/sec
36. (a) ratb œ axatbbi ayatbbj; where xatb œ a145 cos 23‰ 14bt and yatb œ 2.5 a145 sin 23‰ bt 16t2 .
(b) ymax œ
av0 sin !b2
2g
2.5 œ
a145sin 23‰ b2
64
2.5 ¸ 52.655 feet, which is reached at t œ
‰
v0 sin !
g
œ
145sin 23‰
32
¸ 1.771 seconds.
(c) For the time, solve y œ 2.5 a145 sin 23 bt 16t œ 0 for t, using the quadratic formula
tœ
145 sin 23‰ Éa145 sin 23‰ b2 160
32
2
¸ 3.585 sec. Then the range at t ¸ 3.585 is about x œ a145 cos 23‰ 14ba3.585b
¸ 428.311 feet.
(d) For the time, solve y œ 2.5 a145 sin 23‰ bt 16t2 œ 20 for t, using the quadratic formula
tœ
145 sin 23‰ Éa145 sin 23‰ b2 1120
32
‰
¸ 0.342 and 3.199 seconds. At those times the ball is about
xa0.342b œ a145 cos 23 14ba0.342b ¸ 40.860 feet from home plate and xa3.199b œ a145 cos 23‰ 14ba3.199b
¸ 382.195 feet from home plate.
(e) Yes. According to part (d), the ball is still 20 feet above the ground when it is 382 feet from home plate.
37.
d2 r
dt2
k ddtr œ gj Ê Patb œ k and Qatb œ gj Ê ' Patb dt œ kt Ê vatb œ e' Patb dt œ ekt Ê
dr
dt
œ
1
vatb
' vatb Qatb dt
œ gekt ' ekt j dt œ gekt ek j C1 ‘ œ gk j Cekt , where C œ gC1 ; apply the initial condition:
kt
dr
dt ¹tœ0
œ av0 cos !bi av0 sin !bj œ gk j C Ê C œ av0 cos !bi ˆ kg v0 sin !‰j
Ê
œ ˆv0 ekt cos !‰i ˆ gk ekt ˆ gk v0 sin !‰‰j, r œ ' c ˆv0 ekt cos !‰i ˆ gk ekt ˆ gk v0 sin !‰‰j ddt
dr
dt
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 13.2 Integrals of Vector Functions; Projectile Motion
gt
v
œ ˆ k0 ekt cos !‰i Š k
eckt ˆ g
k
k
v0 sin !‰‹j C2 ; apply the initial condition:
v0 sin ! ‰
j C2 Ê C2 œ ˆ vk0 cos !‰i ˆ kg2
k
ˆ vk0 ˆ1 ekt ‰sin ! kg2 ˆ1 kt ekt ‰‰j
ra0b œ 0 œ ˆ vk0 cos !‰i ˆ kg2
Ê ratb œ ˆ vk0 ˆ1 ekt ‰cos !‰i
769
v0 sin ! ‰
j
k
152 ‰
38. (a) ratb œ axatbbi ayatbbj; where xatb œ ˆ 0.12
a1 e0.12t bacos 20‰ b and
152 ‰
32 ‰
0.12t
yatb œ 3 ˆ 0.12
a1 e0.12t basin 20‰ b ˆ 0.12
b
2 a1 0.12t e
(b) Solve graphically using a calculator or CAS: At t ¸ 1.484 seconds the ball reaches a maximum height of about 40.435
feet.
(c) Use a graphing calculator or CAS to find that y œ 0 when the ball has traveled for ¸ 3.126 seconds. The range is
152 ‰ˆ
about xa3.126b œ ˆ 0.12
1 e0.12a3.126b ‰acos 20‰ b ¸ 372.311 feet.
(d) Use a graphing calculator or CAS to find that y œ 30 for t ¸ 0.689 and 2.305 seconds, at which times the ball is about
xa0.689b ¸ 94.454 feet and xa2.305b ¸ 287.621 feet from home plate.
(e) Yes, the batter has hit a home run since a graph of the trajectory shows that the ball is more than 14 feet above the
ground when it passes over the fence.
39. (a)
'ab kr(t) dt œ 'ab [kf(t)i kg(t)j kh(t)k] dt œ 'ab [kf(t)] dt i
'a [kg(t)] dt j 'a [kh(t)] dt k
b
b
œ k Œ'a f(t) dt i 'a g(t) dt j 'a h(t) dt k œ k 'a r(t) dt
b
(b)
b
b
b
'ab [r" (t) „ r# (t)] dt œ 'ab acf" (t)i g" (t)j h" (t)kd „ cf# (t)i g# (t)j h# (t)kdb dt
b
œ 'a acf" (t) „ f# (t)d i [g" (t) „ g# (tb] j [h" (t) „ h# (t)] k) dt
b
b
b
œ 'a cf" (t) „ f# (t)d dt i 'a cg" (t) „ g# (t)d dt j 'a ch" (t) „ h# (t)d dt k
œ ”'a f" (t) dt i „ 'a f# (t) dt i• ”'a g" (t) dt j „ 'a g# (t) dt j • ”'a h" (t) dt k „ 'a h# (t) dt k•
b
b
b
b
b
b
œ 'a r" (t) dt „ 'a r# (t) dt
b
b
(c) Let C œ c" i c# j c$ k. Then 'a C † r(t) dt œ 'a cc" f(t) c# g(t) c$ h(t)d dt
b
b
'ab f(t) dt c# 'ab g(t) dt c$ 'ab h(t) dt = C † 'ab r(t) dt;
'ab C ‚ r(t) dt œ 'ab cc# h(t) c$ g(t)d i cc$ f(t) c" h(t)d j cc" g(t) c# f(t)d k dt
œ c"
œ ”c#
'ab
h(t) dt c$
'ab
g(t) dt• i ”c$
'ab
f(t) dt c"
'ab h(t) dt• j ”c" 'ab g(t) dt c# 'ab f(t) dt• k
œ C ‚ 'a r(t) dt
b
40. (a) Let u and r be continuous on [aß b]. Then lim u(t)r(t) œ lim [u(t)f(t)i u(t)g(t)j u(t)h(t)k]
t Ä t!
t Ä t!
œ u(t! )f(t! )i u(t! )g(t! )j u(t! )h(t! )k œ u(t! )r(t! ) Ê ur is continuous for every t! in [aß b].
(b) Let u and r be differentiable. Then dtd (ur) œ dtd [u(t)f(t)i u(t)g(t)j u(t)h(t)k]
dg
df ‰
du
dh ‰
ˆ du
œ ˆ du
dt f(t) u(t) dt i Š dt g(t) u(t) dt ‹j dt h(t) u(t) dt k
df
œ [f(t)i g(t)j h(t)k] du
dt u(t) Š dt i
dg
dt
j
dh
dt
dr
k‹ œ r du
dt u dt
41. (a) If R" (t) and R# (t) have identical derivatives on I, then
œ
d R#
dt
Ê
df"
dt
œ
df#
dt
,
dg"
dt
œ
dg#
dt
,
dh"
dt
œ
dh#
dt
d R"
dt
œ
df"
dt
i
dg"
dt
j
dh"
dt
kœ
df#
dt
i
dg#
dt
j
dh#
dt
k
Ê f" (t) œ f# (t) c" , g" (t) œ g# (t) c# , h" (t) œ h# (t) c$
Ê f" (t)i g" (t)j h" (t)k œ [f# (t) c" ]i [g# (t) c# ]j [h# (t) c$ ]k Ê R" (t) œ R# (t) C, where
C œ c" i c# j c$ k.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
770
Chapter 13 Vector-Valued Functions and Motion in Space
(b) Let R(t) be an antiderivative of r(t) on I. Then Rw (t) œ r(t). If U(t) is an antiderivative of r(t) on I, then
Uw (t) œ r(t). Thus Uw (t) œ Rw (t) on I Ê U(t) œ R(t) C.
42.
'at r(7 ) d7 œ dtd 'at [f(7 )i g(7 )j h(7 )k] d7 œ dtd 'at f(7 ) d7 i dtd 'at g(7 ) d7 j dtd 'at h(7 ) d7 k
t
t
œ f(t)i g(t)j h(t)k œ r(t). Since dtd 'a r(7 ) d7 œ r(t), we have that 'a r(7 ) d7 is an antiderivative of
t
a
r. If R is any antiderivative of r, then R(t) œ 'a r(7 ) d7 C by Exercise 41(b). Then R(a) œ 'a r(7 ) d7 C
t
b
œ 0 C Ê C œ R(a) Ê 'a r(7 ) d7 œ R(t) C œ R(t) R(a) Ê 'a r(7 ) d7 œ R(b) R(a).
d
dt
1 ‰
43. (a) ratb œ axatbbi ayatbbj; where xatb œ ˆ 0.08
a1 e0.08t ba152 cos 20‰ 17.6b and
152 ‰
32 ‰
0.08t
yatb œ 3 ˆ 0.08
a1 e0.08t basin 20‰ b ˆ 0.08
b
2 a1 0.08t e
(b) Solve graphically using a calculator or CAS: At t ¸ 1.527 seconds the ball reaches a maximum height of about 41.893
feet.
(c) Use a graphing calculator or CAS to find that y œ 0 when the ball has traveled for ¸ 3.181 seconds. The range is
1 ‰ˆ
about xa3.181b œ ˆ 0.08
1 e0.08a3.181b ‰a152 cos 20‰ 17.6b ¸ 351.734 feet.
(d) Use a graphing calculator or CAS to find that y œ 35 for t ¸ 0.877 and 2.190 seconds, at which times the ball is about
xa0.877b ¸ 106.028 feet and xa2.190b ¸ 251.530 feet from home plate.
(e) No; the range is less than 380 feet. To find the wind needed for a home run, first use the method of part (d) to find that
1 ‰ˆ
y œ 20 at t ¸ 0.376 and 2.716 seconds. Then define xawb œ ˆ 0.08
1 e0.08a2.716b ‰a152 cos 20‰ wb, and solve
xawb œ 380 to find w ¸ 12.846 ft/sec.
44. ymax œ
(v! sin !)#
2g
Ê
3
4
ymax œ
3(v! sin !)#
and
8g
# #
y œ (v! sin !)t "# gt# Ê
3(v! sin !)#
8g
œ (v! sin !)t "# gt#
Ê 3(v! sin !)# œ (8gv! sin !)t 4g t Ê 4g# t# (8gv! sin !)t 3(v! sin !)# œ 0 Ê 2gt 3v! sin ! œ 0 or
sin !
!
2gt v! sin ! œ 0 Ê t œ 3v!2gsin ! or t œ v! 2g
. Since the time it takes to reach ymax is tmax œ v! sin
,
g
then the time it takes the projectile to reach
3
4
of ymax is the shorter time t œ
v! sin !
2g
or half the time it takes
to reach the maximum height.
13.3 ARC LENGTH IN SPACE
1. r œ (2 cos t)i (2 sin t)j È5tk Ê v œ (2 sin t)i (2 cos t)j È5k
#
Ê kvk œ Ê(2 sin t)# (2 cos t)# ŠÈ5‹ œ È4 sin# t 4 cos# t 5 œ 3; T œ
œ ˆ 23 sin t‰ i ˆ 23 cos t‰ j
È5
3
1
v
kv k
1
k and Length œ '0 kvk dt œ '0 3 dt œ c3td 1! œ 31
2. r œ (6 sin 2t)i (6 cos 2t)j 5tk Ê v œ (12 cos 2t)i (12 sin 2t)j 5k
Ê kvk œ È(12 cos 2t)# (12 sin 2t)# 5# œ È144 cos# 2t 144 sin# 2t 25 œ 13; T œ
‰
ˆ 12
‰
œ ˆ 12
13 cos 2t i 13 sin 2t j
5
13
1
1
v
kv k
k and Length œ '0 kvk dt œ '0 13 dt œ c13td 1! œ 131
#
3. r œ ti 23 t$Î# k Ê v œ i t"Î# k Ê kvk œ É1# at"Î# b œ È1 t ; T œ
)
and Length œ '0 È1 t dt œ 23 (1 t)$Î# ‘ ! œ
8
v
kvk
œ
"
È1 t
i
Èt
È1 t
52
3
4. r œ (2 t)i (t 1)j tk Ê v œ i j k Ê kvk œ È1# (1)# 1# œ È3 ; T œ
v
kv k
œ
$
and Length œ '0 È3 dt œ ’È3t“ œ 3È3
3
k
!
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
È3
i
1
È3
j
"
È3
k
Section 13.3 Arc Length in Space
771
5. r œ acos$ tb j asin$ tb k Ê v œ a3 cos# t sin tb j a3 sin# t cos tb k Ê kvk
œ Éa3 cos# t sin tb# a3 sin# t cos tb# œ Èa9 cos# t sin# tb acos# t sin# tb œ 3 kcos t sin tk ;
Tœ
v
kvk
œ
3 cos# t sin t
3 kcos t sin tk
1Î2
Length œ '0
j
3 sin# t cos t
3 kcos t sin tk
k œ ( cos t)j (sin t)k , if 0 Ÿ t Ÿ
1Î2
1Î2
3 kcos t sin tk dt œ '0 3 cos t sin t dt œ '0
3
#
1
#
, and
1Î#
sin 2t dt œ 34 cos 2t‘ !
œ
3
#
6. r œ 6t$ i 2t$ j 3t$ k Ê v œ 18t# i 6t# j 9t# k Ê kvk œ Éa18t# b# a6t# b# a9t# b# œ È441t% œ 21t# ;
Tœ
v
kvk
œ
"8t#
21t#
i
6t#
21t#
j
7. r œ (t cos t)i (t sin t)j
9t#
21t#
kœ
6
7
2È2 $Î#
k
3 t
i 27 j 37 k and Length œ '1 21t# dt œ c7t$ d " œ 49
2
#
Ê v œ (cos t t sin t)i (sin t t cos t)j ŠÈ2 t"Î# ‹ k
#
Ê kvk œ Ê(cos t t sin t)# (sin t t cos t)# ŠÈ2 t‹ œ È1 t# 2t œ È(t 1)# œ kt 1k œ t 1, if t
Tœ
v
kvk
œ ˆ cos tt t1sin t ‰ i ˆ sin ttt1cos t ‰ j Š
È2 t"Î#
t1 ‹ k
1
1
and Length œ '0 (t 1) dt œ ’ t2 t“ œ
#
!
1#
2
0;
1
8. r œ (t sin t cos t)i (t cos t sin t)j Ê v œ (sin t t cos t sin t)i (cos t t sin t cos t)j
œ (t cos t)i (t sin t)j Ê kvk œ È(t cos t)# (t sin t)# œ Èt# œ ktk œ t if È2 Ÿ t Ÿ 2; T œ v
kvk
t‰
t‰
œ ˆ t cos
i ˆ t sin
j œ (cos t)i (sin t)j and Length œ 'È2 t dt œ ’ t2 “
t
t
2
#
#
È#
œ1
9. Let P(t! ) denote the point. Then v œ (5 cos t)i (5 sin t)j 12k and 261 œ '0 È25 cos# t 25 sin# t 144 dt
t!
œ '0 13 dt œ 13t! Ê t! œ 21, and the point is P(21) œ (5 sin 21ß 5 cos 21ß 241) œ (0ß 5ß 241)
t!
10. Let P(t! ) denote the point. Then v œ (12 cos t)i (12 sin t)j 5k and
131 œ '0 È144 cos# t 144 sin# t 25 dt œ '0 13 dt œ 13t! Ê t! œ 1, and the point is
t!
t!
P(1) œ (12 sin (1)ß 12 cos (1)ß 51) œ (0ß 12ß 51)
11. r œ (4 cos t)i (4 sin t)j 3tk Ê v œ (4 sin t)i (4 cos t)j 3k Ê kvk œ È(4 sin t)# (4 cos t)# 3#
œ È25 œ 5 Ê s(t) œ '0 5 d7 œ 5t Ê Length œ s ˆ 1# ‰ œ
t
51
#
12. r œ (cos t t sin t)i (sin t t cos t)j Ê v œ (sin t sin t t cos t)i (cos t cos t t sin t)j
t
œ (t cos t)i (t sin t)j Ê kvk œ È(t cos t)# (t cos t)# œ œ Èt# œ t, since 1# Ÿ t Ÿ 1 Ê s(t) œ '0 7 d7 œ
Ê Length œ s(1) s ˆ 1# ‰ œ
1#
#
ˆ 1# ‰#
#
œ
t#
#
31 #
8
13. r œ aet cos tb i aet sin tb j et k Ê v œ aet cos t et sin tb i aet sin t et cos tb j et k
Ê kvk œ Éaet cos t et sin tb# aet sin t et cos tb# aet b# œ œ È3e2t œ È3 et Ê s(t) œ '0 È3 e7 d7
t
œ È3 et È3 Ê Length œ s(0) s( ln 4) œ 0 ŠÈ3 e ln 4 È3‹ œ
3È 3
4
14. r œ (1 2t)i (1 3t)j (6 6t)k Ê v œ 2i 3j 6k Ê kvk œ È2# 3# (6)# œ 7 Ê s(t) œ '0 7 d7 œ 7t
t
Ê Length œ s(0) s(1) œ 0 (7) œ 7
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
772
Chapter 13 Vector-Valued Functions and Motion in Space
#
#
15. r œ ŠÈ2t‹ i ŠÈ2t‹ j a1 t# b k Ê v œ È2i È2j 2tk Ê kvk œ ÊŠÈ2‹ ŠÈ2‹ (2t)# œ È4 4t#
œ 2È1 t# Ê Length œ '0 2È1 t# dt œ ’2 Š 2t È1 t#
1
"
#
"
ln Št È1 t# ‹‹“ œ È2 ln Š1 È2‹
!
16. Let the helix make one complete turn from t œ 0 to t œ 21.
Note that the radius of the cylinder is 1 Ê the
circumference of the base is 21. When t œ 21, the point P is
(cos 21ß sin 21ß 21) œ (1ß 0ß 21) Ê the cylinder is 21 units
high. Cut the cylinder along PQ and flatten. The resulting
rectangle has a width equal to the circumference of the
cylinder œ 21 and a height equal to 21, the height of the
cylinder. Therefore, the rectangle is a square and the portion
of the helix from t œ 0 to t œ 21 is its diagonal.
17. (a) r œ (cos t)i (sin t)j (" cos t)k, 0 Ÿ t Ÿ 21 Ê x œ cos t, y œ sin t, z œ 1 cos t Ê x# y#
œ cos# t sin# t œ 1, a right circular cylinder with the z-axis as the axis and radius œ 1. Therefore
P(cos tß sin tß 1 cos t) lies on the cylinder x# y# œ 1; t œ 0 Ê P(1ß 0ß 0) is on the curve; t œ 1# Ê Q(!ß 1ß 1)
Ä
Ä
is on the curve; t œ 1 Ê R(1ß 0ß 2) is on the curve. Then PQ œ i j k and PR œ 2i 2k
j k×
Ô i
Ä
Ä
Ê PQ ‚ PR œ 1 " " œ 2i 2k is a vector normal to the plane of P, Q, and R. Then the
Õ 2 0 2 Ø
plane containing P, Q, and R has an equation 2x 2z œ 2(1) 2(0) or x z œ 1. Any point on the curve
will satisfy this equation since x z œ cos t (1 cos t) œ 1. Therefore, any point on the curve lies on the
intersection of the cylinder x# y# œ 1 and the plane x z œ 1 Ê the curve is an ellipse.
(b) v œ ( sin t)i (cos t)j (sin t)k Ê kvk œ Èsin# t cos# t sin# t œ È1 sin# t Ê T œ kvvk
œ
( sin t)i (cos t)j (sin t)k
È1 sin# t
Ê T(0) œ j , T ˆ 1# ‰ œ
ik
È2
, T(1) œ j , T ˆ 3#1 ‰ œ
ik
È2
(c) a œ ( cos t)i (sin t)j (cos t)k ; n œ i k is
normal to the plane x z œ 1 Ê n † a œ cos t cos t
œ 0 Ê a is orthogonal to n Ê a is parallel to the
plane; a(0) œ i k , a ˆ 1# ‰ œ j , a a1b œ i k ,
‰œj
a ˆ 31
#
21
(d) kvk œ È1 sin# t (See part (b) Ê L œ '0 È1 sin# t dt
(e) L ¸ 7.64 (by Mathematica)
18. (a) r œ (cos 4t)i (sin 4t)j 4tk Ê v œ (4 sin 4t)i (4 cos 4t)j 4k Ê kvk œ È(4 sin 4t)# (4 cos 4t)# 4#
1Î2
œ È32 œ 4È2 Ê Length œ '0 4È2 dt œ ’4È2 t“
1Î#
!
œ 21È2
(b) r œ ˆcos #t ‰ i ˆsin #t ‰ j #t k Ê v œ ˆ #" sin #t ‰ i ˆ #" cos #t ‰ j #" k
#
#
#
Ê kvk œ Ɉ "# sin #t ‰ ˆ #" cos #t ‰ ˆ #" ‰ œ É 4"
"
4
œ
È2
#
41
Ê Length œ '0
È2
#
dt œ ’
È2
2
t“
%1
!
œ 21 È 2
(c) r œ (cos t)i (sin t)j tk Ê v œ ( sin t)i (cos t)j k Ê kvk œ È( sin t)# ( cos t)# (1)# œ È1 1
œ È2 Ê Length œ 'c21 È2 dt œ ’È2 t“
0
!
#1
œ 21 È 2
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 13.4 Curvature and Normal Vectors of a Curve
773
19. nPQB œ nQOB œ t and PQ œ arc (AQ) œ t since
PQ œ length of the unwound string œ length of arc (AQ);
thus x œ OB BC œ OB DP œ cos t t sin t, and
y œ PC œ QB QD œ sin t t cos t
20. r œ acos t t sin tbi asin t t cos tbj Ê v œ asin t t cos t sin tbi acos t atasin tb cos tbbj
œ at cos tbi at sin tbj Ê kvk œ Éat cos tb2 at sin tb2 œ Èt2 œ ktk œ t, t
0ÊTœ
v
kvk
œ
t cos t
t i
t sin t
t j
œ cos t i sin t j
21. v œ
d
dt ax0
t u 1 bi
d
dt ay0
t u 2 bj
d
dt az0
t u3 bk œ u1 i u2 j u3 k œ u, so satb œ '0 lvldt œ '0 luld7 œ '0 1 d7 œ t
t
t
t
22. ratb œ t i t2 j t3 k Ê vatb œ i 2t j 3t2 k Ê lvatbl œ Éa1b2 a2tb2 a3t2 b2 œ È1 4t2 9t4 . a0, 0, 0b Ê t œ 0
and a2, 4, 8b Ê t œ 2. Thus L œ '0 lvatbl dt œ '0 È1 4t2 9t4 dt. Using Simpson's rule with n œ 10 and
2
?x œ
20
10
œ 0.2 Ê L ¸
0.2
3 Šlva0bl
2
4lva0.2bl 2lva0.4bl 4lva0.6bl 2lva0.8bl 4lva1bl 2lva1.2bl 4lva1.4bl
2lva1.6bl 4lva1.8bl lva2bl‹ ¸
0.2
3 Š1
4a1.0837b 2a1.3676b 4a1.8991b 2a2.6919b 4a3.7417b
2a5.0421b 4a6.5890b 2a8.3800b 4a10.4134b 12.6886‹ œ
0.2
3 a143.5594b
¸ 9.5706
13.4 CURVATURE AND NORMAL VECTORS OF A CURVE
sin t ‰
È1# ( tan t)# œ Èsec# t œ ksec tk œ sec t, since
1. r œ ti ln (cos t)j Ê v œ i ˆ cos
t j œ i (tan t)j Ê kvk œ
tan t ‰
dT
1# t 1# Ê T œ kvvk œ ˆ sec" t ‰ i ˆ sec
t j œ (cos t)i (sin t)j ; dt œ ( sin t)i (cos t)j
Ê ¸ ddtT ¸ œ È( sin t)# ( cos t)# œ 1 Ê N œ
ˆ ddtT ‰
¸ ddtT ¸
œ ( sin t)i (cos t)j ; , œ
1
kv k
† ¸ ddtT ¸ œ
"
sec t
† 1 œ cos t.
t tan t ‰
2. r œ ln (sec t)i tj Ê v œ ˆ secsec
i j œ (tan t)i j Ê kvk œ È( tan t)# 1# œ Èsec# t œ ksec tk œ sec t,
t
1
1
v
tan
since # t # Ê T œ kvk œ ˆ sec tt ‰ i ˆ sec1 t ‰ j œ (sin t)i (cos t)j ; ddtT œ (cos t)i (sin t)j
Ê ¸ ddtT ¸ œ È(cos t)# ( sin t)# œ 1 Ê N œ
ˆ ddtT ‰
¸ ddtT ¸
œ (cos t)i (sin t)j ; , œ
1
kv k
† ¸ ddtT ¸ œ
"
sec t
† 1 œ cos t.
3. r œ (2t 3)i a5 t# b j Ê v œ 2i 2tj Ê kvk œ È2# (2t)# œ 2È1 t# Ê T œ kvvk œ È 2 # i
2 1t
Í
#
#
Í
Í
"
t
"
T
"
t
dT
d
t
¸ dt ¸ œ
œ È # i È # j ; dt œ
$i
$ j Ê
$
$
1 t
1 t
ŠÈ1 t# ‹
ŠÈ1 t# ‹
ŠÈ1 t# ‹
Ì ŠÈ1 t# ‹
œ É a1 "t# b# œ
"
1t#
Ê Nœ
ˆ ddtT ‰
¸ ddtT ¸
œ
t
È1 t#
i
"
È1 t#
j; , œ
1
kvk
† ¸ ddtT ¸ œ
"
#È1 t#
†
"
1 t#
œ
2t
2È1 t#
j
"
# a1 t# b3/2
4. r œ (cos t t sin t)i (sin t t cos t)j Ê v œ (t cos t)i (t sin t)j Ê kvk œ È( t cos t)# (t sin t)# œ Èt# œ ktk œ t, since
t 0 Ê T œ kvvk œ (t cos t)it (t sin t)j œ (cos t)i (sin t)j ; ddtT œ ( sin t)i (cos t)j Ê ¸ ddtT ¸ œ È( sin t)# (cos t)#
œ1ÊNœ
ˆ ddtT ‰
¸ ddtT ¸
œ ( sin t)i (cos t)j ; , œ
1
kv k
† ¸ ddtT ¸ œ
"
t
†1œ
"
t
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
774
Chapter 13 Vector-Valued Functions and Motion in Space
5. (a) ,axb œ
1
kvaxbk
† ¹ dTdtaxb ¹. Now, v œ i f w axbj Ê kvaxbk œ É1 c f w axb d2 Ê T œ
1Î2
œ Š1 c f w axb d2 ‹
Ê
Ta b
¹ d dtx ¹
w
dT
dt axb
ww
ww
ww
œ
f axbf axb
w
ww
Š1 c f axb d ‹
axb d2 Š1 c f axb d2 ‹
w
3
Î
2
Š1 c f axb d ‹
w
w
1
a1 Òf axbÓ2 b1
w
2
Î
†
k f a x bk
k1 Òf axbÓ2 k
ww
w
œ cos x, since 1# x
œ
i
f ax b
ww
3Î2
2
Š1 c f axb d ‹
j
w
kf axbk
ww
2
¹1 c f axb d ¹
w
kf axbk
ww
œ
3Î2
2
Š1 c f axb d ‹
w
œ ˆ cos" x ‰ ( sin x) œ tan x Ê
dy
dx
3Î2
2
w
Î
w
(b) y œ ln (cos x) Ê
"
sec x
j. Thus
Í
2
Í
2
f axbf axb
cf
Í
f ax b
œ ” Š1 c f axb d2 ‹3 2 •
3 2 œ Ë
2
Ì
Š1 c f axb d ‹
Thus ,axb œ
œ
1Î2
i f w axbŠ1 c f w axb d2 ‹
v
kvk
d# y
dx#
œ sec# x Ê , œ
k sec# xk
c1 ( tan x)# d$Î#
œ
sec# x
ksec$ xk
1
#
(c) Note that f ww (x) œ 0 at an inflection point.
Þ
Þ
Þ
Þ
6. (a) r œ f(t)i g(t)j œ xi yj Ê v œ xi yj Ê kvk œ Èx# y# Ê T œ
dT
dt
œ
Þ Þ ÞÞ Þ ÞÞ
Þ Þ ÞÞ Þ ÞÞ
yay x x yb
xax y y x b
dT
Þ # Þ # 3/2 i Þ # Þ # 3/2 j Ê ¸ dt ¸
ax y b
ax y b
Þ ÞÞ Þ ÞÞ
k y x x yk
1
1
dT
Þ
Þ ; , œ kvk † ¸ dt ¸ œ È Þ # Þ #
kx# y# k
x y
œ
Þ Þ ÞÞ Þ ÞÞ
yay x x yb 2
Þ
Þ 3/2 “ ’
ax# y# b
Þ ÞÞ Þ ÞÞ
Þ ÞÞ Þ ÞÞ
k y x x yk
lxyyx l
Þ # Þ # œ Þ # Þ # 3/2 .
kx y k
ax y b
œ Ê’
†
v
kv k
Þ Þ ÞÞ Þ ÞÞ
xax y y xb 2
Þ
Þ 3/2 “
ax# y# b
œ
Þ
Þ
y
x
ÈxÞ # yÞ # i ÈxÞ # yÞ # j
œÊ
Þ
Þ Þ ÞÞ Þ ÞÞ
ay# x# bay x x yb2
Þ
Þ 3
ax# y# b
Þ
ÞÞ
Þ
(b) r(t) œ ti ln (sin t)j , 0 t 1 Ê x œ t and y œ ln (sin t) Ê x œ 1, x œ 0; y œ
Ê ,œ
k csc# t 0k
a1 cot# t)b$Î#
"
œ
csc# t
csc$ t
cos t
sin t
ÞÞ
œ cot t, y œ csc# t
œ sin t
Þ
t
"
(sinh t)i ln (cosh t)j Ê x œ tan" (sinh t) and y œ ln (cosh t) Ê x œ 1 cosh
sinh# t œ cosh t
$
#
ÞÞ
Þ
ÞÞ
ksech t sech t tanh tk
sinh t
#
œ sech t, x œ sech t tanh t; y œ cosh
œ ksech tk œ sech t
asech# t tanh# tb
t œ tanh t, y œ sech t Ê , œ
(c) r(t) œ tan
7. (a) r(t) œ f(t)i g(t)j Ê v œ f w (t)i gw (t)j is tangent to the curve at the point (f(t)ß g(t));
n † v œ c gw (t)i f w (t)jd † cf w (t)i gw (t)jd œ gw (t)f w (t) f w (t)gw (t) œ 0; n † v œ (n † v) œ 0; thus, n and n are
both normal to the curve at the point
(b) r(t) œ ti e2t j Ê v œ i 2e2t j Ê n œ 2e2t i j points toward the concave side of the curve; N œ knnk and
knk œ È4e4t 1 Ê N œ
2e2t
È1 4e4t
(c) r(t) œ È4 t# i tj Ê v œ
Nœ
n
knk
and knk œ É1
t#
4 t#
i
t
È4 t#
œ
"
È1 4e4t
j
i j Ê n œ i
Ê Nœ
2
È4 t#
"
#
t
È4 t#
j points toward the concave side of the curve;
ŠÈ4 t# i tj‹
8. (a) r(t) œ ti "3 t$ j Ê v œ i t# j Ê n œ t# i j points toward the concave side of the curve when t 0 and
n œ t# i j points toward the concave side when t 0 Ê N œ
Nœ
"
È1 t%
2ktk
;
1 t%
at# i jb for t 0 and
at# i jb for t 0
(b) From part (a), kvk œ È1 t% Ê T œ
œ
"
È1 t%
Nœ
ˆ ddtT ‰
¸ ddtT ¸
œ
2t$
1 t%
2ktk Š a1 t% b$Î# i
"
È1 t% i
t#
È1 t% j
2t
$Î# j‹
a1 t% b
œ
Ê
dT
dt
t$
i
ktkÈ1 t%
œ
2t$
$Î# i
a1 t% b
t
j;
ktkÈ1 t%
¸ ddsT ¸ œ ¸ ddtT
dT ¸
dt tœ0 œ 0 so the curvature , œ
œ "3 t$ Ê y œ 3" x$ , the curve is the
curve has a point of inflection;
undefined. Since x œ t and y
2t
$Î# j
a1 t% b
6
2
Ê ¸ ddtT ¸ œ É a4t1 t%4tb$
t Á 0. N does not exist at t œ 0, where the
†
dt ¸
ds
œ 0 at t œ 0 Ê N œ
" dT
, ds
is
cubic power curve which is concave down for
x œ t 0 and concave up for x œ t 0.
9. r œ (3 sin t)i (3 cos t)j 4tk Ê v œ (3 cos t)i (3 sin t)j 4k Ê kvk œ È(3 cos t)# (3 sin t)# 4# œ È25
œ 5 Ê T œ kvvk œ ˆ 35 cos t‰ i ˆ 35 sin t‰ j 45 k Ê ddtT œ ˆ 53 sin t‰ i ˆ 53 cos t‰ j
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 13.4 Curvature and Normal Vectors of a Curve
#
#
Ê ¸ ddtT ¸ œ Ɉ 35 sin t‰ ˆ 35 cos t‰ œ
Ê Nœ
3
5
ˆ ddtT ‰
¸ ddtT ¸
œ ( sin t)i (cos t)j ; , œ
1
5
†
3
5
œ
775
3
25
10. r œ (cos t t sin t)i (sin t t cos t)j 3k Ê v œ (t cos t)i (t sin t)j Ê kvk œ È(t cos t)# (t sin t)# œ Èt#
œ ktk œ t, if t 0 Ê T œ
v
kvk
œ (cos t)i (sin t)j , t 0 Ê
ˆ ddtT ‰
¸ ddtT ¸
Ê ¸ ddtT ¸ œ È( sin t)# (cos t)# œ 1 Ê N œ
œ ( sin t)i (cos t)j
dT
dt
"
t
œ ( sin t)i (cos t)j ; , œ
†1œ
"
t
11. r œ aet cos tb i aet sin tb j 2k Ê v œ aet cos t et sin tb i aet sin t et cos tb j Ê
kvk œ Éaet cos t et sin tb# aet sin t et cos tb# œ È2e2t œ et È2 ;
Tœ
v
kvk
cos t
t sin t
œ Š cos È
‹ i Š sin tÈ
‹j Ê
2
2
#
dT
dt
t sin t
œ Š sinÈt 2 cos t ‹ i Š cos È
‹j
2
#
t sin t
Ê ¸ ddtT ¸ œ ÊŠ sinÈt cos t ‹ Š cos È
‹ œ1 Ê Nœ
2
,œ
1
kvk
† ¸ ddtT ¸ œ
2
†1œ
1
et È2
ˆ ddtT ‰
¸ ddtT ¸
œ Š cosÈt sin t ‹ i Š sinÈt cos t ‹ j ;
2
2
1
et È2
12. r œ (6 sin 2t)i (6 cos 2t)j 5tk Ê v œ (12 cos 2t)i (12 sin 2t)j 5k Ê kvk œ È(12 cos 2t)# (12 sin 2t)# 5#
5
dT
‰
ˆ 12
‰
ˆ 24
‰
ˆ 24
‰
œ È169 œ 13 Ê T œ kvvk œ ˆ 12
13 cos 2t i 13 sin 2t j 13 k Ê dt œ 13 sin 2t i 13 cos 2t j
‰# ˆ 24
‰# œ
Ê ¸ ddtT ¸ œ Ɉ 24
13 sin 2t
13 cos 2t
,œ
1
kvk
† ¸ ddtT ¸ œ
$
#
†
1
13
24
13
œ
24
13
ˆ ddtT ‰
¸ ddtT ¸
Ê Nœ
œ ( sin 2t)i (cos 2t)j ;
24
169 .
13. r œ Š t3 ‹ i Š t# ‹ j , t 0 Ê v œ t# i tj Ê kvk œ Èt% t# œ tÈt# 1, since t 0 Ê T œ
œ
t
Èt# t
i
1
Èt# 1
#
t
œ É at1#
œ
1 b$
j Ê
"
t# 1
œ
dT
dt
Ê Nœ
i
1
at# 1b$Î#
ˆ ddtT ‰
¸ ddtT ¸
14. r œ acos$ tb i asin$ tb j , 0 t
œ
1
#
1
Èt# 1
t
at# 1b$Î#
i
j Ê ¸ ddtT ¸ œ ÊŠ
t
Èt# 1
j; , œ
1
kvk
#
"
‹
at# 1b$Î#
† ¸ ddtT ¸ œ
Š
1
tÈt# 1
†
t
‹
at# 1b$Î#
1
t# 1
œ
v
kv k
#
"
.
t at# 1b$Î#
Ê v œ a3 cos# t sin tb i a3 sin# t cos tb j
Ê kvk œ Éa3 cos# t sin tb# a3 sin# t cos tb# œ È9 cos% t sin# t 9 sin% t cos# t œ 3 cos t sin t, since 0 t
Ê Tœ
v
kv k
œ ( cos t)i (sin t)j Ê
œ (sin t)i (cos t)j; , œ
1
kvk
† ¸ ddtT ¸ œ
dT
dt
œ (sin t)i (cos t)j Ê ¸ ddtT ¸ œ Èsin# t cos# t œ 1 Ê N œ
1
3 cos t sin t
†1œ
v
kv k
œ ˆsech at ‰ i ˆtanh at ‰ j Ê
Ê ¸ ddtT ¸ œ É a"# sech# ˆ at ‰ tanh# ˆ at ‰
,œ
1
kv k
† ¸ ddtT ¸ œ
1
cosh
t
a
†
"
a
sech ˆ at ‰ œ
"
a
"
a#
dT
dt
ˆ ddtT ‰
¸ ddtT ¸
1
3 cos t sin t .
15. r œ ti ˆa cosh at ‰ j , a 0 Ê v œ i ˆsinh at ‰ j Ê kvk œ É1 sinh# ˆ at ‰ œ Écosh# ˆ at ‰ œ cosh
Ê Tœ
1
#
œ ˆ "a sech
sech% ˆ at ‰ œ
"
a
t
a
t
a
tanh at ‰ i ˆ "a sech# at ‰ j
sech ˆ at ‰ Ê N œ
ˆ ddtT ‰
¸ ddtT ¸
œ ˆ tanh at ‰ i ˆsech at ‰ j ;
sech# ˆ at ‰.
16. r œ (cosh t)i (sinh t)j tk Ê v œ (sinh t)i (cosh t)j k Ê kvk œ Èsinh# t ( cosh t)# 1 œ È2 cosh t
Ê Tœ
Ê
,œ
¸ ddtT ¸
1
kv k
v
kv k
œ
œ Š È" tanh t‹ i
2
É "#
† ¸ ddtT ¸ œ
sech% t
1
È2 cosh t
†
"
#
"
È2
j Š È" sech t‹ k Ê
2
sech# t tanh# t œ
"
È2
sech t œ
"
#
"
È2
dT
dt
œ Š È" sech# t‹ i Š È" sech t tanh t‹ k
sech t Ê N œ
2
ˆ ddtT ‰
¸ ddtT ¸ œ
2
(sech t)i (tanh t)k ;
sech# t.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
776
Chapter 13 Vector-Valued Functions and Motion in Space
17. y œ ax# Ê yw œ 2ax Ê yww œ 2a; from Exercise 5(a), ,(x) œ
Ê ,w (x) œ 3# k2ak a1 4a# x# b
&Î#
k2ak
a1 4a# x# b$Î#
œ k2ak a1 4a# x# b
$Î#
a8a# xb ; thus, ,w (x) œ 0 Ê x œ 0. Now, ,w (x) 0 for x 0 and ,w (x) 0 for
x 0 so that ,(x) has an absolute maximum at x œ 0 which is the vertex of the parabola. Since x œ 0 is the
only critical point for ,(x), the curvature has no minimum value.
18. r œ (a cos t)i (b sin t)j
â
i
j
â
â
œ â a sin t b cos t
â
â a cos t b sin t
Ê v œ (a sin t)i (b cos t)j Ê a œ (a cos t)i (b sin t)j Ê v ‚ a
â
kâ
â
0 â œ abk Ê kv ‚ ak œ kabk œ ab, since a b 0; , (t) œ kvkv‚k$ak
â
0â
$Î#
œ ab aa# sin# t b# cos# tb
&Î#
; ,w (t) œ #3 (ab) aa# sin# t b# cos# tb
œ 3# (ab) aa# b# b (sin 2t) aa# sin# t b# cos# tb
points on the major axis, or t œ 1# ,
1
#
0t
and for 1 t
31
# ;
w
31
#
&Î#
a2a# sin t cos t 2b# sin t cos tb
; thus, ,w (t) œ 0 Ê sin 2t œ 0 Ê t œ 0, 1 identifying
identifying points on the minor axis. Furthermore, ,w (t) 0 for
, (t) 0 for
1
#
t 1 and
31
#
t 21. Therefore, the points associated
with t œ 0 and t œ 1 on the major axis give absolute maximum curvature and the points associated with t œ
and t œ
19. , œ
31
#
on the minor axis give absolute minimum curvature.
d,
da
Ê
a
a# b #
a b and
1
#
d,
da
a # b #
a a # b # b#
œ
;
d,
da
œ 0 Ê a# b# œ 0 Ê a œ „ b Ê a œ b since a, b
0 if a b Ê , is at a maximum for a œ b and ,(b) œ
b
b# b#
œ
"
2b
0. Now,
d,
da
0 if
is the maximum value of ,.
20. (a) From Example 5, the curvature of the helix r(t) œ (a cos t)i (a sin t)j btk, a, b 0 is , œ a# a b# ; also
kvk œ Èa# b# . For the helix r(t) œ (3 cos t)i (3 sin t)j tk, 0 Ÿ t Ÿ 41, a œ 3 and b œ 1 Ê , œ 3# 3 1# œ
41
and kvk œ È10 Ê K œ '0
3
10
È10 dt œ ’
3
È10
t“
%1
!
3
10
121
È10
œ
(b) y œ x# Ê x œ t and y œ t# , _ t _ Ê r(t) œ ti t# j Ê v œ i 2tj Ê kvk œ È1 4t# ;
Tœ
,œ
1
È1 4t# i
1
È1 4t#
œaÄ
lim
_
†
2t
dT
È1 4t# j; dt
2
1 4t#
'a0
œ
2
14t#
œ
4t
i
a1 4t# b3/2
2
3.
ŠÈ1 4t# ‹
dt lim
bÄ_
œaÄ
lim
a tan" 2ab lim
_
¸ ddtT ¸ œ É
_
Then K œ 'c_
'0b 1 24t
bÄ_
2
j;
a1 4t# b3/2
#
2
$
ŠÈ1 4t# ‹
16t2 4
a1 4t# b3
œ
2
1 4t# .
ŠÈ1 4t# ‹ dt œ ' _ 124t# dt
!
dt œ a Ä
lim
ctan" 2td a lim
_
a tan" 2bb œ
1
#
1
#
Thus
_
bÄ_
œ1
ctan" 2td 0
b
21. r œ ti (sin t)j Ê v œ i (cos t)j Ê kvk œ È1# (cos t)# œ È1 cos# t Ê ¸v ˆ 1# ‰¸ œ É1 cos# ˆ 1# ‰ œ 1; T œ
œ
i cos t j
È1 cos2 t
Ê 3œ
"
1
Ê
dT
dt
œ
sin t cos t
i
a1 cos2 tb3/2
sin t
j
a1 cos2 tb3/2
Ê ¸ ddtT ¸ œ
ksin tk
1 cos2 t ;
¸ ddtT ¸
tœ 12
œ
¸sin 12 ¸
1 cos2 ˆ 12 ‰
œ
1
1
œ 1. Thus ,ˆ 12 ‰ œ
1
1
†1œ1
#
œ 1 and the center is ˆ 1# ß 0‰ Ê ˆx 1# ‰ y# œ 1
2
22. r œ (2 ln t)i ˆt "t ‰ j Ê v œ ˆ 2t ‰ i ˆ1 t"# ‰ j Ê kvk œ É t42 ˆ1 t12 ‰ œ
dT
dt
œ
2ˆt2 1‰
i
at2 1b2
œ
"
#
Ê 3œ
"
,
at2
4t
j
1 b2
2
Ê ¸ ddtT ¸ œ Ê 4at
1b2 16t2
at2 1b4
œ
2
t2 1 .
Thus , œ
1
kv k
t2 1
t2
† ¸ ddtT ¸ œ
ÊTœ
t2
t2 1
†
2
t2 1
v
kvk
œ
œ
2t2
at2 1b2
2t
t2 1 i
t2 1
t2 1 j;
Ê ,a1b œ
2
22
œ 2. The circle of curvature is tangent to the curve at P(0ß 2) Ê circle has same tangent as the curve
Ê v(1) œ 2i is tangent to the circle Ê the center lies on the y-axis. If t Á 1 (t 0), then (t 1)# 0
#
Ê t# 2t 1 0 Ê t# 1 2t Ê t t 1 2 since t 0 Ê t "t 2 Ê ˆt "t ‰ 2 Ê y 2 on both
sides of (0ß 2) Ê the curve is concave down Ê center of circle of curvature is (0ß 4) Ê x# (y 4)# œ 4
is an equation of the circle of curvature
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
v
kvk
Section 13.4 Curvature and Normal Vectors of a Curve
23. y œ x# Ê f w (x) œ 2x and f ww (x) œ 2
Ê ,œ
24. y œ
x%
4
k2 k
a1 (2x)# b$Î#
œ
2
a1 4x# b$Î#
Ê f w (x) œ x$ and f ww (x) œ 3x#
Ê ,œ
k3x# k
#
$Î#
Š1 ax$ b ‹
œ
3x#
a1 x' b$Î#
25. y œ sin x Ê f w (x) œ cos x and f ww (x) œ sin x
Ê ,œ
k sin xk
a1 cos# xb$Î#
œ
ksin xk
a1 cos# xb$Î#
26. y œ ex Ê f w (x) œ ex and f ww (x) œ ex
Ê ,œ
kex k
# $Î#
Š1 aex b ‹
œ
ex
ˆ1 e2x ‰$Î#
27-34. Example CAS commands:
Maple:
with( plots );
r := t -> [3*cos(t),5*sin(t)];
lo := 0;
hi := 2*Pi;
t0 := Pi/4;
P1 := plot( [r(t)[], t=lo..hi] ):
display( P1, scaling=constrained, title="#27(a) (Section 13.4)" );
CURVATURE := (x,y,t) ->simplify(abs(diff(x,t)*diff(y,t,t)-diff(y,t)*diff(x,t,t))/(diff(x,t)^2+diff(y,t)^2)^(3/2));
kappa := eval(CURVATURE(r(t)[],t),t=t0);
UnitNormal := (x,y,t) ->expand( [-diff(y,t),diff(x,t)]/sqrt(diff(x,t)^2+diff(y,t)^2) );
N := eval( UnitNormal(r(t)[],t), t=t0 );
C := expand( r(t0) + N/kappa );
OscCircle := (x-C[1])^2+(y-C[2])^2 = 1/kappa^2;
evalf( OscCircle );
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
777
778
Chapter 13 Vector-Valued Functions and Motion in Space
P2 := implicitplot( (x-C[1])^2+(y-C[2])^2 = 1/kappa^2, x=-7..4, y=-4..6, color=blue ):
display( [P1,P2], scaling=constrained, title="#27(e) (Section 13.4)" );
Mathematica: (assigned functions and parameters may vary)
In Mathematica, the dot product can be applied either with a period "." or with the word, "Dot".
Similarly, the cross product can be applied either with a very small "x" (in the palette next to the arrow) or with the word,
"Cross". However, the Cross command assumes the vectors are in three dimensions
For the purposes of applying the cross product command, we will define the position vector r as a three dimensional vector
with zero for its z-component. For graphing, we will use only the first two components.
Clear[r, t, x, y]
r[t_]={3 Cos[t], 5 Sin[t] }
t0= 1 /4; tmin= 0; tmax= 21;
r2[t_]= {r[t][[1]], r[t][[2]]}
pp=ParametricPlot[r2[t], {t, tmin, tmax}];
mag[v_]=Sqrt[v.v]
vel[t_]= r'[t]
speed[t_]=mag[vel[t]]
acc[t_]= vel'[t]
curv[t_]= mag[Cross[vel[t],acc[t]]]/speed[t]3 //Simplify
unittan[t_]= vel[t]/speed[t]//Simplify
unitnorm[t_]= unittan'[t] / mag[unittan'[t]]
ctr= r[t0] + (1 / curv[t0]) unitnorm[t0] //Simplify
{a,b}= {ctr[[1]], ctr[[2]]}
To plot the osculating circle, load a graphics package and then plot it, and show it together with the original curve.
<;
t0 := sqrt(3);
rr := eval( r, t=t0 );
v := map( diff, r, t );
vv := eval( v, t=t0 );
a := map( diff, v, t );
aa := eval( a, t=t0 );
s := simplify(Norm( v, 2 )) assuming t::real;
ss := eval( s, t=t0 );
T := v/s;
TT := vv/ss ;
q1 := map( diff, simplify(T), t ):
NN := simplify(eval( q1/Norm(q1,2), t=t0 ));
BB := CrossProduct( TT, NN );
kappa := Norm(CrossProduct(vv,aa),2)/ss^3;
tau := simplify( Determinant(< vv, aa, eval(map(diff,a,t),t=t0) >)/Norm(CrossProduct(vv,aa),2)^3 );
a_t := eval( diff( s, t ), t=t0 );
a_n := evalf[4]( kappa*ss^2 );
Mathematica: (assigned functions and value for t0 will vary)
Clear[t, v, a, t]
mag[vector_]:=Sqrt[vector.vector]
Print["The position vector is ", r[t_]={t Cos[t], t Sin[t], t}]
Print["The velocity vector is ", v[t_]= r'[t]]
Print["The acceleration vector is ", a[t_]= v'[t]]
Print["The speed is ", speed[t_]= mag[v[t]]//Simplify]
Print["The unit tangent vector is ", utan[t_]= v[t]/speed[t] //Simplify]
Print["The curvature is ", curv[t_]= mag[Cross[v[t],a[t]]] / speed[t]3 //Simplify]
Print["The torsion is ", torsion[t_]= Det[{v[t], a[t], a'[t]}] / mag[Cross[v[t],a[t]]]2 //Simplify]
Print["The unit normal vector is ", unorm[t_]= utan'[t] / mag[utan'[t]] //Simplify]
Print["The unit binormal vector is ", ubinorm[t_]= Cross[utan[t],unorm[t]] //Simplify]
Print["The tangential component of the acceleration is ", at[t_]=a[t].utan[t] //Simplify]
Print["The normal component of the acceleration is ", an[t_]=a[t].unorm[t] //Simplify]
You can evaluate any of these functions at a specified value of t.
t0= Sqrt[3]
{utan[t0], unorm[t0], ubinorm[t0]}
N[{utan[t0], unorm[t0], ubinorm[t0]}]
{curv[t0], torsion[t0]}
N[{curv[t0], torsion[t0]}]
{at[t0], an[t0]}
N[{at[t0], an[t0]}]
To verify that the tangential and normal components of the acceleration agree with the formulas in the book:
at[t]== speed'[t] //Simplify
an[t]==curv [t] speed[t]2 //Simplify
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
783
784
Chapter 13 Vector-Valued Functions and Motion in Space
13.6 VELOCITY AND ACCELERATION IN POLAR COORDINATES
1.
d)
dt
Þ
ÞÞ
.
..
œ 3 œ ) Ê ) œ 0, r œ aa1 cos )b Ê r œ a sin ) ddt) œ 3a sin ) Ê r œ 3a cos ) ddt) œ 9a cos )
v œ a3a sin )bur aaa1 cos )bba3bu) œ a3a sin )bur 3aa1 cos )bu)
a œ Š9a cos ) aa1 cos )ba3b2 ‹ur aaa1 cos )b † 0 2a3a sin )ba3bbu)
œ a9a cos ) 9a 9a cos )bur a18a sin )bu) œ 9aa2 cos ) 1bur a18a sin )bu)
2.
d)
dt
Þ
ÞÞ
.
..
œ 2t œ ) Ê ) œ 2, r œ a sin 2) Ê r œ a cos 2) † 2 ddt) œ 4ta cos 2) Ê r œ 4ta ˆsin 2) † 2 ddt) ‰ 4a cos 2)
œ 16t2 a sin 2) 4a cos 2)
v œ a4ta cos 2)bur aa sin 2)ba2tbu) œ a4ta cos 2)bur a2ta sin 2)bu)
a œ ’a 16t2 a sin 2) 4a cos 2)b aa sin 2)ba2tb2 “ur aa sin 2)ba2b 2a4ta cos 2)ba2tb‘u)
œ ’16t2 a sin 2) 4a cos 2) 4t2 a sin 2)“ur 2a sin 2) 16t2 a cos 2)‘u)
œ ’20t2 a sin 2) 4a cos 2)“ur 2a sin 2) 16t2 a cos 2)‘u) œ 4aacos 2) 5t2 sin 2)bur 2aasin 2) 8t2 cos 2)bu)
3.
Þ
ÞÞ
.
..
œ 2 œ ) Ê ) œ 0, r œ ea ) Ê r œ ea ) † a ddt) œ 2a ea ) Ê r œ 2a ea ) † a ddt) œ 4a2 ea )
v œ ˆ2a ea ) ‰ur ˆea ) ‰a2bu) œ ˆ2a ea ) ‰ur ˆ2ea ) ‰u)
d)
dt
a œ ’ˆ 4a2 ea ) ‰ ˆea ) ‰a2b2 “ur ’ˆea ) ‰a0b 2ˆ2a ea ) ‰a2b“u) œ ’4a2 ea ) 4ea ) “ur ’0 8a ea ) “u)
œ 4ea ) aa2 1bur ˆ8a ea ) ‰u)
Þ
ÞÞ
.
..
4. ) œ 1 et Ê ) œ et Ê ) œ et , r œ aa1 sin tb Ê r œ a cos t Ê r œ a sin t
v œ aa cos tbur aaa1 sin tbbaet bu) œ aa cos tbur a et a1 sin tbu)
a œ ’a a sin tb aaa1 sin tbbaet b “ur ’aaa1 sin tbbaet b 2aa cos tbaet b“u)
2
œ ’a sin t a e2t a1 sin tb“ur ’a et a1 sin tb 2a et cos t“u)
œ aasin t e2t a1 sin tbbur a et aa1 sin tb 2cos tbu)
œ aasin t e2t a1 sin tbbur a et a2cos t 1 sin tbu)
Þ
ÞÞ
.
..
5. ) œ 2t Ê ) œ 2 Ê ) œ 0, r œ 2 cos 4t Ê r œ 8 sin 4t Ê r œ 32 cos 4t
v œ a8 sin 4tbur a2 cos 4tba2bu) œ 8asin 4tbur 4acos 4tbu)
a œ Ša32 cos 4tb a2 cos 4tba2b2 ‹ur aa2 cos 4tb † 0 2a8 sin 4tba2bbu)
œ a32 cos 4t 8 cos 4tbur a0 32sin 4tbu) œ 40acos 4tbur 32asin 4tbu)
6. e œ
r! v#!
GM
1 Ê v#! œ
GM(e 1)
r!
Ê v! œ É GM(er! 1) ;
Circle: e œ 0 Ê v! œ É GM
r!
2GM
Ellipse: 0 e 1 Ê É GM
r! v! É r!
Parabola: e œ 1 Ê v! œ É 2GM
r!
Hyperbola: e 1 Ê v! É 2GM
r!
7. r œ
GM
v#
Ê v# œ
GM
r
Ê v œ É GM
r which is constant since G, M, and r (the radius of orbit) are constant
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 13 Practice Exercises
8. ?A œ
"
#
kr(t ?t) ‚ r(t)k Ê
œ
"
#
¹ r(t ??t)t r(t) ‚ r(t)
œ
"
#
¸ ddtr ‚ r(t)¸ œ
"
#
?A
?t
?t)
‚ r(t)¹ œ
¹ r(t
?t
"
#
œ
"
#
r(t) r(t)
‚ r(t)¹
¹ r(t ?t) ?
t
r(t) ‚ r(t)¹ œ #" ¹ r(t ??t)t r(t) ‚ r(t)¹ Ê
¸r(t) ‚ ddtr ¸ œ "# kr ‚ rÞ k
"
?t
#
# %
dA
dt
?t Ä 0 #
¹ r(t ??t)t r(t) ‚ r(t)¹
#
r v#
# %
"
œ lim
! !
9. T œ Š 2r!1va! ‹ È1 e# Ê T# œ Š 4r1# va# ‹ a1 e# b œ Š 4r1# va# ‹ ”1 Š GM
1‹ • (from Equation 5)
! !
! !
r# v %
# %
r v#
# %
! !
œ Š 4r1# va# ‹ ’ G!# M!# 2 Š GM
‹“ œ Š 4r1# va# ‹ ’
!
!
!
œ a41# a% b Š
2GM r! v#!
ˆ 2 ‰
2r! GM ‹ GM
!
2GMr! v!# r!# v!%
“
G# M#
œ
ˆ41# a% ‰ a2GM r! v#! b
r! G# M#
" ‰ˆ 2 ‰
#
œ a41# a% b ˆ 2a
GM (from Equation 10) Ê T œ
4 1 # a$
GM
Ê
T#
a$
œ
41 #
GM
minutes
seconds
7
10. r œ 365.256 days œ 365.256 days ‚ 24 hours
day ‚ 60 hour ‚ 60 minute œ 31,558,118.4 seconds ¸ 3.16 ‚ 10 ,
G œ 6.6726 ‚ 1011 Nkg†m# , and the mass of the sun M œ 1.99 ‚ 1030 kg.
2
Ê a3 œ a3.16 ‚ 107 b
2 ˆ6.6726‚10c11 ‰ˆ1.99‚1030 ‰
41 2
T2
a3
œ
41 2
GM
Ê a3 œ T2 GM
41 2
3
¸ 3.35863335 ‚ 1033 Ê a œ È
3.35863335 ‚ 1033
¸ 149757138111 m ¸ 149.757 billion km
CHAPTER 13 PRACTICE EXERCISES
1. r(t) œ (4 cos t)i ŠÈ2 sin t‹ j Ê x œ 4 cos t
and y œ È2 sin t Ê
x#
16
y#
#
œ 1;
v œ (4 sin t)i ŠÈ2 cos t‹ j and
a œ (4 cos t)i ŠÈ2 sin t‹ j ; r(0) œ 4 i , v(0) œ È2j ,
a(0) œ 4i ; r ˆ 14 ‰ œ 2È2i j , v ˆ 14 ‰ œ 2È2i j ,
a ˆ 1 ‰ œ 2È2i j ; kvk œ È16 sin# t 2 cos# t
4
Ê aT œ
d
dt
kvk œ
14 sin t cos t
È16 sin# t2 cos# t
; at t œ 0: aT œ 0, aN œ Ékak# 0 œ 4, a œ 0T 4N œ 4N, , œ
aN
kv k #
œ
4È 2
27
2. r(t) œ ŠÈ3 sec t‹ i ŠÈ3 tan t‹ j Ê x œ È3 sec t and y œ È3 tan t Ê
x#
3
at t œ 14 : aT œ
7
È 8 1
œ
7
3
, aN œ É9
49
9
œ
4È 2
3
, a œ 37 T
4È 2
3 N,
,œ
y#
3
and
a œ ŠÈ3 sec t tan# t È3 sec$ t‹ i Š2È3 sec# t tan t‹ j ;
r(0) œ È3i , v(0) œ È3j , a(0) œ È3i ;
kvk œ È3 sec# t tan# t 3 sec% t
d
dt
kvk œ
6 sec# t tan$ t 18 sec% t tan t
2È3 sec# t tan# t 3 sec% t
;
at t œ 0: aT œ 0, aN œ Ékak# 0 œ È3,
a œ 0T È3N œ È3N, , œ
aN
kvk#
œ
È3
3
œ
œ
œ sec# t tan# t œ 1;
Ê x# y# œ 3; v œ ŠÈ3 sec t tan t‹ i ŠÈ3 sec# t‹ j
Ê aT œ
aN
kv k #
"
È3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
4
2
œ 2;
785
786
Chapter 13 Vector-Valued Functions and Motion in Space
"
È1 t#
3. r œ
œ
"
1 t#
t 0,
i
t
È1 t#
j Ê v œ t a1 t # b
d kvk
dt
. We want to maximize kvk :
2t
a1 t# b#
$Î#
œ
i a 1 t# b
2t
a1 t# b#
and
$Î#
d kv k
dt
#
#
j Ê kvk œ Ê’t a1 t# b$Î# “ ’a1 t# b$Î# “
œ0 Ê
2t
a1 t# b#
œ 0 Ê t œ 0. For t 0,
2t
a1 t# b#
0; for
0 Ê kvk max occurs when t œ 0 Ê kvk max œ 1
4. r œ aet cos tb i aet sin tb j Ê v œ aet cos t et sin tb i aet sin t et cos tb j
Ê a œ aet cos t et sin t et sin t et cos tb i aet sin t et cos t et cos t et sin tb j
œ a2et sin tb i a2et cos tb j . Let ) be the angle between r and a . Then ) œ cos" Š krrk†kaak ‹
œ cos"
2e2t sin t cos t2e2t sin t cos t
Éaet cos tb# aet sin tb# Éa2et sin tb# a2et cos tb#
1
#
œ cos" Š 2e02t ‹ œ cos" 0 œ
for all t
â
â
â i j kâ
â
â
5. v œ 3i 4j and a œ 5i 15j Ê v ‚ a œ â 3 4 0 â œ 25k Ê kv ‚ ak œ 25; kvk œ È3# 4# œ 5
â
â
â 5 "5 0 â
Ê ,œ
6. , œ
kv‚ak
kv k $
kyww k
$Î#
1 ayw b# ‘
œ ex a1 e2x b
d,
dx
œ
25
5$
œ
"
5
œ ex a1 e2x b
$Î#
$Î#
3e3x a1 e2x b
7. r œ xi yj Ê v œ
#
&Î#
œ 0 Ê a1 2e2x b œ 0 Ê e2x œ
maximum at the point Š ln È2ß
#
x y œ 1, 2x
dx
dt
dx
dt
i
2y
dy
dt
dy
dt
d,
dx
Ê
œ ex a1 e2x b
œ ex a1 e2x b
$Î#
ex ’ 3# a1 e2x b
&Î#
&Î#
a2e2x b“
&Î#
ca1 e2x b 3e2x d œ ex a1 e2x b
a1 2e2x b ;
Ê 2x œ ln 2 Ê x œ "# ln 2 œ ln È2 Ê y œ È" ; therefore , is at a
"
#
2
"
È2 ‹
j and v † i œ y Ê
œ0 Ê
dy
dt
œ
x dx
y dt
dx
dt
œ y. Since the particle moves around the unit circle
Ê
dy
dt
œ xy (y) œ x. Since
dx
dt
œ y and
dy
dt
œ x, we have
v œ yi xj Ê at (1ß 0), v œ j and the motion is clockwise.
dy
dy
" # dx
# dx
dt œ 3x dt Ê dt œ 3 x dt . If r œ xi yj , where x and y are differentiable functions of
dy
dy
dx
" # dx
"
#
then v œ dx
dt i dt j. Hence v † i œ 4 Ê dt œ 4 and v † j œ dt œ 3 x dt œ 3 (3) (4) œ 12 at (3ß 3). Also,
#
#
#
#
#
#
‰ ˆ 3" x# ‰ ddt#x . Hence a † i œ 2 Ê ddt#x œ 2 and
a œ ddt#x i ddt#y j and ddt#y œ ˆ 32 x‰ ˆ dx
dt
#
a † j œ ddt#y œ 23 (3)(4)# "3 (3)# (2) œ 26 at the point (xß y) œ (3ß 3).
8. 9y œ x$ Ê 9
9.
dr
dt
orthogonal to r Ê 0 œ
dr
dt
†rœ
" dr
# dt
† r "# r †
dr
dt
œ
" d
# dt
#
t,
(r † r) Ê r † r œ K, a constant. If r œ xi yj , where
x and y are differentiable functions of t, then r † r œ x# y Ê x# y# œ K, which is the equation of a circle
centered at the origin.
10. (a)
(b) v œ (1 1 cos 1t)i (1 sin 1t)j
Ê a œ a1# sin 1tb i a1# cos 1tb j ;
v(0) œ 0 and a(0) œ 1# j ;
v(1) œ 21i and a(1) œ 1# j ;
v(2) œ 0 and a(2) œ 1# j ;
v(3) œ 21i and a(3) œ 1# j
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 13 Practice Exercises
(c) Forward speed at the topmost point is kv(1)k œ kv(3)k œ 21 ft/sec; since the circle makes
"
#
787
revolution per
second, the center moves 1 ft parallel to the x-axis each second Ê the forward speed of C is 1 ft/sec.
11. y œ y! (v! sin !)t "# gt# Ê y œ 6.5 (44 ft/sec)(sin 45°)(3 sec) "# a32 ft/sec# b (3 sec)# œ 6.5 66È2 144
¸ 44.16 ft Ê the shot put is on the ground. Now, y œ 0 Ê 6.5 22È2t 16t# œ 0 Ê t ¸ 2.13 sec (the
positive root) Ê x ¸ (44 ft/sec)(cos 45°)(2.13 sec) ¸ 66.27 ft or about 66 ft, 3 in. from the stopboard
12. ymax œ y!
(v! sin !)#
#g
œ 7 ft
[(80 ft/sec)(sin 45°)]#
(2) a32 ft/sec# b
¸ 57 ft
(v! sin !)t "# gt#
(v sin !) " gt
y
œ ! v! cos ! #
x œ
(v! cos !)t
2v! sin ! 2v! cos ! tan 9
, which is the time when
g
13. x œ (v! cos !)t and y œ (v! sin !)t "# gt# Ê tan 9 œ
Ê v! cos ! tan 9 œ v! sin ! "# gt Ê t œ
the golf ball
hits the upward slope. At this time x œ (v! cos !) Š 2v! sin ! 2vg ! cos ! tan 9 ‹ œ Š g2 ‹ av#! sin ! cos ! v#! cos# ! tan 9b .
Now OR œ
x
cos 9
Ê OR œ Š g2 ‹ Š
v#! sin ! cos ! v#! cos# ! tan 9
‹
cos 9
œŠ
2v#! cos !
sin !
‹ Š cos
g
9
œŠ
2v#! cos !
9 cos ! sin 9
‹ Š sin ! cos cos
‹
#9
g
œŠ
2v#! cos !
g cos# 9 ‹ [sin (!
cos ! tan 9
cos 9 ‹
9)]. The distance OR is maximized
when x is maximized:
dx
d!
œŠ
2v#!
g ‹(cos
2! sin 2! tan 9) œ 0
Ê (cos 2! sin 2! tan 9) œ 0 Ê cot 2! tan 9 œ 0 Ê cot 2! œ tan (9) Ê 2! œ
1
#
9 Ê!œ
9
#
1
4
5
14. (a) x œ v! (cos 40°)t and y œ 6.5 v! (sin 40°)t "# gt# œ 6.5 v! (sin 40°)t 16t# ; x œ 262 12
ft and y œ 0 ft
5
Ê 262 12
œ v! (cos 40°)t or v! œ
262.4167
#
#
and 0 œ 6.5 ’ (cos
40°)t “ (sin 40°)t 16t Ê t œ 14.1684
262.4167
(cos 40°)t
Ê t ¸ 3.764 sec. Therefore, 262.4167 ¸ v! (cos 40°)(3.764 sec) Ê v! ¸
(b) ymax œ y!
(v! sin !)#
2g
¸ 6.5
a(91)(sin 40°)b2
(2)(32)
262.4167
(cos 40°)(3.764 sec)
Ê v! ¸ 91 ft/sec
¸ 60 ft
15. r œ (2 cos t)i (2 sin t)j t# k Ê v œ (2 sin t)i (2 cos t)j 2tk Ê kvk œ È(2 sin t)# (2 cos t)# (2t)#
œ 2È1 t# Ê Length œ '0 2È1 t# dt œ ’tÈ1 t# ln ¹t È1 t# ¹“
1Î4
1Î%
!
œ
1
4
É1
1#
16
ln Š 14 É1
1#
16 ‹
16. r œ (3 cos t)i (3 sin t)j 2t$Î# k Ê v œ (3 sin t)i (3 cos t)j 3t"Î# k Ê kvk œ É(3 sin t)# (3 cos t)# a3t"Î# b
$
œ È9 9t œ 3È1 t Ê Length œ '0 3È1 t dt œ 2(1 t)$Î# ‘ ! œ 14
3
17. r œ
4
9
(1 t)$Î# i 49 (1 t)$Î# j "3 tk Ê v œ
#
2
3
(1 t)"Î# i 32 (1 t)"Î# j 3" k
#
#
Ê kvk œ É 23 (1 t)"Î# ‘ 23 (1 t)"Î# ‘ ˆ 3" ‰ œ 1 Ê T œ
i 32 j 3" k ;
"
3
(1 t)"Î# i
â
â
â
Ê N(0) œ È"2 i È"2 j ; B(0) œ T(0) ‚ N(0) œ ââ
â
â
Ê T(0) œ
aœ
"
3
2
3
dT
dt
œ
(1 t)"Î# i "3 (1 t)"Î# j Ê a(0) œ
"
3
2
3
(1 t)"Î# i 32 (1 t)"Î# j 3" k
"
3
(1 t)"Î# j Ê ddtT (0) œ 3" i 3" j Ê ¸ ddtT (0)¸ œ
â
i
j
kâ
2
" â
"
"
4
â
23
3
3 â œ È i È j È k;
3 2
3 2
3 2
"
"
0 ââ
È2
È2
i "3 j and v(0) œ
2
3
i 23 j "3 k Ê v(0) ‚ a(0)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
È2
3
#
788
Chapter 13 Vector-Valued Functions and Motion in Space
â i
â
â
œ â 23
â "
â 3
k ââ
" â
3 ⠜
â
0â
j
23
"
3
"
9
i 9" j 94 k Ê kv ‚ ak œ
È2
3
Ê ,(0) œ
k v ‚a k
kv k $
Þ
Þ
a œ "6 (1 t)$Î# i "6 (1 t)$Î# j Ê a(0) œ "6 i "6 j Ê 7 (0) œ
œ
â 2
â 3
â
â "
â 3
â "
â 6
Š
È2 ‹
3
1$
23
"
3
"
6
kv‚ak
#
œ
" ââ
3 â
0 ââ
0 ââ
È2
3
;
2 ‰
ˆ 3" ‰ ˆ 18
œ
Š
È2 ‹# œ
"
6
3
18. r œ aet sin 2tb i aet cos 2tb j 2et k Ê v œ aet sin 2t 2et cos 2tb i aet cos 2t 2et sin 2tb j 2et k
Ê kvk œ Éaet sin 2t 2et cos 2tb# aet cos 2t 2et sin 2tb# a2et b# œ 3et Ê T œ
œ ˆ "3 sin 2t
dT
dt
2
3
cos 2t‰ i ˆ 3" cos 2t
œ ˆ 32 cos 2t
Ê N(0) œ
2
3
sin 2t‰ j 23 k Ê T(0) œ
sin 2t‰ i ˆ 32 sin 2t
4
3
4
3
cos 2t‰ j Ê
dT
dt
È
Š2 3 5‹
i 3" j 32 k ;
i 34 j Ê ¸ ddtT (0)¸ œ 32 È5
â
j
kâ
1
2 â
4
2
5
â
3
3 â œ È i È j È k;
3 5
3 5
3 5
2
â
È5 0 â
(0) œ
â
â i
â 2
"
2
œ È5 i È5 j ; B(0) œ T(0) ‚ N(0) œ ââ 3
â "
â È5
ˆ 23 i 43 j‰
2
3
2
3
v
kvk
a œ a4et cos 2t 3et sin 2tb i a3et cos 2t 4et sin 2tb j 2et k Ê a(0) œ 4i 3j 2k and v(0) œ 2i j 2k
â
â
j kâ
âi
â
â
Ê v(0) ‚ a(0) œ â 2 " 2 â œ 8i 4j 10k Ê kv ‚ ak œ È64 16 100 œ 6È5 and kv(0)k œ 3
â
â
â 4 3 2 â
Ê ,(0) œ
6È 5
3$
œ
2È 5
9
t
â
â 2
â
â 4
â
â 2
1
3
11
;
Þ
t
a œ a4e cos 2t 8e sin 2t 3et sin 2t 6et cos 2tb i a3et cos 2t 6et sin 2t 4et sin 2t 8et cos 2tb j 2et k
Þ
œ a2et cos 2t 11et sin 2tb i a11 et cos 2t 2et sin 2tb j 2et k Ê a(0) œ 2i 11j 2k
Ê 7 (0) œ
kv‚ak#
â
2â
â
2â
â
2â
œ
80
180
œ 49
19. r œ ti "# e2t j Ê v œ i e2t j Ê kvk œ È1 e4t Ê T œ
dT
dt
œ
2 e
ˆ1 e4t ‰$Î#
4t
i
2t
2e
ˆ1 e4t ‰$Î#
j Ê
dT
dt
(ln 2) œ
32
17È17
i
8
17È17
"
È1 e4t
i
e2t
È1 e4t
j Ê T (ln 2) œ
j Ê N (ln 2) œ È417 i
"
È17
"
È17
i
4
È17
j;
j;
â i
j
k ââ
â
â "
4
0 ââ œ k ; a œ 2e2t j Ê a(ln 2) œ 8j and v(ln 2) œ i 4j
È17
B (ln 2) œ T(ln 2) ‚ N(ln 2) œ ââ È17
â
"
â 4
â
â È17 È17 0 â
â
â
â i j kâ
â
â
Þ
8
Ê v(ln 2) ‚ a(ln 2) œ â " 4 0 â œ 8k Ê kv ‚ ak œ 8 and kv(ln 2)k œ È17 Ê ,(ln 2) œ 17È
; a œ 4e2t j
17
â
â
â0 8 0â
Þ
Ê a(ln 2) œ 16j Ê 7 (ln 2) œ
â
â1
â
â0
â
â0
4
8
16
k v ‚a k #
â
0â
â
0â
â
0â
œ0
20. r œ (3 cosh 2t)i (3 sinh 2t)j 6tk Ê v œ (6 sinh 2t)i (6 cosh 2t)j 6k
Ê kvk œ È36 sinh# 2t 36 cosh# 2t 36 œ 6È2 cosh 2t Ê T œ kvvk œ Š È"2 tanh 2t‹ i
Ê T(ln 2) œ
#
15
17È2
i
"
È2
j
8
17È2
k;
8 ‰
8 ‰ ˆ 15 ‰
œ Š È22 ‹ ˆ 17
i Š È22 ‹ˆ 17
17 k œ
dT
dt
œ Š È22 sech# 2t‹ i Š È22 sech 2t tanh 2t‹ k Ê
128
289È2
i
240
289È2
"
È2
j Š È"2 sech 2t‹ k
dT
dt
(ln 2)
#
#
128
240
k Ê ¸ ddtT (ln 2)¸ œ ÊŠ 289
È2 ‹ Š 289È2 ‹ œ
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
8È 2
17
Chapter 13 Practice Exercises
â
j
â i
â 15
"
8
15
â
Ê N(ln 2) œ 17 i 17 k ; B(ln 2) œ T(ln 2) ‚ N(ln 2) œ â 17È2 È2
â 8
0
â 17
‰
ˆ 15 ‰
a œ (12 cosh 2t)i (12 sinh 2t)j Ê a(ln 2) œ 12 ˆ 17
8 i 12 8 j œ
â
k â
8 â
15
â
17È2 â œ 17È2 i
15 â
17 â
i
45
# j and
â i
j
â
â 45 51
15 ‰
17 ‰
45
51
ˆ
ˆ
v(ln 2) œ 6 8 i 6 8 j 6k œ 4 i 4 j 6k Ê v(ln 2) ‚ a(ln 2) œ â 4
4
â 51 45
â 2
#
œ 135i 153j 72k Ê kv ‚ ak œ 153È2 and kv(ln 2)k œ
51
4
51
#
È2 Ê ,(ln 2) œ
Þ
Þ
a œ (24 sinh 2t)i (24 cosh 2t)j Ê a(ln 2) œ 45i 51j Ê 7 (ln 2) œ
â 45
â 4
â 5"
â 2
â
â 45
51
4
45
2
51
kv ‚a k #
â
6â
â
0â
â
0â
œ
"
È2
j
8
17È2
k;
k ââ
6 ââ
â
0â
153È2
$
È 2‹
Š 51
4
œ
32
867
;
32
867
21. r œ a2 3t 3t# b i a4t 4t# b j (6 cos t)k Ê v œ (3 6t)i (4 8t)j (6 sin t)k
Ê kvk œ È(3 6t)# (4 8t)# (6 sin t)# œ È25 100t 100t# 36 sin# t
"Î#
"
#
a25 100t 100t# 36 sin# tb
(100 200t 72 sin t cos t) Ê aT (0) œ ddtkvk (0) œ 10;
a œ 6i 8j (6 cos t)k Ê kak œ È6# 8# (6 cos t)# œ È100 36 cos# t Ê ka(0)k œ È136
Ê
d kv k
dt
œ
Ê aN œ Ékak# a#T œ È136 10# œ È36 œ 6 Ê a(0) œ 10T 6N
22. r œ (2 t)i at 2t# b j a1 t# b k Ê v œ i (1 4t)j 2tk Ê kvk œ È1# (1 4t)# (2t)#
"Î#
œ È2 8t 20t# Ê ddtkvk œ "# a2 8t 20t# b
(8 40t) Ê aT œ ddtkvk (0) œ 2È2; a œ 4j 2k
#
Ê kak œ È4# 2# œ È20 Ê aN œ Ékak# a#T œ Ê20 Š2È2‹ œ È12 œ 2È3 Ê a(0) œ 2È2T 2È3N
23. r œ (sin t)i ŠÈ2 cos t‹ j (sin t)k Ê v œ (cos t)i ŠÈ2 sin t‹ j (cos t)k
#
Ê kvk œ Ê(cos t)# ŠÈ2 sin t‹ (cos t)# œ È2 Ê T œ
v
kvk
œ Š È"2 cos t‹ i (sin t)j Š È"2 cos t‹ k ;
#
#
œ Š È" sin t‹ i (cos t)j Š È" sin t‹ k Ê ¸ ddtT ¸ œ ÊŠ È" sin t‹ ( cos t)# Š È" sin t‹ œ 1
2
2
2
2
â
â
i
j
k
â
â
â
â
d
T
"
"
ˆ dt ‰
cos
t
sin
t
cos
t
"
"
â
â
È
È
Ê N œ ¸ dT ¸ œ Š È sin t‹ i (cos t)j Š È sin t‹ k ; B œ T ‚ N œ â 2
2
â
2
2
dt
â " sin t cos t " sin t â
â È2
â
È2
â i
â
j
k â
â
â
â
"
"
È
È
œ È2 i È2 k ; a œ ( sin t)i Š 2 cos t‹ j (sin t)k Ê v ‚ a œ â cos t 2 sin t cos t â
â
â
â sin t È2 cos t sin t â
Þ
œ È2 i È2 k Ê kv ‚ ak œ È4 œ 2 Ê , œ kvkv‚k$ak œ 2 $ œ È"2 ; a œ ( cos t)i ŠÈ2 sin t‹ j (cos t)k
dT
dt
ŠÈ2‹
Ê 7œ
â
â cos t
â
â sin t
â
â
â cos t
â
È2 sin t
cos t ââ
È
2 cos t sin t ââ
È2 sin t cos t ââ
k v ‚ a k#
œ
(cos t) ŠÈ2‹ ŠÈ2 sin t‹ (0) (cos t) ŠÈ2‹
4
œ0
24. r œ i (5 cos t)j (3 sin t)k Ê v œ (5 sin t)j (3 cos t)k Ê a œ (5 cos t)j (3 sin t)k
Ê v † a œ 25 sin t cos t 9 sin t cos t œ 16 sin t cos t; v † a œ 0 Ê 16 sin t cos t œ 0 Ê sin t œ 0 or cos t œ 0
Ê t œ 0, 1# or 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
789
790
Chapter 13 Vector-Valued Functions and Motion in Space
25. r œ 2i ˆ4 sin #t ‰ j ˆ3 1t ‰ k Ê 0 œ r † (i j) œ 2(1) ˆ4 sin #t ‰ (1) Ê 0 œ 2 4 sin
Ê tœ
1
3
t
#
Ê sin
t
#
œ
"
#
Ê
t
#
œ
1
6
(for the first time)
26. r(t) œ ti t# j t$ k Ê v œ i 2tj 3t# k Ê kvk œ È1 4t# 9t% Ê kv(1)k œ È14
Ê T(1) œ È"14 i È214 j È314 k , which is normal to the normal plane
"
È14
Ê
(x 1)
2
È14
(y 1)
3
È14
(z 1) œ 0 or x 2y 3z œ 6 is an equation of the normal plane. Next we
calculate N(1) which is normal to the rectifying plane. Now, a œ 2j 6tk Ê a(1) œ 2j 6k Ê v(1) ‚ a(1)
â
â
â i j kâ
#
È76
È19
â
â
œ â " 2 3 â œ 6i 6j 2k Ê kv(1) ‚ a(1)k œ È76 Ê ,(1) œ
œ kv(t)k Ê ddt#s ¹
; ds
$ œ
È
dt
7
14
È
â
â
tœ1
Š 14‹
â0 2 6â
œ
"
#
œ
22
È14
a1 4t# 9t% b
"Î#
2j3k
Š iÈ
‹
14
a8t 36t$ b¹
È19
7È14
#
tœ1
œ
22
È14
ŠÈ14‹ N Ê N œ
, so a œ
È14
2È19
d# s
dt#
#
‰ N Ê 2j 6k
T , ˆ ds
dt
8
9 ‰
11
8
9
ˆ 11
7 i 7 j 7 k Ê 7 (x 1) 7 (y 1) 7 (z 1)
œ 0 or 11x 8y 9z œ 10 is an equation of the rectifying plane. Finally, B(1) œ T(1) ‚ N(1)
â
â
j kâ
â i
È14
â
â
2 3 â œ È" (3i 3j k) Ê 3(x 1) 3(y 1) (z 1) œ 0 or 3x 3y z
œ Š 2È19 ‹ Š È" ‹ ˆ "7 ‰ â "
19
14
â
â
â 11 8 9 â
œ 1 is an equation of the osculating plane.
" ‰
27. r œ et i (sin t)j ln (1 t)k Ê v œ et i (cos t)j ˆ 1
t k Ê v(0) œ i j k ; r(0) œ i Ê (1ß 0ß 0) is on the line
Ê x œ 1 t, y œ t, and z œ t are parametric equations of the line
28. r œ ŠÈ2 cos t‹ i ŠÈ2 sin t‹ j tk Ê v œ ŠÈ2 sin t‹ i ŠÈ2 cos t‹ j k Ê v ˆ 14 ‰
œ ŠÈ2 sin 14 ‹ i ŠÈ2 cos 14 ‹ j k œ i j k is a vector tangent to the helix when t œ
is parallel to v ˆ 14 ‰ ; also r ˆ 14 ‰ œ ŠÈ2 cos 14 ‹ i ŠÈ2 sin 14 ‹ j
Ê x œ 1 t, y œ 1 t, and z œ
1
4
1
4
#
29. x# œ av#! cos# !b t# and ˆy "# gt# ‰ œ av#! sin# !b t# Ê x# ˆy "# gt# ‰ œ v#! t#
Þ ÞÞ Þ ÞÞ
ÞÞ# ÞÞ# ÞÞ# ÞÞ# ÞÞ# axÞ ÞÞx yÞ ÞÞyb#
x x y y
Ê
x y s œ x y xÞ # yÞ #
Þ
Þ
#
#
Èx y
ÞÞ# ÞÞ# Þ # Þ #
Þ # ÞÞ#
Þ ÞÞ Þ ÞÞ Þ # ÞÞ#
Þ ÞÞ Þ ÞÞ
Þ ÞÞ
Þ ÞÞ
Þ ÞÞ Þ ÞÞ
ax y b ax y b ax x 2x x y y y y b
ax y y x b#
x# y# y# x # 2x x y y
œ
œ
œ
Þ# Þ#
Þ# Þ#
Þ
Þ
x y
x y
x# y#
Þ ÞÞ Þ ÞÞ
Þ # Þ # $Î#
Þ
Þ
#
#
kx y y xk
ax y b
ÞÞ
ÞÞ
ÞÞ
x y
Ê È x# y# s # œ È Þ # Þ # Ê ÈÞÞ# ÞÞ# ÞÞ# œ kxÞ ÞÞy yÞ ÞÞxk œ ," œ 3
x y
x y s
ÞÞ
30. s œ
d
dt
ÈxÞ # yÞ # œ
31. s œ a) Ê ) œ
s
a
Ê 9œ
s
a
1
#
Ê
d9
ds
œ
"
a
Ê , œ ¸ "a ¸ œ
"
a
Ê the tangent line
k Ê the point ˆ1ß 1ß 14 ‰ is on the line
t are parametric equations of the line
#
1
4
since a 0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 13 Additional and Advanced Exercises
32. (1) ?SOT ¸ ?TOD Ê
Ê y! œ
6380#
6817
DO
OT
œ
Ê
OT
SO
œ
y!
6380
6380
6380437
Ê y! ¸ 5971 km;
(2) VA œ '5971 21x Ê1 Š dx
dy ‹ dy
#
6380
6380
œ 21'5971 È6380# y# Š È6380
# y# ‹ dy
6817
œ 21 '5971 6380 dy œ 21 c6380yd ')"(
&*("
6817
œ 16,395,469 km# ¸ 1.639 ‚ 10( km# ;
(3) percentage visible ¸
16,395,469 km#
41(6380 km)#
¸ 3.21%
CHAPTER 13 ADDITIONAL AND ADVANCED EXERCISES
1. (a) r()) œ (a cos ))i (a sin ))j b)k Ê
d)
dt
œ Èa# b#
(b)
d)
dt
Ê
)
œ É a#2gb
b# Ê
d)
dt
d)
È)
dr
dt
œ [(a sin ))i (a cos ))j bk]
)
É a#2gb
œ É a#2gz
b# œ
b# Ê
gbt#
2 aa# b# b
; z œ b) Ê z œ
œ [(a sin ))i (a cos ))j bk]
d)
dt
i (a cos ))j bk
Ê v(t) œ ’ (a sin ))È
“ Š È gbt
#
#
#
a b#
a b
d# r
dt#
d) ¸
dt )œ#1
d)
dt
; kvk œ È2gz œ ¸ ddtr ¸
œ É a#41gbb# œ 2É a#1gbb#
"Î#
œ É a#2gb
œ É a#2gb
b# dt Ê 2)
b# t C; t œ 0 Ê ) œ 0 Ê C œ 0
Ê 2)"Î# œ É a#2gb
b# t Ê ) œ
(c) v(t) œ
dr
dt
gb# t#
2 aa# b# b
œ [(a sin ))i (a cos ))j bk] Š a# gbt
b# ‹ , from part (b)
‹œ
gbt
È a# b#
T;
#
œ [(a cos ))i (a sin ))j] ˆ ddt) ‰ [(a sin ))i (a cos ))j bk]
#
d# )
dt#
gb
œ Š a# gbt
b# ‹ [(a cos ))i (a sin ))j] [(a sin ))i (a cos ))j bk] Š a# b# ‹
i (a cos ))j bk
œ ’ (a sin ))È
“ ŠÈ
# #
a
œ
gb
È a# b #
b
gb
‹
b#
a#
#
a Š a# gbt
b# ‹ [( cos ))i (sin ))j]
#
T a Š a# gbt
b# ‹ N (there is no component in the direction of B).
2. (a) r()) œ (a) cos ))i (a) sin ))j b)k Ê
kvk œ È2gz œ ¸ ddtr ¸ œ aa# a# )# b# b
(b) s œ '0 kvk dt œ '0 aa# a# )#
t
t
œ '0 aÉ
)
a#
a#
(1 e)r!
1 e cos )
Ê
dr
d)
"Î#
"Î#
b# b ddt)
œ [(a cos ) a) sin ))i (a sin ) a) cos ))j bk]
ˆ ddt) ‰ Ê
d)
dt
œ
œ
c#
#
)
ln ¹u Èc# u# ¹“ œ
(1 e)r! (e sin ))
(1 e cos ))#
!
;
dr
d)
œ0 Ê
Ê sin ) œ 0 Ê ) œ 0 or 1. Note that
dr
d)
a
#
"Î#
d) œ '0 aa# a# u# b# b
)
"
#
"Î#
du
È a# b #
k ak
Š)Èc# )# c# ln ¹) Èc# )# ¹ c# ln c‹
(1 e)r! (e sin ))
(1 e cos ))#
œ 0 Ê (1 e)r! (e sin )) œ 0
0 when sin ) 0 and
dr
d)
0 when sin ) 0. Since sin ) 0 on
1 ) 0 and sin ) 0 on 0 ) 1, r is a minimum when ) œ 0 and r(0) œ
4. (a) f(x) œ x 1
;
È a# a# ) # b#
dt œ '0 aa# a# )# b# b
u# du œ a '0 Èc# u# du, where c œ
d)
dt
È2gb)
t
)
b#
Ê s œ a ’ u# Èc# u#
3. r œ
dr
dt
sin x œ 0 Ê f(0) œ 1 and f(2) œ 2 1
"
#
sin 2
"
#
(1 e)r!
1 e cos 0
œ r!
since ksin 2k Ÿ 1; since f is continuous
on [0ß 2], the Intermediate Value Theorem implies there is a root between 0 and 2
(b) Root ¸ 1.4987011335179
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
791
792
Chapter 13 Vector-Valued Functions and Motion in Space
.
.
.
.
.
.
.
5. (a) v œ x i y j and v œ r ur r ) u) œ arb [(cos ))i (sin ))j] ˆr )‰ [( sin ))i (cos ))j] Ê v † i œ x and
.
.
.
.
.
. .
.
v † i œ r cos ) r ) sin ) Ê x œ r cos ) r ) sin ); v † j œ y and v † j œ r sin ) r ) cos )
.
. .
Ê y œ r sin ) r ) cos )
.
.
(b) ur œ (cos ))i (sin ))j Ê v † ur œ x cos ) y sin )
.
.
.
.
œ ˆr cos ) r ) sin )‰ (cos )) ˆr sin ) r ) cos )‰ (sin )) by part (a),
.
. .
.
Ê v † ur œ r ; therefore, r œ x cos ) y sin );
.
.
u) œ (sin ))i (cos ))j Ê v † u) œ x sin ) y cos )
.
.
.
.
.
œ ˆr cos ) r ) sin )‰ ( sin )) ˆr sin ) r ) cos )‰ (cos )) by part (a) Ê v † u) œ r ) ;
.
.
.
therefore, r ) œ x sin ) y cos )
6. r œ f()) Ê
dr
dt
œ f w () )
d)
dt
Ê
d# r
dt#
#
œ f ww ()) ˆ ddt) ‰ f w ())
d# )
dt#
;vœ
dr
dt
ur r
d)
dt
u)
"Î#
"Î#
d) ‰
ˆ dr ‰# r# ˆ ddt) ‰# “ œ ’af w b# f # “
r sin ) ddt) ‰ i ˆsin ) dr
dt r cos ) dt j Ê kvk œ ’ dt
Þ ÞÞ Þ ÞÞ
d)
dr
kv ‚ ak œ kx y y xk , where x œ r cos ) and y œ r sin ). Then dx
dt œ (r sin )) dt (cos )) dt
œ ˆcos )
dr
dt
#
#
d) dr
d)
ˆ d) ‰# (r sin )) ddt#) (cos )) ddt#r ; dy
dt dt (r cos )) dt
dt œ (r cos )) dt (sin
#
#
#
ˆ d) ‰ (r cos )) ddt#) (sin )) ddt#r . Then kv ‚ ak
Ê
œ (2 cos )) ddt) dr
dt (r sin )) dt
#
$
$
d) d# r
d) ˆ dr ‰#
œ (after much algebra) r# ˆ ddt) ‰ r ddt#) dr
œ ˆ ddt) ‰ Šf 2 f † f ww 2af w b2 ‹
dt r dt dt# 2 dt dt
Ê
d# x
dt#
d# y
dt#
Ê ,œ
œ (2 sin ))
kv‚ak
kvk
œ
dr
dt
dr
dt
f 2 f†f ww 2af w b2
$Î#
af w b# f # ‘
7. (a) Let r œ 2 t and ) œ 3t Ê
vœ
))
ˆ ddt) ‰ ;
dr
dt
œ 1 and
d)
dt
d# r
dt#
œ3 Ê
#
ur r ddt) u) Ê v(1) œ ur 3u) ; a œ ’ ddt#r
œ
d# )
dt#
#
r ˆ ddt) ‰ “ ur
œ 0. The halfway point is (1ß 3) Ê t œ 1;
#
’r ddt#) 2 dr
dt
d)
dt “ u)
Ê a(1) œ 9ur 6u)
(b) It takes the beetle 2 min to crawl to the origin Ê the rod has revolved 6 radians
#
#
Ê L œ '0 É[f())]# cf w ())d# d) œ '0 Ɉ2 3) ‰ ˆ 3" ‰ d) œ '0 É4
4)
3
œ '0 É 37 129 ) ) d) œ
"
#
6
6
6
œ È37
#
"
6
"
3
'0
6
È() 6)# 1 d) œ
6
"
3
’ ()#6) È() 6)# 1
)#
9
"
9
d)
ln ¸) 6 È() 6)# 1¸“
ln ŠÈ37 6‹ ¸ 6.5 in.
8. (a) x œ r cos ) Ê dx œ cos ) dr r sin ) d); y œ r sin ) Ê dy œ sin ) dr r cos ) d); thus
dx# œ cos# ) dr# 2r sin ) cos ) dr d) r# sin# ) d)# and
dy# œ sin# ) dr# 2r sin ) cos ) dr d) r# cos# ) d)# Ê ds2 œ dx# dy# dz# œ dr# r# d)# dz#
(b)
(c) r œ e) Ê dr œ e) d)
Ê L œ '0
ln 8
Èdr# r# d)# dz#
œ '0 Èe#) e#) e#) d)
ln 8
œ '0 È3e) d) œ ’È3 e) “
ln 8
ln 8
0
œ 8È3 È3 œ 7È3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
'
!
Chapter 13 Additional and Advanced Exercises
â
â i
â
9. (a) ur ‚ u) œ â cos )
â
â sin )
j
sin )
cos )
793
â
kâ
â
0 â œ k Ê a right-handed frame of unit vectors
â
0â
œ ( sin ))i (cos ))j œ u) and ddu)) œ ( cos ))i (sin ))j œ ur
Þ
ÞÞ
Þ Þ
Þ
ÞÞ
ÞÞ
ÞÞ
Þ
Þ
ÞÞ
(c) From Eq. (7), v œ rur r)u) zk Ê a œ v œ a r ur r ur b ˆr )u) r) u) r) u) ‰ z k
Þ#
ÞÞ
ÞÞ
ÞÞ
ÞÞ
œ Š r r) ‹ ur ˆr) 2r )‰ u) z k
(b)
dur
d)
10. L(t) œ r(t) ‚ mv(t) Ê
œ ma Ê
dL
dt
dL
dt
œ ˆ ddtr ‚ mv‰ Šr ‚ m
d# r
dt# ‹
Ê
dL
dt
œ (v ‚ mv) (r ‚ ma) œ r ‚ ma ; F œ ma Ê krck$ r
œ r ‚ ma œ r ‚ Š krck$ r‹ œ krck$ (r ‚ r) œ 0 Ê L œ constant vector
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
794
Chapter 13 Vector-Valued Functions and Motion in Space
NOTES:
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
CHAPTER 14 PARTIAL DERIVATIVES
14.1 FUNCTIONS OF SEVERAL VARIABLES
1. (a) fa0, 0b œ 0
(d) fa3, 2b œ 33
2. (a) fˆ2, 16 ‰ œ
(b) fa1, 1b œ 0
È3
2
(b) fˆ3,
1‰
12
(c) fa2, 3b œ 58
œ È12
(c) fˆ1, 14 ‰ œ
1
È2
(d) fˆ 12 , 7‰ œ 1
3. (a) fa3, 1, 2b œ
(b) fˆ1, 12 , 14 ‰ œ
4
5
8
5
(c) fˆ0, 13 , 0‰ œ 3
(d) fa2, 2, 100b œ 0
4. (a) fa0, 0, 0b œ 7
(d) fŠ È42 ,
5
6
È2 , È2 ‹
(b) fa2, 3, 6b œ 0
(c) fa1, 2, 3b œ È35
œ É 21
2
5. Domain: all points axß yb on or above the line
yœx2
6. Domain: all points axß yb outside the circle
x2 y2 œ 4
7. Domain: all points axß yb not liying on the graph
of y œ x or y œ x3
8. Domain: all points axß yb not liying on the graph
of x2 y2 œ 25
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
796
Chapter 14 Partial Derivatives
9. Domain: all points axß yb satisfying
x2 1 Ÿ y Ÿ x2 1
10. Domain: all points axß yb satisfying
ax 1bay 1b 0
11. Domain: all points axß yb satisfying
ax 2bax 2bay 3bay 3b 0
12. Domain: all points axß yb inside the circle
x2 y2 œ 4 such that x2 y2 Á 3
13.
14.
15.
16.
17. (a) Domain: all points in the xy-plane
(b) Range: all real numbers
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.1 Functions of Several Variables
(c)
(d)
(e)
(f)
18. (a)
(b)
(c)
(d)
level curves are straight lines y x œ c parallel to the line y œ x
no boundary points
both open and closed
unbounded
Domain: set of all axß yb so that y x 0 Ê y x
Range: z 0
level curves are straight lines of the form y x œ c where c
boundary is Èy x œ 0 Ê y œ x, a straight line
0
(e) closed
(f) unbounded
19. (a) Domain: all points in the xy-plane
(b) Range: z 0
(c) level curves: for f(xß y) œ 0, the origin; for faxß yb œ c 0, ellipses with center a0ß 0b and major and minor
axes along the x- and y-axes, respectively
(d) no boundary points
(e) both open and closed
(f) unbounded
20. (a) Domain: all points in the xy-plane
(b) Range: all real numbers
(c) level curves: for faxß yb œ 0, the union of the lines y œ „ x; for faxß yb œ c Á 0, hyperbolas centered at
a0ß 0b with foci on the x-axis if c 0 and on the y-axis if c 0
(d) no boundary points
(e) both open and closed
(f) unbounded
21. (a) Domain: all points in the xy-plane
(b) Range: all real numbers
(c) level curves are hyperbolas with the x- and y-axes as asymptotes when faxß yb Á 0, and the x- and y-axes
when f(xß y) œ 0
(d) no boundary points
(e) both open and closed
(f) unbounded
22. (a) Domain: all axß yb Á a0ß yb
(b) Range: all real numbers
(c) level curves: for faxß yb œ 0, the x-axis minus the origin; for faxß yb œ c Á 0, the parabolas y œ c x# minus the
origin
(d) boundary is the line x œ 0
(e) open
(f) unbounded
23. (a) Domain: all axß yb satisfying x# y# 16
(b) Range: z "4
(c) level curves are circles centered at the origin with radii r 4
(d) boundary is the circle x# y# œ 16
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
797
798
Chapter 14 Partial Derivatives
(e) open
(f) bounded
24. (a)
(b)
(c)
(d)
(e)
(f)
Domain: all axß yb satisfying x# y# Ÿ 9
Range: 0 Ÿ z Ÿ 3
level curves are circles centered at the origin with radii r Ÿ 3
boundary is the circle x# y# œ 9
closed
bounded
25. (a)
(b)
(c)
(d)
(e)
(f)
Domain: axß yb Á a0ß 0b
Range: all real numbers
level curves are circles with center a0ß 0b and radii r 0
boundary is the single point a0ß 0b
open
unbounded
26. (a)
(b)
(c)
(d)
(e)
(f)
Domain: all points in the xy-plane
Range: 0 z Ÿ 1
level curves are the origin itself and the circles with center a0ß 0b and radii r 0
no boundary points
both open and closed
unbounded
27. (a) Domain: all axß yb satisfying 1 Ÿ y x Ÿ 1
(b) Range: 1# Ÿ z Ÿ 1#
(c)
(d)
(e)
(f)
level curves are straight lines of the form y x œ c where 1 Ÿ c Ÿ 1
boundary is the two straight lines y œ 1 x and y œ 1 x
closed
unbounded
28. (a) Domain: all axß yb, x Á 0
(b) Range: 1# z 1#
(c)
(d)
(e)
(f)
level curves are the straight lines of the form y œ c x, c any real number and x Á 0
boundary is the line x œ 0
open
unbounded
29. (a)
(b)
(c)
(d)
(e)
(f)
Domain: all points axß yb outside the circle x# y# œ 1
Range: all reals
Circles centered ar the origin with radii r 1
Boundary: the cricle x# y# œ 1
open
unbounded
30. (a)
(b)
(c)
(d)
Domain: all points axß yb inside the circle x# y# œ 9
Range: z ln 9
Circles centered ar the origin with radii r 9
Boundary: the cricle x# y# œ 9
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.1 Functions of Several Variables
(e) open
(f) bounded
31. f
32. e
33. a
34. c
35. d
36. b
37. (a)
(b)
38. (a)
(b)
39. (a)
(b)
40. (a)
(b)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
799
800
Chapter 14 Partial Derivatives
41. (a)
(b)
42. (a)
(b)
43. (a)
(b)
44. (a)
(b)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.1 Functions of Several Variables
45. (a)
(b)
46. (a)
(b)
47. (a)
(b)
48. (a)
(b)
#
#
801
49. faxß yb œ 16 x# y# and Š2È2ß È2‹ Ê z œ 16 Š2È2‹ ŠÈ2‹ œ 6 Ê 6 œ 16 x# y# Ê x# y# œ 10
50. faxß yb œ Èx# 1 and a1ß 0b Ê z œ È1# 1 œ 0 Ê x# 1 œ 0 Ê x œ 1 or x œ 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
802
Chapter 14 Partial Derivatives
51. faxß yb œ Èx y2 3 and a3, 1b Ê z œ É3 a1b2 3 œ 1 Ê x y2 3 œ 1 Ê x y2 œ 4
52. faxß yb œ
2y x
xy1
and a1ß 1b Ê z œ
#a 1 b a 1 b
a1b 1 + 1
œ3 Ê 3œ
53.
54.
55.
56.
57.
58.
59.
60.
2y x
xy1
Ê y œ 4x 3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.1 Functions of Several Variables
803
61. faxß yß zb œ Èx y ln z at a3ß 1ß 1b Ê w œ Èx y ln z; at a3ß 1ß 1b Ê w œ È3 a1b ln 1 œ 2
Ê Èx y ln z œ 2
62. faxß yß zb œ ln ax# y z# b at a"ß #ß "b Ê w œ ln ax# y z# b ; at a"ß #ß "b Ê w œ ln a1 2 1b œ ln 4
Ê ln 4 œ ln ax# y z# b Ê x# y z# œ 4
63. gaxß yß zb œ Èx# y2 z# at Š1ß 1ß È2‹ Ê w œ Èx# y2 z# ; at Š1ß 1ß È2‹ Ê w œ Ê1# a1b2 ŠÈ2‹
#
œ 2 Ê 2 œ Èx# y2 z# Ê x# y2 z# œ 4
xyz
2x y z
64. gaxß yß zb œ
at a1ß 0ß 2b Ê w œ
xyz
2x y z ;
at a1ß 0ß 2b Ê w œ
1 0 a2b
2a1b 0 a2b
œ 14 Ê 14 œ
xyz
2x y z
Ê 2x y z œ 0
_
n
65. faxß yb œ ! Š xy ‹ œ
n œ0
1
1 Š xy ‹
œ
y
yx
for
¹ xy ¹ 1 Ê Domain: all points ax, yb satisfying lxl lyl;
at a1, 2b Ê since ¹ 12 ¹ 1 Ê z œ
Ê
y
yx
2
21
œ2
œ 2 Ê y œ 2x
_
66. gaxß yß zb œ !
n œ0
(x b y)n
n! zn
œ eÐxyÑÎz Ê Domain: all points ax, y, zb satisfying z Á 0; at aln 4ß ln 9ß 2b
Ê w œ eÐln 4 ln 9ÑÎ2 œ eÐln 36ÑÎ2 œ eln 6 œ 6 Ê 6 œ eÐxyÑÎz Ê
67. faxß yb œ 'x
y
d)
È1 )#
xy
z
œ ln 6
œ sin1 y sin1 x Ê Domain: all points
ax, yb satisfying 1 Ÿ x Ÿ 1 and 1 Ÿ y Ÿ 1;
at a0, 1b Ê sin1 1 sin1 0 œ 12 Ê sin1 y sin1 x
œ 12 . Since 12 Ÿ sin1 y Ÿ
1
2
and 12 Ÿ sin1 x Ÿ 12 , in
order for sin1 y sin1 x to equal 12 , 0 Ÿ sin1 y Ÿ
12
1
1
2
and
Ÿ sin x Ÿ 0; that is 0 Ÿ y Ÿ 1 and 1 Ÿ x Ÿ 0. Thus
y œ sinˆ 1 sin1 x‰ œ È1 x2 , x Ÿ 0
2
68. gaxß yß zb œ 'x
y
dt
1 t#
'0
z
d)
È4 )#
œ tan1 y tan1 x sin1 ˆ 2z ‰ Ê Domain: all points ax, y, zb satisfying 2 Ÿ z Ÿ 2;
at Š0ß 1ß È3‹ Ê tan1 1 tan1 0 sin1 Š
1
12
Ÿ tan1 y tan1 x Ÿ
131
12
È3
2 ‹
œ
71
12
Ê tan1 y tan1 x sin1 ˆ 2z ‰ œ
Ê z œ 2 sinˆ 7121 tan1 y tan1 x‰,
1
12
71
12 .
Ÿ tan1 y tan1 x Ÿ
Since 12 Ÿ sin1 ˆ 2z ‰ Ÿ 12 ,
131
12
69-72. Example CAS commands:
Maple:
with( plots );
f := (x,y) -> x*sin(y/2) + y*sin(2*x);
xdomain := x=0..5*Pi;
ydomain := y=0..5*Pi;
x0,y0 := 3*Pi,3*Pi;
plot3d( f(x,y), xdomain, ydomain, axes=boxed, style=patch, shading=zhue, title="#69(a) (Section 14.1)" );
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
804
Chapter 14 Partial Derivatives
plot3d( f(x,y), xdomain, ydomain, grid=[50,50], axes=boxed, shading=zhue, style=patchcontour, orientation=[-90,0],
title="#69(b) (Section 14.1)" );
# (b)
L := evalf( f(x0,y0) );
# (c)
plot3d( f(x,y), xdomain, ydomain, grid=[50,50], axes=boxed, shading=zhue, style=patchcontour, contours=[L],
orientation=[-90,0], title="#45(c) (Section 13.1)" );
73-76. Example CAS commands:
Maple:
eq := 4*ln(x^2+y^2+z^2)=1;
implicitplot3d( eq, x=-2..2, y=-2..2, z=-2..2, grid=[30,30,30], axes=boxed, title="#73 (Section 14.1)" );
77-80. Example CAS commands:
Maple:
x := (u,v) -> u*cos(v);
y := (u,v) ->u*sin(v);
z := (u,v) -> u;
plot3d( [x(u,v),y(u,v),z(u,v)], u=0..2, v=0..2*Pi, axes=boxed, style=patchcontour, contours=[($0..4)/2], shading=zhue,
title="#77 (Section 14.1)" );
69-60. Example CAS commands:
Mathematica: (assigned functions and bounds will vary)
For 69 - 72, the command ContourPlot draws 2-dimensional contours that are z-level curves of surfaces z = f(x,y).
Clear[x, y, f]
f[x_, y_]:= x Sin[y/2] y Sin[2x]
xmin= 0; xmax= 51; ymin= 0; ymax= 51; {x0, y0}={31, 31};
cp= ContourPlot[f[x,y], {x, xmin, xmax}, {y, ymin, ymax}, ContourShading Ä False];
cp0= ContourPlot[[f[x,y], {x, xmin, xmax}, {y, ymin, ymax}, Contours Ä {f[x0,y0]}, ContourShading Ä False,
PlotStyle Ä {RGBColor[1,0,0]}];
Show[cp, cp0]
For 73 - 76, the command ContourPlot3D will be used. Write the function f[x, y, z] so that when it is equated to zero, it
represents the level surface given.
For 73, the problem associated with Log[0] can be avoided by rewriting the function as x2 + y2 +z2 - e1/4
Clear[x, y, z, f]
f[x_, y_, z_]:= x2 y2 z2 Exp[1/4]
ContourPlot3D[f[x, y, z], {x, 5, 5}, {y, 5, 5}, {z, 5, 5}, PlotPoints Ä {7, 7}];
For 77 - 80, the command ParametricPlot3D will be used. To get the z-level curves here, we solve x and y in terms of z
and either u or v (v here), create a table of level curves, then plot that table.
Clear[x, y, z, u, v]
ParametricPlot3D[{u Cos[v], u Sin[v], u}, {u, 0, 2}, {v, 0, 2p}];
zlevel= Table[{z Cos[v], z sin[v]}, {z, 0, 2, .1}];
ParametricPlot[Evaluate[zlevel],{v, 0, 21}];
14.2 LIMITS AND CONTINUITY IN HIGHER DIMENSIONS
3x# y# 5
1.
lim
#
#
Ðxß yÑ Ä Ð0ß 0Ñ x y 2
2.
lim
Ðxß yÑ Ä Ð0ß 4Ñ Èy
x
œ
0
È4
œ
3(0)# 0# 5
0# 0# 2
œ
5
#
œ0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.2 Limits and Continuity in Higher Dimensions
3.
4.
5.
6.
7.
8.
9.
10.
lim
Ðxß yÑ Ä Ð3ß 4Ñ
Èx# y# 1 œ È3# 4# 1 œ È24 œ 2È6
#
lim
sec x tan y œ (sec 0) ˆtan 14 ‰ œ (1)(1) œ 1
lim
cos Š xxy y 1 ‹ œ cos Š 000 0 1 ‹ œ cos 0 œ 1
Ðxß yÑ Ä Ð2ß 3Ñ
Ðxß yÑ Ä ˆ0ß 14 ‰
Ðxß yÑ Ä Ð0ß 0Ñ
lim
#
lim
Ðxß yÑ Ä Ð1ß 1Ñ
Ðxß yÑ Ä Ð1Î27ß 13 Ñ
œ
12.
lim
y sin x
Ðxß yÑ Ä ˆ 12 ß 0‰
15.
16.
17.
1†sinˆ 16 ‰
1# 1
x# 2xy y#
xy
lim
x# y#
xy
lim
xy y 2x 2
x1
œ
œ
œ
œ
lim
Ðxß yÑ Ä Ð1ß 1Ñ
œ
œ
(x y)(x y)
xy
œ
lim
Ðx ß y Ñ Ä Ð 1 ß 1Ñ
xÁ1
y4
x y 2È x 2È y
Èx Èy
œ
œ
œ 2
(x y)#
xy
lim
1
2
1
4
11
1
Ðx ß y Ñ Ä Ð 1 ß 1Ñ
lim
#
#
Ðxß yÑ Ä Ð2ß 4Ñ x y xy 4x 4x
#
y Á 4, x Á x
lim
xÄ0
1Î2
2
œ
acos 0b "
0 sin ˆ 1# ‰
œ
lim
Ðxß yÑ Ä Ð0ß 0Ñ
xÁy
"
#
aey b ˆ sinx x ‰ œ e! † lim ˆ sinx x ‰ œ 1 † 1 œ 1
lim
Ðxß yÑ Ä Ð0ß 0Ñ
œ
cos y 1
Ðxß yÑ Ä Ð1ß 1Ñ
xÁ1
$
3
1 ‰ 3
3 xy œ cos É
ˆ 27
cos È
1 œ cos ˆ 13 ‰ œ
lim
Ðxß yÑ Ä Ð1ß 1Ñ
xÁy
"
36
ln k1 x# y# k œ ln k1 (1)# (1)# k œ ln 2
ey sin x
x
Ðxß yÑ Ä Ð0ß 0Ñ
Ðxß yÑ Ä Ð1ß 1Ñ
xÁy
#
exy œ e0 ln 2 œ eln ˆ 2 ‰ œ
lim
lim
$
1
Ðxß yÑ Ä Ð0ß ln 2Ñ
x sin y
#
Ðxß yÑ Ä Ð1ß 1Î6Ñ x 1
14.
#
Š x" y" ‹ œ #" ˆ "3 ‰‘ œ ˆ 6" ‰ œ
11.
13.
#
lim
lim
(x y) œ (" 1) œ 0
lim
(x y) œ (1 1) œ 2
Ðxß yÑ Ä Ð1ß 1Ñ
Ðxß yÑ Ä Ð1ß 1Ñ
(x 1)(y 2)
x1
œ
lim
Ðxß yÑ Ä Ð1ß 1Ñ
y4
lim
Ðxß yÑ Ä Ð2ß 4Ñ x(x 1)(y 4)
y Á 4, x Á x#
lim
Ðx ß y Ñ Ä Ð 0 ß 0Ñ
xÁy
œ
(y 2) œ (1 2) œ 1
1
lim
Ðxß yÑ Ä Ð2ß 4Ñ x(x 1)
x Á x#
ˆÈ x È y ‰ ˆ È x È y 2 ‰
Èx Èy
œ
lim
Ðxß yÑ Ä Ð0ß 0Ñ
œ
"
#(2 1)
Note: (xß y) must approach (0ß 0) through the first quadrant only with x Á y.
xy4
lim
Ðxß yÑ Ä Ð2ß 2Ñ Èx y 2
xyÁ4
œ
lim
Ðxß yÑ Ä Ð2ß 2Ñ
xyÁ4
ˆÈx y 2‰ ˆÈx y 2‰
Èx y 2
œ
lim
Ðxß yÑ Ä Ð2ß 2Ñ
xyÁ4
"
#
ˆÈ x È y 2 ‰
œ ŠÈ0 È0 2‹ œ 2
18.
œ
ˆÈ x y 2 ‰
œ ŠÈ2 2 2‹ œ 2 2 œ 4
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
805
806
Chapter 14 Partial Derivatives
19.
lim
œ
20.
22.
23.
24.
25.
26.
27.
28.
29.
30.
"
È(2)(2) 0 #
"
22
œ
"
È4 È3 1
œ
"
22
È2x y 2
œ
lim
Ðxß yÑ Ä Ð2ß 0Ñ ˆÈ2x y 2‰ ˆÈ2x y 2‰
2x y Á 4
œ
"
4
È x È y 1
xy1
lim
Ðxß yÑ Ä Ð4ß 3Ñ
xyÁ1
œ
21.
È2x y 2
2x y 4
Ðxß yÑ Ä Ð2ß 0Ñ
2x y Á 4
œ
œ
Èx Èy 1
lim
Ðxß yÑ Ä Ð4ß 3Ñ ˆÈx Èy 1‰ ˆÈx Èy 1‰
xyÁ1
sinax# y# b
x# y#
œ lim
sinar# b
r#
lim
1 cosaxyb
xy
œ lim
1 cos u
u
Ðxß yÑ Ä Ð0ß 0Ñ
x3 y3
Ðxß yÑ Ä Ð1ß "Ñ x y
lim
xy
lim
4
4
Ðx ß y Ñ Ä Ð 2 ß 2 Ñ x y
lim
T Ä Ð1ß 3ß 4Ñ
Š "x
lim
T Ä Ð 1 ß 1 ß 1Ñ
lim
T Ä Ð3ß 3ß 0Ñ
lim
lim
lim
œ
uÄ0
œ lim
rÄ0
œ lim
2r†cosar# b
2r
uÄ0
sin u
1
rÄ0
œ0
ax ybˆx2 xy y2 ‰
xy
Ðxß yÑ Ä Ð1ß "Ñ
xy
lim
2
2
Ðxß yÑ Ä Ð2ß 2Ñ ax ybax ybax y b
œ
"
lim
Ðxß yÑ Ä Ð4ß 3Ñ Èx Èy 1
œ lim cosar# b œ 1
œ
lim
"z ‹ œ
2xy yz
x # z#
œ
"
1
"
3
"
4
œ
2(1)(1) (1)(1)
1# (1)#
12 4 3
12
œ
2 "
11
œ
œ
lim
Ðxß yÑ Ä Ð1ß "Ñ
ax2 xy y2 b œ Š12 a1ba1b a1b2 ‹ œ 3
1
lim
2
2
Ðxß yÑ Ä Ð2ß 2Ñ ax ybax y b
œ
1
a2 2ba22 22 b
œ
1
32
19
12
œ #"
asin# x cos# y sec# zb œ asin# 3 cos# 3b sec# 0 œ 1 1# œ 2
T Ä ˆ 14 ß 12 ß 2‰
T Ä Ð1ß 0ß 3Ñ
"
y
rÄ0
œ
"
lim
Ðxß yÑ Ä Ð2ß 0Ñ È2x y #
"
4
lim
Ðxß yÑ Ä Ð0ß 0Ñ
œ
ze
T Ä Ð2 ß 3 ß 6 Ñ
tan" (xyz) œ tan" ˆ "4 †
2y
1
#
† 2‰ œ tan" ˆ 14 ‰
cos 2x œ 3e 2Ð0Ñ cos 21 œ (3)(1)(1) œ 3
ln Èx# y# z# œ ln È2# (3)# 6# œ ln È49 œ ln 7
31. (a) All axß yb
(b) All axß yb except a0ß 0b
32. (a) All axß yb so that x Á y
(b) All axß yb
33. (a) All axß yb except where x œ 0 or y œ 0
(b) All axß yb
34. (a) All axß yb so that x# 3x 2 Á 0 Ê ax 2bax 1b Á 0 Ê x Á 2 and x Á 1
(b) All axß yb so that y Á x#
35. (a) All axß yß zb
(b) All axß yß zb except the interior of the cylinder x# y# œ 1
36. (a) All axß yß zb so that xyz 0
(b) All axß yß zb
37. (a) All axß yß zb with z Á 0
(b) All axß yß zb with x# z# Á 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.2 Limits and Continuity in Higher Dimensions
38. (a) All axß yß zb except axß 0ß 0b
(b) All axß yß zb except a0ß yß 0b or axß 0ß 0b
39. (a) All axß yß zb such that z x2 y2 1
(b) All axß yß zb such that z Á Èx2 y2
807
40. (a) All axß yß zb such that x2 y2 z2 Ÿ 4
(b) All axß yß zb such that x2 y2 z2 9 except when x2 y2 z2 œ 25
41.
lim
x
È x# y#
œ lim b Èx#x x# œ lim b È2x kxk œ lim b Èx2 x œ lim b È"2 œ È"2 ;
xÄ0
xÄ0
xÄ0
xÄ0
lim
x
È x# y#
œ lim c È2x kxk œ lim c È2(xx) œ lim c
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ x
x0
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ x
x0
42.
43.
44.
45.
46.
47.
48.
49.
50.
lim
x%
x% y#
œ lim
lim
x% y#
x% y#
œ lim
lim
xy
kxyk
lim
xy
xy
lim
x2 y
xy
œ lim
lim
x# y
y
œ lim
lim
x# y
x4 y2
œ lim
lim
xy2 1
y1
œ lim
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ 0
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ kx#
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ kx
kÁ0
Ðx ß y Ñ Ä Ð 0 ß 0 Ñ
along y œ kx
k Á 1
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ kx
kÁ1
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ kx#
kÁ0
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ kx#
Ðxß yÑ Ä Ð1ß 1Ñ
along x œ 1
lim
Ðxß yÑ Ä Ð1ß 1Ñ
along y œ 1
œ
xÄ0
xÄ0
x%
œ 1;
%
#
x Ä 0 x 0
x% akx# b
#
œ lim
x(kx)
x Ä 0 kx(kx)k
œ lim
x kx
x Ä 0 x kx
1k
1k
x# kx#
kx#
kx4
y2 1
y Ä 1 y1
xk
œ
k
1k
#
x Ä 0 x% ax# b
"
È2
œ lim
x%
%
x Ä 0 2x
œ
"
#
Ê different limits for different values of k
; if k 0, the limit is 1; but if k 0, the limit is 1
k
1 k2
Ê different limits for different values of k
yÄ1
1
x Ä 1 x1
œ lim
Ê different limits for different values of k, k Á 1
Ê different limits for different values of k, k Á 0
œ lim ay 1b œ 2;
x 1
2
x Ä 1 x 1
œ lim
1k
k
œ
k
x Ä 0 kkk
x%
œ lim
1 k#
1 k#
œ
œ lim
x Ä 0 1k
œ
x%
x% y#
œ
Ê different limits for different values of k, k Á 1
œ lim
4
2 4
x Ä 0 x k x
xy 1
x2 y2
kx#
#
x Ä 0 kkx k
œ
x % k# x%
%
# %
x Ä 0 x k x
œ lim
x2 kx
x Ä 0 x kx
xÄ0
lim
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ x#
œ lim
#
x Ä 0 x% akx# b
"
È2
xÄ0
lim
Ðx ß y Ñ Ä Ð 1 ß 1Ñ
along y œ x
œ 21 ;
lim
xy2 1
y1
Ð x ß y Ñ Ä Ð 1 ß 1Ñ
along y œ x2
y3 1
y Ä 1 y1
œ lim
xy 1
x2 y2
œ lim ay2 y 1b œ 3
x 3 1
2
4
x Ä 1 x x
œ lim
3
2
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
yÄ1
x2 x 1
a
x
1bax2 1b
xÄ1
œ lim
808
Chapter 14 Partial Derivatives
Ú 1 if y x%
51. fax, yb œ Û 1 if y Ÿ 0
Ü 0 otherwise
(a)
(b)
(c)
fax, yb œ 1 since any path through a0, 1b that is close to a0, 1b satisfies y
lim
fax, yb œ 0 since any path through a2, 3b that is close to a2, 3b does not satisfiy either y
lim
fax, yb œ 1 and
(b)
(c)
Ðx ß y Ñ Ä Ð 0 ß 0 Ñ
along x œ 0
x2
x3
lim
fax, yb œ 0 Ê
lim
Ðx ß y Ñ Ä Ð 0 ß 0Ñ
along y œ x2
fax, yb does not exist
lim
Ðxß yÑ Ä Ð0ß 0Ñ
if x 0
if x 0
fax, yb œ 32 œ 9 since any path through a3, 2b that is close to a3, 2b satisfies x
0
Ðxß yÑ Ä Ð3ß 2Ñ
fax, yb œ a2b3 œ 8 since any path through a2, 1b that is close to a2, 1b satisfies x 0
lim
Ðxß yÑ Ä Ð2ß 1Ñ
fax, yb œ 0 since the limit is 0 along any path through a0, 0b with x 0 and the limit is also zero along
lim
Ðxß yÑ Ä Ð0ß 0Ñ
any path through a0, 0b with x
0
53. First consider the vertical line x œ 0 Ê
2x2 y
4 y2
x
Ðxß yÑ Ä Ð0ß 0Ñ
lim
2a0b2 y
a
b4 y2
0
yÄ0
œ lim
along x œ 0
œ lim 0 œ 0. Now consider any nonvertical
yÄ0
through a0, 0b. The equation of any line through a0, 0b is of the form y œ mx Ê
œ
x% or y Ÿ 0
Ðxß yÑ Ä Ð2ß 3Ñ
52. fax, yb œ œ
(a)
x%
lim
Ðxß yÑ Ä Ð0ß 1Ñ
2
lim 2x amxb 2
x Ä 0 x4 amxb
œ
3
lim 4 2mx 2 2
x Ä 0 x m x
54. If f is continuous at (x! ß y! ), then
3
lim 2 2mx
2
2
x Ä 0 x ax m b
œ
lim
Ðxß yÑ Ä Ðx! ß y! Ñ
œ
lim
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ mx
œ 0. Thus
lim 22mx 2
x Ä 0 ax m b
faxß yb œ
2x2 y
4 y2
x
Ðxß yÑ Ä Ð0ß 0Ñ
lim
along y
lim
Ðxß yÑ Ä Ð0ß 0Ñ
any line though a0, 0b
2x2 y
x4 y2
œ mx
œ 0.
f(xß y) must equal f(x! ß y! ) œ 3. If f is not continuous at
(x! ß y! ), the limit could have any value different from 3, and need not even exist.
55.
lim
Ðxß yÑ Ä Ð0ß 0Ñ
Š1
x# y#
3 ‹
œ 1 and
lim
Ðx ß y Ñ Ä Ð 0 ß 0Ñ
1œ1 Ê
# #
56. If xy 0,
2 kxyk Š x 6y ‹
lim
kxyk
Ðx ß y Ñ Ä Ð 0 ß 0Ñ
tan " xy
xy
lim
Ðxß yÑ Ä Ð0ß 0Ñ
œ 1, by the Sandwich Theorem
# #
œ
2xy Š x 6y ‹
lim
xy
Ðxß yÑ Ä Ð0ß 0Ñ
œ
lim
Ðxß yÑ Ä Ð0ß 0Ñ
ˆ2
xy ‰
6
œ 2 and
# #
2 kxyk
Ðxß yÑ Ä Ð0ß 0Ñ kxyk
lim
œ
lim
Ðx ß y Ñ Ä Ð 0 ß 0 Ñ
œ
lim
Ðxß yÑ Ä Ð0ß 0Ñ
ˆ2
xy ‰
6
2 œ 2; if xy 0,
œ 2 and
lim
Ðx ß y Ñ Ä Ð 0 ß 0Ñ
lim
2 kxyk Š x 6y ‹
kxyk
Ðxß yÑ Ä Ð0ß 0Ñ
2 kxyk
kxyk
œ2 Ê
lim
Ðxß yÑ Ä Ð0ß 0Ñ
# #
œ
2xy Š x 6y ‹
lim
xy
Ðxß yÑ Ä Ð0ß 0Ñ
4 4 cos Èkxyk
kxyk
œ 2, by the Sandwich Theorem
57. The limit is 0 since ¸sin ˆ "x ‰¸ Ÿ 1 Ê 1 Ÿ sin ˆ x" ‰ Ÿ 1 Ê y Ÿ y sin ˆ x" ‰ Ÿ y for y 0, and y y sin ˆ "x ‰
y Ÿ 0. Thus as (xß y) Ä (!ß !), both y and y approach 0 Ê y sin ˆ "x ‰ Ä 0, by the Sandwich Theorem.
58. The limit is 0 since ¹cos Š "y ‹¹ Ÿ 1 Ê 1 Ÿ cos Š y" ‹ Ÿ 1 Ê x Ÿ x cos Š y" ‹ Ÿ x for x
0, and x
x cos Š y" ‹
for x Ÿ 0. Thus as (xß y) Ä (!ß !), both x and x approach 0 Ê x cos Š "y ‹ Ä 0, by the Sandwich Theorem.
59. (a) f(xß y)k yœmx œ
2m
1 m#
œ
2 tan )
1 tan# )
œ sin 2). The value of f(xß y) œ sin 2) varies with ), which is the line's
angle of inclination.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
y for
x
Section 14.2 Limits and Continuity in Higher Dimensions
(b) Since f(xß y)k yœmx œ sin 2) and since 1 Ÿ sin 2) Ÿ 1 for every ),
lim
Ðxß yÑ Ä Ð0ß 0Ñ
809
f(xß y) varies from 1 to 1
along y œ mx.
60. kxy ax# y# bk œ kxyk kx# y# k Ÿ kxk kyk kx# y# k œ Èx# Èy# kx# y# k Ÿ Èx# y# Èx# y# kx# y# k
#
#
#
œ ax# y# b Ê ¹ xyxa#xy#y b ¹ Ÿ
Ê
61.
62.
63.
lim
Ðxß yÑ Ä Ð0ß 0Ñ
x$ xy#
œ lim
lim
#
#
Ðxß yÑ Ä Ð0ß 0Ñ x y
lim
Ðxß yÑ Ä Ð0ß 0Ñ
lim
Ðxß yÑ Ä Ð0ß 0Ñ
x# y#
x# y# ‹
Šxy
rÄ0
$
ax# y# b
x# y#
#
œ x# y# Ê ax # y # b Ÿ
œ 0 by the Sandwich Theorem, since
r$ cos$ ) (r cos )) ar# sin# )b
r# cos# ) r# sin# )
$
œ lim
rÄ0
$
$
$
y#
x# y#
r# sin# )
r#
rÄ0
œ lim
lim
Ðx ß y Ñ Ä Ð 0 ß 0Ñ
r acos$ ) cos ) sin# )b
1
y
r cos ) r sin )
cos Š xx#
y# ‹ œ lim cos Š r# cos# ) r# sin# ) ‹ œ lim cos ’
$
rÄ0
xy ax# y# b
x# y#
rÄ0
Ÿ a x# y# b
„ ax# y# b œ 0; thus, define fa0ß 0b œ 0
œ0
r acos$ ) sin$ )b
“
1
œ cos 0 œ 1
œ lim asin# )b œ sin# ); the limit does not exist since sin# ) is between
rÄ0
0 and 1 depending on )
64.
65.
lim
Ðxß yÑ Ä Ð0ß 0Ñ
2r cos )
œ lim
2x
lim
#
#
Ðxß yÑ Ä Ð0ß 0Ñ x x y
#
r Ä 0 r r cos )
œ lim
2 cos )
r Ä 0 r cos )
œ
2 cos )
cos )
ky k
krk akcos )k ksin )kb
" kr cos )k kr sin )k
tan" ’ kxx#k
’
“ œ lim tan" ’
“;
y# “ œ lim tan
r#
r#
rÄ0
if r Ä 0 , then lim b
rÄ!
rÄ0
tan" ’ krk akcos )rk# ksin )kb “
œ lim b tan" ’ kcos )k r ksin )k “ œ
rÄ!
lim tan" ’ krk akcos )rk# ksin )kb “ œ lim c tan" Š kcos )kr ksin )k ‹ œ
rÄ!
r Ä !c
66.
; the limit does not exist for cos ) œ 0
x# y#
œ lim
lim
#
#
Ðxß yÑ Ä Ð0ß 0Ñ x y
rÄ0
r# cos# ) r# sin# )
r#
1
#
1
#
Ê the limit is
; if r Ä 0 , then
1
#
œ lim acos# ) sin# )b œ lim (cos 2)) which ranges between
rÄ0
rÄ0
1 and 1 depending on ) Ê the limit does not exist
67.
lim
Ðx ß y Ñ Ä Ð 0 ß 0Ñ
ln Š 3x
#
x# y# 3y#
‹
x# y#
œ lim ln Š 3r
rÄ0
#
cos# ) r% cos# ) sin# ) 3r# sin# )
‹
r#
œ lim ln a3 r# cos# ) sin# )b œ ln 3 Ê define f(0ß 0) œ ln 3
rÄ0
68.
lim
Ðxß yÑ Ä Ð0ß 0Ñ
3xy#
x# y#
(3r cos )) ar# sin# )b
r#
rÄ0
œ lim
œ lim 3r cos ) sin# ) œ 0 Ê define f(0ß 0) œ 0
rÄ0
69. Let $ œ 0.1. Then Èx# y# $ Ê Èx# y# 0.1 Ê x# y# 0.01 Ê kx# y# 0k 0.01
Ê kf(xß y) f(!ß !)k 0.01 œ %.
70. Let $ œ 0.05. Then kxk $ and kyk $ Ê kfaxß yb fa0ß 0bk œ ¸ x# y 1 0¸ œ ¸ x# y 1 ¸ Ÿ kyk 0.05 œ %.
71. Let $ œ 0.005. Then kxk $ and kyk $ Ê kfaxß yb fa0ß 0bk œ ¸ xx#y1 0¸ œ ¸ xx#y1 ¸ Ÿ kx yk kxk kyk
0.005 0.005 œ 0.01 œ %.
kx yk
"
72. Let $ œ 0.01. Since 1 Ÿ cos x Ÿ 1 Ê 1 Ÿ 2 cos x Ÿ 3 Ê "3 Ÿ #cos
Ÿ ¸ 2 x cosy x ¸ Ÿ kx yk
x Ÿ 1 Ê
3
Ÿ kxk kyk . Then kxk $ and kyk $ Ê kfaxß yb fa0ß 0bk œ ¸ 2 x cosy x 0¸ œ ¸ 2 x cosy x ¸ Ÿ kxk kyk 0.01 0.01
œ 0.02 œ %.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
810
Chapter 14 Partial Derivatives
y2
x2 y2
73. Let $ œ 0.04. Since y2 Ÿ x2 y2 Ê
Ÿ1Ê
lxly2
x2 y2
Ÿ lxl œ Èx2 Ÿ Èx2 y2 $ Ê kfaxß yb fa0ß 0bk
2
œ ¹ x2xy y2 0¹ 0.04 œ %.
74. Let $ œ 0.01. If lyl Ÿ 1, then y2 Ÿ lyl œ Èy2 Ÿ Èx2 y2 , so lxl œ Èx2 Ÿ Èx2 y2 Ê lxl y2 Ÿ 2Èx2 y2 . Since
x2
x2 y2
x2 Ÿ x 2 y 2 Ê
Ÿ 1 and y2 Ÿ x2 y2 Ê
y2
x2 y2
Ÿ 1. Then
lx3 y4 l
x2 y2
Ÿ
x2
x2 y2 lxl
y2
2
x2 y2 y
Ÿ lxl y2 2$
y
Ê kfaxß yb fa0ß 0bk œ ¹ xx2
y2 0¹ 2a0.01b œ 0.002 œ % .
3
4
75. Let $ œ È0.015. Then Èx# y# z# $ Ê kf(xß yß z) f(!ß 0ß 0)k œ kx# y# z# 0k œ kx# y# z# k
#
#
œ ŠÈx# t# x# ‹ ŠÈ0.015‹ œ 0.015 œ %.
76. Let $ œ 0.2. Then kxk $ , kyk $ , and kzk $ Ê kf(xß yß z) f(!ß 0ß 0)k œ kxyz 0k œ kxyzk œ kxk kyk kzk (0.2)$
œ 0.008 œ %.
77. Let $ œ 0.005. Then kxk $ , kyk $ , and kzk $ Ê kf(xß yß z) f(!ß 0ß 0)k œ ¹ x# x y# yz#z 1 0¹
œ ¹ x# x y# yz#z 1 ¹ Ÿ kx y zk Ÿ kxk kyk kzk 0.005 0.005 0.005 œ 0.015 œ %.
78. Let $ œ tan" (0.1). Then kxk $ , kyk $ , and kzk $ Ê kf(xß yß z) f(!ß 0ß 0)k œ ktan# x tan# y tan# zk
Ÿ ktan# xk ktan# yk ktan# zk œ tan# x tan# y tan# z tan# $ tan# $ tan# $ œ 0.01 0.01 0.01 œ 0.03 œ %.
79.
f(xß yß z) œ
lim
Ðxß yß zÑ Ä Ðx! ß y! ß z! Ñ
lim
(x y z) œ x! y! z! œ f(x! ß y! ß z! ) Ê f is continuous at
lim
ax# y# z# b œ x!# y!# z!# œ f(x! ß y! ß z! ) Ê f is continuous at
Ðxß yß zÑ Ä Ðx! ß y! ß z! Ñ
every (x! ß y! ß z! )
80.
f(xß yß z) œ
lim
Ðxß yß zÑ Ä Ðx! ß y! ß z! Ñ
Ðxß yß zÑ Ä Ðx! ß y! ß z! Ñ
every point (x! ß y! ß z! )
14.3 PARTIAL DERIVATIVES
1.
`f
`x
œ 4x,
`f
`y
3.
`f
`x
œ 2x(y 2),
5.
`f
`x
œ 2y(xy 1),
7.
`f
`x
œ
9.
`f
`x
œ (x " y)# †
10.
`f
`x
œ
11.
`f
`x
œ œ
2.
`f
`x
œ 2x y,
4.
`f
`x
œ 5y 14x 3,
œ 2x(xy 1)
6.
`f
`x
œ 6(2x 3y)# ,
y
È x# y#
8.
`f
`x
œ
œ 3
`f
`y
x
`f
È x# y# , ` y
œ x# 1
`f
`y
œ
`
`x
(x y) œ (x " y)# ,
ax# y# b (1) x(2x)
ax# y# b#
œ
y# x#
ax# y# b#
(xy 1)(1) (x y)(y)
(xy 1)#
œ
,
`f
`y
y# 1
(xy 1)#
œ
,
`f
`y
œ (x " y)# †
ax# y# b (0) x(2y)
ax# y# b#
`f
`y
œ
`
`y
2x#
$ $
É
x ˆ #y ‰
`f
`y
,
œ x 2y
`f
`y
`f
`y
œ 5x 2y 6
`f
`y
œ 9(2x 3y)#
œ
"
$ $
3É
x ˆ #y ‰
(x y) œ (x " y)#
œ ax# 2xy
y # b#
(xy ")(1) (x y)(x)
(xy 1)#
œ
x # "
(xy 1)#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.3 Partial Derivatives
12.
`f
`x
œ
13.
`f
`x
œ eÐxy1Ñ †
14.
`f
`x
œ ex sin (x y) ex cos (x y),
15.
`f
`x
œ
16.
`f
`x
œ exy †
`f
`x
`f
`y
œ 2 sin (x 3y) †
`f
`x
œ 2 cos a3x y# b †
17.
18.
"
#
1 ˆ xy ‰
"
xy
†
`
`x
†
`
`x
`
`x
ˆ yx ‰ œ
`
`x
œ x# y y# ,
y
#
x# ’1 ˆ xy ‰ “
(x y 1) œ eÐxy1Ñ ,
(x y) œ
"
xy
,
`f
`y
(xy) † ln y œ yexy ln y,
`
`x
`
`y
œ 2 sin (x 3y) †
"
xy
œ
`f
`y
`f
`y
`f
`y
†
`f
`y
"
#
1 ˆ xy ‰
œ
œ eÐxy1Ñ †
`
`y
†
`
`y
1
#
x ’1 ˆ xy ‰ “
œ
x
x# y#
(x y 1) œ eÐxy1Ñ
œ ex cos (x y)
`
`y
"
xy
(x y) œ
œ exy †
`
`y
(xy) † ln y exy †
sin (x 3y) œ 2 sin (x 3y) cos (x 3y) †
sin (x 3y) œ 2 sin (x 3y) cos (x 3y) †
`
`x
ˆ yx ‰ œ
"
y
œ xexy ln y
`
`x
`
`y
(x 3y) œ 2 sin (x 3y) cos (x 3y),
exy
y
(x 3y) œ 6 sin (x 3y) cos (x 3y)
cos a3x y# b œ 2 cos a3x y# b sin a3x y# b †
œ 6 cos a3x y# b sin a3x y# b ,
`f
`
#
#
#
#
` y œ 2 cos a3x y b † ` y cos a3x y b œ 2 cos a3x y b sin a3x y b †
`
`x
a3x y# b
`
`y
a3x y# b
œ 4y cos a3x y# b sin a3x y# b
19.
`f
`x
œ yxyc1 ,
21.
`f
`x
œ g(x),
`f
`y
œ xy ln x
`f
`y
20. f(xß y) œ
Ê
`f
`x
"
x ln y
œ
and
`f
`y
œ
ln x
y(ln y)#
œ g(y)
_
22. f(xß y) œ ! (xy)n , kxyk 1 Ê f(xß y) œ
n œ0
`f
`y
ln x
ln y
œ (1 "xy)# †
`
`y
(1 xy) œ
"
1 xy
Ê
`f
`x
œ (1 "xy)# †
`
`x
(1 xy) œ
y
(1 xy)#
and
x
(1 xy)#
23. fx œ y# , fy œ 2xy, fz œ 4z
24. fx œ y z, fy œ x z, fz œ y x
25. fx œ 1, fy œ Èy#y z# , fz œ Èy#z z#
26. fx œ x ax# y# z# b
27. fx œ
yz
È 1 x # y# z#
28. fx œ
"
kx yzk È(x yz)# 1
29. fx œ
"
x 2y 3z
30. fx œ yz †
"
xy
†
, fy œ
, fy œ
`
`x
xz
È 1 x # y# z#
, fy œ
(xy) œ
#
, fy œ y ax# y# z# b
#
`
`z
, fz œ
(yz)(y)
xy
, fz œ
œ
yz
x
$Î#
, fz œ z ax# y# z# b
$Î#
xy
È 1 x# y# z#
z
kx yzk È(x yz)# 1
2
x 2y 3z
fz œ y ln (xy) yz †
#
$Î#
, fz œ
y
kx yzk È(x yz)# 1
3
x 2y 3z
, fy œ z ln (xy) yz †
`
`y
ln (xy) œ z ln (xy)
yz
xy
†
`
`y
(xy) œ z ln (xy) z,
ln (xy) œ y ln (xy)
#
#
#
#
#
#
31. fx œ 2xe ax y z b , fy œ 2ye ax y z b , fz œ 2ze ax y z b
32. fx œ yzexyz , fy œ xzexyz , fz œ xyexyz
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
811
812
Chapter 14 Partial Derivatives
33. fx œ sech# (x 2y 3z), fy œ 2 sech# (x 2y 3z), fz œ 3 sech# (x 2y 3z)
34. fx œ y cosh axy z# b , fy œ x cosh axy z# b , fz œ 2z cosh axy z# b
35.
`f
`t
œ 21 sin (21t !),
36.
`g
`u
œ v# eÐ2uÎvÑ †
37.
`h
`3
œ sin 9 cos ),
38.
`g
`r
œ 1 cos ),
`
`u
`f
`!
œ sin (21t !)
`g
`v
Ð2uÎvÑ
ˆ 2u
‰
,
v œ 2ve
`h
`9
`g
`)
39. Wp œ V, Wv œ P
`h
`)
œ 3 cos 9 cos ),
`g
`z
œ r sin ),
$ v#
2g ,
W$ œ
Vv#
2g
m `A
q , `m
œ
2V$ v
2g
, Wv œ
#
V$ v
g
, Wg œ V#$gv#
`A
`h
œ
q
#
41.
`f
`x
œ 1 y,
`f
`y
œ 1 x,
42.
`f
`x
œ y cos xy,
43.
`g
`x
œ 2xy y cos x,
44.
`h
`x
œ ey ,
45.
`r
`x
œ
46.
`s
`x
œ”
` #s
` x#
œ
`w
`x
œ 2x tanaxyb x2 sec2 axyb † y œ 2x tanaxyb x2 y sec2 axyb,
47.
` #w
` x#
`h
`y
`f
`y
œ x cos xy,
`g
`y
œ xey 1,
"
`r
x y , ` y
"
#•
1 ˆ xy ‰
y(2x)
ax# y# b#
†
œ
`
`x
` #f
` y#
` #f
` x#
œ km
q#
` #f
` y` x
œ 0,
` #h
` x#
œ 0,
œ
` #h
` y#
"
(xy)#
,
` #r
` y#
` #s
` y#
,
œ
œ
` #f
` x` y
` #f
` y#
œ
` #h
` x` y
` #w
` x#
`w
`x
` #w
` x#
` #w
` y#
œ
` #w
` x` y
œ yex
2
,
` #f
` x` y
œ cos y,
œ cos xy xy sin xy
` #g
` y` x
œ
` #g
` x` y
œ 2x cos x
` #r
` x` y
`s
`y
œ ax# 2xy
,
y # b#
œ
"
(xy)#
œ”
"
#•
1 ˆ xy ‰
` #s
` y` x
œ
` #s
` x` y
`w
`y
†
`
`y
ˆ xy ‰ œ ˆ 1x ‰ ”
"
#•
1 ˆ xy ‰
ax# y# b (1) y(2y)
ax # y # b #
œ
œ
œ
x
x # y#
,
y# x#
ax # y # b #
œ x2 sec2 axyb † x œ x3 sec2 axyb,
œ x3 a2secaxybsecaxyb tanaxyb † xb œ 2x4 sec2 axyb tanaxyb
† 2x œ 2xy ex
y
2xyŠex
` #w
` y` x
2
2
y `w
, `y
y
` #w
` x` y
2
y
2
y
œ a1bex
† 2x‹ œ 2yex
2
y
a1 2x2 b,
† a1b œ ex
` #w
` y#
œ Šex
y
œ sinax2 yb x cosax2 yb † 2xy œ sinax2 yb 2x2 ycosax2 yb,
`w
`y
ay 2b,
œ
œ Šex
2
y
yex
2
2
y
2
` #w
` y#
œ 3x2 sec2 axyb x3 a2secaxybsecaxyb tanaxyb † yb œ 3x2 sec2 axyb x3 y sec2 axyb tanaxyb
y
œ 2y ex
œ ex
49.
` #g
` y#
œ
œ 2tanaxyb 2x sec2 axyb † y 2xy sec2 axyb x2 y a2secaxybsecaxyb tanaxyb † yb
` #w
` y` x
`w
`x
` #f
` y` x
œ ey
œ
œ 2tanaxyb 4xy sec2 axyb 2x2 y2 sec2 axyb tanaxyb,
48.
œ1
œ 2y y sin x,
y
x# y#
œ
h
#
œ x# sin xy,
"
` #r
(xy)# , ` y` x
"
#•
1 ˆ xy ‰
x(2y)
ax # y # b#
` #g
` x#
` #h
` y` x
œ xey ,
ˆ xy ‰ œ ˆ xy# ‰ ”
2xy
ax # y # b #
œ
œ y# sin xy,
œ x# sin y sin x,
"
` #r
x y , ` x #
œ
c,
k
q
œ 0,
`A
`q
œ
œ m,
` #f
` x#
Ð2uÎvÑ
ˆ 2u
‰
2ueÐ2uÎvÑ
v œ 2ve
œ 1
`A
`c
œ
`
`v
œ 3 sin 9 sin )
40.
,
`A
`k
œ 2veÐ2uÎvÑ v# eÐ2uÎvÑ †
† 2x‹a1 yb œ 2x ex
2
2
y
y
a1 yb ,
† a1b‹a1 yb ex
2
y
a1b
a1 yb
œ x cosax2 yb † x2 œ x3 cosax2 yb,
œ cosax2 yb † 2xy 4xy cosax2 yb 2x2 y sinax2 yb † 2xy œ 6xy cosax2 yb 4x3 y2 sinax2 yb,
œ x3 sinax2 yb † x2 œ x5 sinax2 yb,
` #w
` y` x
œ
` #w
` x` y
œ 3x2 cosax2 yb x3 sinax2 yb † 2xy œ 3x2 cosax2 yb 2x4 y sinax2 yb
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.3 Partial Derivatives
50.
`w
`x
` #w
` x#
` #w
` y#
œ
œ
œ
ˆx2 y‰ ax yba2xb
ax2 yb2
œ
x2 2xy y ` w
, `y
ax 2 y b 2
œ
ˆx2 y‰a1b ax yb
ax2 yb2
ˆx2 y‰2 a2x 2yb ˆx2 2xy y‰2ˆx2 y‰a2xb
2
2
’ax2 yb “
2
ˆx y‰ † 0 ˆx x‰2ˆx2 y‰† 1
2
2
2
’ax2 yb “
2
œ
œ
2x2 2x ` # w
,
ax 2 y b 3 ` y ` x
œ
` #w
` x` y
2x3 3x2 2xy y
ax2 yb3
51.
`w
`x
œ
52.
`w
`x
œ ex ln y yx ,
53.
`w
`x
œ y# 2xy$ 3x# y% ,
`w
`y
œ 2xy 3x# y# 4x$ y$ ,
54.
`w
`x
œ sin y y cos x y,
`w
`y
œ x cos y sin x x,
,
`w
`y
œ
55. (a) x first
3
2x 3y
`w
`y
` #w
` y` x
,
œ
x
y
57. fx a1ß 2b œ lim
hÄ0
hÄ0
` #w
` y` x
, and
"
y
œ œ
` #w
` x` y
œ
hÄ0
2
’ax2 yb “
` #w
` x` y
` #w
` y` x
` #w
` y` x
2
"
y
œ
"
x
#
œ 2y 6xy# 12x# y$ , and ``x`wy œ 2y 6xy# 12x# y$
` #w
` x` y
œ cos y cos x 1
(e) y first
(f) y first
œ cos y cos x 1, and
(d) x first
(b) y first three times
œ lim
ˆx2 y‰2 a2x 1b ˆx2 2xy y‰2ˆx2 y‰† 1
6
(2x 3y)#
œ
x" , and
(c) x first
f(1 hß 2) f(1ß 2)
h
13h 6h#
h
6
(2x 3y)#
ln x,
(b) y first
56. (a) y first three times
œ lim
œ
x 2 x
,
ax2 yb2
2ˆx3 3x2 y 3 xy y2 ‰
,
ax 2 y b 3
œ
2
2x 3y
œ
(c) y first twice
c1 (1 h) 2 6(1 h)# d (2 6)
h
(d) x first twice
h 6 a1 2h h# b 6
h
œ lim
hÄ0
œ lim (13 6h) œ 13,
hÄ0
f(1ß 2 h) f(1ß 2)
h
hÄ0
fy (1ß 2) œ lim
œ lim (2) œ 2
c1 1 (2 h) 3(2 h)d (2 6)
h
œ lim
hÄ0
(2 6 2h) (2 6)
h
œ lim
hÄ0
hÄ0
58. fx a2ß 1b œ lim
hÄ0
œ lim
hÄ0
fa2 hß 1b fa2ß 1b
h
a2h 1 hb 1
h
œ lim
hÄ0
c4 2a2 hb 3 a2 hbd a3 2b
h
œ lim 1 œ 1,
hÄ0
4 4 3a1 hb 2a1 hb# ‘ a3 2b
fy a2ß 1b œ lim fa2ß 1 hhb fa2ß 1b œ lim
h
hÄ0
hÄ0
a3 3h 2 4h 2h# b 1
h 2h#
œ lim
œ lim
œ lim a1 2hb œ 1
h
h
hÄ0
hÄ0
hÄ0
59. fx a2ß 3b œ lim
hÄ0
fa2 hß 3b fa2ß 3b
h
È2h 4 2
h
hÄ0
œ lim
fy a2ß 3b œ lim
hÄ0
œ lim
hÄ0
œ lim Š
hÄ0
œ lim Š
hÄ0
hÄ0
œ lim
œ lim
fa0ß 0 hb fa0ß 0b
h
hÄ0
œ lim
fy a0ß 0b œ lim
hÄ0
hÄ0
hÄ0
œ lim
2
h Ä 0 È2h 4 2
œ 12 ,
È 4 3 a3 h b 1 È 4 9 1
h
È3h 4 2 È3h 4 2
È3h 4 2 ‹
h
fa0 hß 0b fa0ß 0b
h
hÄ0
60. fx a0ß 0b œ lim
È 2 a 2 h b 9 1 È 4 9 1
h
È2h 4 2 È2h 4 2
È2h 4 2 ‹
h
fa2ß 3 hb fa2ß 3b
h
È3h 4 2
h
œ lim
sinŠh3 b 0‹
h2 b 0
0
sinŠ0 b h4 ‹
0 b h2
0
h
h
œ lim
3
h Ä 0 È2h 4 2
œ
3
4
œ lim
sin h3
h3
œ1
œ lim
sin h4
h3
œ lim Šh †
hÄ0
hÄ0
hÄ0
sin h4
h4 ‹
œ0†1œ0
61. (a) In the plane x œ 2 Ê fy axß yb œ 3 Ê fy a2ß 1b œ 3 Ê m œ 3
(b) In the plane y œ 1 Ê fx axß yb œ 2 Ê fy a2ß 1b œ 2 Ê m œ 2
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
813
814
Chapter 14 Partial Derivatives
62. (a) In the plane x œ 1 Ê fy axß yb œ 3y2 Ê fy a1ß 1b œ 3a1b2 œ 3 Ê m œ 3
(b) In the plane y œ 1 Ê fx axß yb œ 2x Ê fy a1ß 1b œ 2a1b œ 2 Ê m œ 2
63. fz ax! ß y! ß z! b œ lim
hÄ0
fz a1ß 2ß 3b œ lim
hÄ0
fax! ß y! ß z! hb fax! , y! ß z! b
h
fa1ß 2ß 3 hb fa1, 2ß 3b
h
64. fy ax! ß y! ß z! b œ lim
hÄ0
hÄ0
fax! ß y! hß z! b fax! , y! ß z! b
h
`z
`x
z$ 2y
œ lim
Ê 2c cos A 2b œ a2bc sin Ab ``Ab Ê
Ê
"2h 2h#
h
œ lim a12 2hb œ 12
hÄ0
œ lim a2h 9b œ 9
hÄ0
(sin A) ``Aa a cos A
sin# A
`A
`b
œ
`A
`a
c cos A b
bc sin A
œ
a
bc sin A
ba csc B cot Bb Ê
œ 2
`x
`z
œ
"
6
; also 0 œ 2b 2c cos A a2bc sin Ab ``Ab
œ 0 Ê asin Ab `` xa a cos A œ 0 Ê
`a
`B
`z
`x
2x‰ `` xz œ x Ê at (1ß 1ß 3) we have (3 1 2) `` xz œ 1 or
y
x
67. a# œ b# c# 2bc cos A Ê 2a œ a2bc sin Ab ``Aa Ê
a
b
sin A œ sin B
ˆ sin" A ‰ ``Ba œ
hÄ0
œ 0 Ê a3xz# 2yb `` xz œ y z$ Ê at (1ß 1ß 1) we have (3 2) `` xz œ 1 1 or
66. ˆ `` xz ‰ z x ˆ yx ‰ `` xz 2x `` xz œ 0 Ê ˆz
68.
œ lim
;
a2h# 9hb 0
h
hÄ0
fa1ß hß 3b fa1, 0ß 3b
h
hÄ0
`z ‰
`x x
2a3 hb# 2a9b
h
œ lim
fy a1ß 0ß 3b œ lim
65. y ˆ3z#
;
`a
`A
œ
a cos A
sin A
; also
œ b csc B cot B sin A
69. Differentiating each equation implicitly gives 1 œ vx ln u ˆ vu ‰ ux and 0 œ ux ln v ˆ uv ‰ vx or
"
º0
aln ub vx ˆ vu ‰ ux œ 1
Ê vx œ
ˆ uv ‰ vx aln vb ux œ 0 Ÿ
º
v
u
ln v º
ln u
u
v
œ
v
u
ln v º
ln v
aln ubaln vb 1
70. Differentiating each equation implicitly gives 1 œ a2xbxu a2ybyu and 0 œ a2xbxu yu or
a2xbxu a2ybyu œ 1
Ê xu œ
a2xbxu
yu œ 0
yu œ
"
0º
#x 4xy
2x
º 2x
œ
2x
2x 4xy
œ
0
0
2y
1 º
2x 2y
º 2x 1 º
2x
2x 4xy
œ 2x Š 2x " 4xy ‹ 2y Š 1 " 2y ‹ œ
71. fx axß yb œ œ
"
º0
œ
"
1 #y
œ
1
1 2y
1
2x 4xy
œ
1
2x 4xy
and
; next s œ x# y# Ê
2y
1 2y
œ
`s
`u
œ 2x
`x
`u
2y
`y
`u
1 2y
1 2y
if y 0
Ê fx axß yb œ 0 for all points ax, yb; at y œ 0, fy axß 0b œ lim fax, 0 hhb fax, 0b œ lim fax, hhb 0
if y 0
h Ä0
h Ä0
œ lim fax,h hb œ 0 because
h Ä0
lim
hÄ0c
fax, hb
h
œ
3
lim h
h Ä0 c h
œ 0 and
limb fax,h hb œ
h Ä0
2
limb hh œ 0 Ê fy axß yb œ œ
h Ä0
3y2
2y
if y 0
;
if y 0
fyx axß yb œ fxy axß yb œ 0 for all points ax, yb
72. At x œ 0, fx a0ß yb œ lim fa0 h, yhb fa0, yb œ lim fah, yhb 0 œ lim fah,h yb which does not exist because
h Ä0
œ
2
limc hh œ 0 and
hÄ0
fy axß yb œ œ
limb fah,h yb œ
h Ä0
h Ä0
Èh
h Ä0 h
limb
hÄ0
1
œ
lim 1 œ _ Ê fx axß yb œ 2Èx
h Ä0 b È h
2x
if x 0
if x 0
lim
hÄ0c
fah, yb
h
;
0 if x 0
Ê fy axß yb œ 0 for all points ax, yb; fyx axß yb œ 0 for all points ax, yb, while fxy axß yb œ 0 for all
0 if x 0
points ax, yb such that x Á 0.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.3 Partial Derivatives
73.
`f
`x
œ 2x,
`f
`y
74.
`f
`x
œ 6xz,
œ 2y,
`f
`y
`f
`z
œ 4z Ê
`f
`z
œ 6yz,
` #f
` x#
` #f
` y#
œ 2,
œ 2,
` #f
` z#
` #f
` x#
œ 6z# 3 ax# y# b ,
` #f
` x#
œ 4 Ê
` #f
` y#
œ 6z,
` #f
` y#
œ 6z,
` #f
` z#
` #f
` z#
815
œ 2 2 (4) œ 0
œ 12z Ê
` #f
` x#
` #f
` y#
` #f
` z#
œ 6z 6z 12z œ 0
`f
`x
œ 2ec2y sin 2x,
76.
`f
`x
œ
77.
`f
`x
œ 3,
78.
`f
`x
œ
75.
`f
`y
œ 2ec2y cos 2x,
œ 4ec2y cos 2x 4ec2y cos 2x œ 0
`f
`x
,
`f
`y
`f
`y
œ
#
1 Š xy ‹
œ
œ 2,
1 Îy
` #f
` x#
Ê
79.
x
x# y#
` #f
` y#
y
x# y#
` #f
` x#
œ 0,
y
y# x#
œ
` #f
` x#
,
,
`f
`y
2xy
ay# x# b2
œ "# ax# y# z# b
œ
y# x#
ax# y# b#
œ 4ec2y cos 2x,
` #f
` x#
` #f
` y#
,
œ
` #f
` y#
œ0 Ê
` #f
` x#
œ
x Îy 2
x
y # x#
2xy
ay# x# b2
$Î#
œ
#
1 Š xy ‹
` #f
` y#
x# y#
ax# y# b#
` #f
` x#
œ 4ec2y cos 2x Ê
` #f
` y#
œ
y# x#
ax# y# b#
` #f
` x#
x# y#
ax# y# b#
œ
ay# x# b†0 y†2x
ay# x# b2
œ
2xy
ay# x# b2
,
` #f
` y#
œ
80.
`f
`x
$Î# ` f
, `y
$Î#
#
a2xb œ x ax# y# z# b
3x# ax# y# z# b
$Î#
3z# ax# y# z# b
`f
`y
œ 3e3x4y cos 5z,
#
&Î#
œ 4e3x4y cos 5z,
` f
` z#
œ 25e3x4y cos 5z Ê
81.
`w
`x
œ cos (x ct),
82.
`w
`x
œ 2 sin (2x 2ct),
Ê
83.
84.
85.
#
` w
` t#
`w
`t
#
` f
` x#
#
` f
` y#
`f
`z
#
` f
` z#
` #w
` x#
œ c cos (x ct);
`w
`t
œ "# ax# y# z# b
œ
&Î#
“ œ 3 ax# y# z# b
œ 5e3x4y sin 5z;
,
`w
`t
œ
c
x ct
;
` #w
` x#
œ
2xy
ay # x # b 2
$Î#
` #f
` x#
1
(x ct)#
a2yb
ax # y # z # b
&Î#
œ sin (x ct),
#
` #w
` t#
,
3y# ax# y# z# b
$Î#
a3x# 3y# 3z# b ax# y# z# b
œ 9e3x4y cos 5z,
` #f
` y#
“
&Î#
œ0
œ 16e3x4y cos 5z,
` #w
` x#
` #w
` t#
œ c# sin (x ct) Ê
œ 4 cos (2x 2ct),
` #w
` t#
` #w
` t#
œ c# [ sin (x ct)] œ c#
œ 4c# cos (2x 2ct)
` w
` x#
,
&Î#
œ 9e3x4y cos 5z 16e3x4y cos 5z 25e3x4y cos 5z œ 0
œ 2c sin (2x 2ct);
œ c# [4 cos (2x 2ct)] œ c#
"
x ct
œ
$Î#
“ ’ ax# y# z# b
`w
`w
` x œ cos (x ct) 2 sin (2x 2ct), ` t œ c cos (x ct) 2c sin (2x
` #w
` #w
#
#
` x# œ sin (x ct) 4 cos (2x 2ct), ` t# œ c sin (x ct) 4c
#
#
Ê `` tw# œ c# [ sin (x ct) 4 cos (2x 2ct)] œ c# `` xw#
`w
`x
œ0
œ0
$Î#
’ ax# y# z# b
` #f
` y#
ay# x# b†0 axb†2y
ay# x# b2
$Î# ` f
$Î#
œ y ax# y# z# b
, ` z œ "# ax# y# z b
a2zb œ z ax# y# z# b
;
` #f
#
#
# $Î#
#
#
#
# &Î# ` # f
#
#
# $Î#
3x ax y z b
, ` y # œ ax y z b
3y#
` x # œ ax y z b
#
#
#
&Î#
` #f
#
#
# $Î#
3z# ax# y# z# b
Ê `` xf# `` yf# `` zf#
` z # œ ax y z b
œ ’ ax# y# z# b
œ00œ 0
` #f
` x#
,
Ê
` #f
` y#
œ
c#
(x ct)#
Ê
` #w
` t#
2ct);
cos (2x 2ct)
"
#
œ c# ’ (x
ct)# “ œ c
` #w
` x#
`w
`w
` #w
#
#
#
` x œ 2 sec (2x 2ct), ` t œ 2c sec (2x 2ct); ` x# œ 8 sec (2x 2ct) tan (2x 2ct),
` #w
` #w
#
#
#
#
` t# œ 8c sec (2x 2ct) tan (2x 2ct) Ê ux ` t# œ c [8 sec (2x 2ct) tan (2x 2ct)] œ
c#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
` #w
` x#
` #w
` x#
816
86.
87.
Chapter 14 Partial Derivatives
#
`w
xbct ` w
, ` t œ 15c sin (3x 3ct) cexbct ; `` xw# œ 45 cos (3x
` x œ 15 sin (3x 3ct) e
#
#
` #w
#
# xbct
Ê `` tw# œ c# c45 cos (3x 3ct) exbct d œ c# `` xw#
` t# œ 45c cos (3x 3ct) c e
`w
`t
œ
`f `u
`u `t
œ
` #f
` u#
Ê
œ a#
`f
`u
(ac) Ê
` #w
` t#
œ a# c#
` #w
` t#
œ (ac) Š `` uf# ‹ (ac) œ a# c#
#
` #f
` u#
œ c# Ša#
` #f
` u# ‹
œ c#
` #f
` u#
;
`w
`x
œ
`f `u
`u `x
œ
`f
`u
†a Ê
3ct) exbct ,
` #w
` x#
#
œ Ša `` uf# ‹ † a
` #w
` x#
88. If the first partial derivatives are continuous throughout an open region R, then by Theorem 3 in this section of the text,
f(xß y) œ f(x! ß y! ) fx (x! ß y! ) ?x fy (x! ß y! ) ?y %" ?x %# ?y, where %" , %# Ä 0 as ?x, ?y Ä 0. Then as
(xß y) Ä (x! ß y! ), ?x Ä 0 and ?y Ä 0 Ê
lim
f(xß y) œ f(x! ß y! ) Ê f is continuous at every point (x! ß y! ) in R.
Ðxß yÑ Ä Ðx! ß y! Ñ
89. Yes, since fxx , fyy , fxy , and fyx are all continuous on R, use the same reasoning as in Exercise 76 with
fx (xß y) œ fx (x! ß y! ) fxx (x! ß y! ) ?x fxy (x! ß y! ) ?y %" ?x %# ?y and
fy (xß y) œ fy (x! ß y! ) fyx (x! ß y! ) ?x fyy (x! ß y! ) ?y s%" ?x s%# ?y. Then
lim
fx (xß y) œ fx (x! ß y! )
Ðxß yÑ Ä Ðx! ß y! Ñ
and
lim
Ðxß yÑ Ä Ðx! ß y! Ñ
fy (xß y) œ fy (x! ß y! ).
90. To find ! and " so that ut œ uxx Ê ut œ " sina! xbe" t and ux œ ! cosa! xbe" t Ê uxx œ !2 sina! xbe" t ; then
ut œ uxx Ê " sina! xbe" t œ !2 sina! xbe" t , thus ut œ uxx only if " œ !2
h†02
04
91. fx a0, 0b œ lim fa0 hß 0hb fa0ß 0b œ lim h2
h Ä0
lim
Ðxß yÑ Ä Ð0ß 0Ñ
along x œ ky2
f ax, yb œ
values of k Ê
0
0†h2
h4
œ lim 0h œ 0; fy a0, 0b œ lim fa0ß 0 hhb fa0ß 0b œ lim 02
h
hÄ0
hÄ0
4
ˆky2 ‰y2
lim
œ lim k2 yky
2
4 y4
y Ä 0 aky2 b y4
yÄ0
lim
Ðxß yÑ Ä Ð0ß 0Ñ
hÄ0
œ
lim 2 k
y Ä 0 k 1
œ
hÄ0
k
k2 1
h
0
œ lim 0h œ 0;
hÄ0
Ê different limits for different
f ax, yb does not exist Ê f ax, yb is not continuous at a0, 0b Ê by Theorem 4, f ax, yb is not
differentiable at a0, 0b.
92. fx a0, 0b œ lim fa0 hß 0hb fa0ß 0b œ lim fahß 0hb 1 œ lim 1 h 1 œ 0; fy a0, 0b œ lim fa0ß 0 hhb fa0ß 0b œ lim fa0ß hhb 1 œ lim 1 h 1 œ 0;
h Ä0
lim
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ x2
h Ä0
h Ä0
f ax, yb œ lim 0 œ 0 but
yÄ0
lim
Ðxß yÑ Ä Ð0ß 0Ñ
along y œ 1.5x2
h Ä0
f ax, yb œ lim 1 œ 1 Ê
yÄ0
h Ä0
lim
Ðxß yÑ Ä Ð0ß 0Ñ
h Ä0
f ax, yb does not exist
Ê f ax, yb is not continuous at a0, 0b Ê by Theorem 4, f ax, yb is not differentiable at a0, 0b.
14.4 THE CHAIN RULE
1. (a)
`w
`x
œ 2x,
`w
`y œ
#
2y,
dx
dt
#
œ sin t,
#
dy
dt
#
œ cos t Ê
œ 0; w œ x y œ cos t sin t œ 1 Ê
(b)
dw
dt
(1 ) œ 0
2. (a)
`w
`x
œ 2x,
`w
`y
œ 2y,
dx
dt
œ sin t cos t,
dy
dt
dw
dt
dw
dt
œ 2x sin t 2y cos t œ 2 cos t sin t 2 sin t cos t
œ0
œ sin t cos t Ê
dw
dt
œ (2x)( sin t cos t) (2y)( sin t cos t)
œ 2(cos t sin t)(cos t sin t) 2(cos t sin t)(sin t cos t) œ a2 cos# t 2 sin# tb a2 cos# t 2 sin# tb
œ 0; w œ x# y# œ (cos t sin t)# (cos t sin t)# œ 2 cos# t 2 sin# t œ 2 Ê dw
dt œ 0
(b)
dw
dt
(0) œ 0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.4 The Chain Rule
3. (a)
`w
`x
œ
Ê
"
z
dw
dt
,
`w
`y
œ
"
z
,
`w
`z
œ
(x y)
z#
œ 2z cos t sin t
2
z
dx
dt
,
œ 2 cos t sin t,
sin t cos t
x y
z# t#
dy
dt
œ 2 sin t cos t,
cos# t sin# t
Š "# ‹ at# b
œ
œ 1; w œ
x
z
dz
dt
œ t"#
y
z
œ
t
(b)
dw
dt
(3) œ 1
4. (a)
`w
`x
œ
2x
x # y # z#
`w
`y
,
œ
2y
x # y# z#
2y cos t
dw
2x sin t
dt œ x# y# z# x# y# z#
œ 11616t ; w œ ln ax# y# z# b
dw
16
dt (3) œ 49
Ê
(b)
5. (a)
`w
`x
œ 2yex ,
`w
`y
œ 2ex ,
`w
`z
,
`w
`z
œ
2z
x # y# z#
4zt "Î#
x# y# z#
#
œ
œ ln acos t
œ "z ,
dx
dt
œ
2t
t# 1
,
,
dx
dt
œ sin t,
dy
dt
6. (a)
œ
dy
dt
"
t# 1
,
œ et Ê
dz
dt
#
t
(4t) atan " tb at# 1b
2 at#t 11b eet œ 4t tan" t 1; w œ 2yex ln
t# 1
"
ˆ 2 ‰ #
Ê dw
tb (2t) 1 œ 4t tan" t 1
dt œ t# 1 at 1b a2 tan
dw
ˆ1‰
dt (1) œ (4)(1) 4 1 œ 1 1
`w
`x
œ y cos xy,
`w
`y
œ x cos xy,
œ (ln t)[cos (t ln t)]
tc1
œe
sin (t ln t) Ê
(b)
(1) œ 1 (1 0)(1) œ 0
7. (a)
`z
`u
œ
`z `y
`y `u
œ 1,
dx
dt
œ 1,
dy
dt
œ
"
t
,
dz
dt
dw
dt
œ
sin# t
Š "t ‹
œt Ê
x
v ‰
4e
œ a4ex ln yb ˆ ucos
cos v Š y ‹ (sin v) œ
dw
dt
4ytex
t# 1
œ
16
1 16t
2ex
t# 1
et
z
z œ a2 tan" tb at# 1b t
œ etc1 Ê
dw
dt
œ y cos xy
4ex ln y
u
x cos xy
t
xy
4ex sin v
y
œ
4(u cos v) ln (u sin v)
v)(sin v)
4(u cos
œ (4 cos v) ln (u sin v) 4 cos v;
u
u sin v
`z
`z `x
`z `y
4ex
x
x
ˆ u sin v ‰
` v œ ` x ` v ` y ` v œ a4e ln yb u cos v Š y ‹ (u cos v) œ a4e
ln yb (tan v)
4ex u cos v
y
4(u cos v)(u cos v)
cos# v
œ (4u sin v) ln (u sin v) 4usin
u sin v
v ;
`z
sin
x
z œ 4e ln y œ 4(u cos v) ln (u sin v) Ê ` u œ (4 cos v) ln (u sin v) 4(u cos v) ˆ u sinvv ‰
v‰
œ (4 cos v) ln (u sin v) 4 cos v; also `` vz œ (4u sin v) ln (u sin v) 4(u cos v) ˆ uu cos
sin v
#
cos v
œ (4u sin v) ln (u sin v) 4usin
v
At ˆ2ß 14 ‰ : `` uz œ 4 cos 14 ln ˆ2 sin 14 ‰ 4 cos 14 œ 2È2 ln È2 2È2 œ È2 (ln 2 2);
(4)(2) ˆcos# 14 ‰
`z
1
1‰
ˆ
œ 4È2 ln È2 4È2 œ 2È2 ln 2 4È2
ˆsin 1 ‰
` v œ (4)(2) sin 4 ln 2 sin 4
œ [4(u cos v) ln (u sin v)](tan v)
(b)
4
8. (a)
`z
`u
`z
`v
œ–
œ–
Š "y ‹
#
Š xy ‹
Š
Š xy ‹
y cos v
x# y#
x sin v
x # y #
œ
(u sin v)(cos v) (u cos v)(sin v)
u#
Š
x
‹
y#
— (u sin v) – Š x ‹# 1 — u cos v œ
1
yu sin v
x# y#
(b) At
xu cos v
x# y#
œ
(u sin v)(u sin v) (u cos v)(u cos v)
u#
y
œ sin# v cos# v œ 1; z œ tan" Š xy ‹ œ tan" (cot v) Ê
œ
œ 0;
y
Š "y ‹
#
x
‹
y#
— cos v – Š x ‹# 1 — sin v œ
1
"
sin# v cos# v œ 1
ˆ1.3ß 16 ‰ : `` uz œ 0
and
`z
`v
`z
`u
œ1
œ 2t"Î#
t cos (t ln t)
etc1 œ (ln t)[cos (t ln t)] cos (t ln t) etc1 ; w œ z sin
t
dw
tc1
[cos (t ln t)] ln t t ˆ "t ‰‘ œ etc1 (1 ln t) cos (t ln t)
dt œ e
dw
dt
`z `x
`x `u
`w
`z
dz
dt
2 cos t sin t 2 sin t cos t 4 ˆ4t"Î# ‰ t "Î#
cos# t sin# t 16t
#
sin t 16tb œ ln (1 16t) Ê dw
dt
œ
(b)
œ cos t,
cos# t
Š "t ‹
œ 0 and
`z
`v
"
#
‰
œ ˆ 1 cot
# v a csc vb
œ 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
etc1
817
818
9. (a)
Chapter 14 Partial Derivatives
`w
`u
œ
`w `x
`x `u
`w `y
`y `u
`w `z
`z `u
œ (y z)(1) (x z)(1) (y x)(v) œ x y 2z v(y x)
œ (u v) (u v) 2uv v(2u) œ 2u 4uv;
`w
`v
œ
`w `x
`x `v
`w `y
`y `v
`w `z
`z `v
œ (y z)(1) (x z)(1) (y x)(u) œ y x (y x)u œ 2v (2u)u œ 2v 2u# ;
w œ xy yz xz œ au# v# b au# v uv# b au# v uv# b œ u# v# 2u# v Ê ``wu œ 2u 4uv and
`w
`v
œ 2v 2u#
(b) At ˆ "# ß 1‰ :
10. (a)
`w
`u
`w
`u
œ 2 ˆ "# ‰ 4 ˆ "# ‰ (1) œ 3 and
`w
`v
#
œ 2(1) 2 ˆ "# ‰ œ #3
2y
2z
v
v
v
v
v
œ Š x# 2x
y# z# ‹ ae sin u ue cos ub Š x# y# z# ‹ ae cos u ue sin ub Š x# y# z# ‹ ae b
v
u
‰ aev sin u uev cos ub
œ ˆ u# e2v sin# u 2ueu# esin
2v cos# u u# e2v
v
cos u
v
‰ v
ˆ u# e2v sin# u 2ue
u# e2v cos# u u# e2v ae cos u ue sin ub
v
‰ aev b œ 2u ;
ˆ u# e2v sin# u u2ue
# e2v cos# u u# e2v
`w
`v
2y
2z
v
v
v
œ Š x# 2x
y# z# ‹ aue sin ub Š x# y# z# ‹ aue cos ub Š x# y# z# ‹ aue b
v
u
‰ auev sin ub
œ ˆ u# e2v sin# u 2ueu# esin
2v cos# u u# e2v
v
cos u
‰ v
ˆ u# e2v sin# u 2ue
u# e2v cos# u u# e2v aue cos ub
‰ auev b œ 2; w œ ln au# e2v sin# u u# e2v cos# u u# e2v b œ ln a2u# e2v b
ˆ u# e2v sin# u u2ue
# e2v cos# u u# e2v
v
œ ln 2 2 ln u 2v Ê
(b) At a2ß 0b:
11. (a)
`w
`u
œ
œ 1 and
2
`w
u and ` v
`w
`v œ 2
œ2
rp
pq
qrrppq
`u `p
`u `q
`u `r
"
œ 0;
` p ` x ` q ` x ` r ` x œ q r (q r)# (q r)# œ
(q r)#
rp
pq
qrrppq
2p 2r
`u
`u `p
`u `q
`u `r
"
œ (q
` y œ ` p ` y ` q ` y ` r ` y œ q r (q r)# (q r)# œ
(q r)#
r)#
(2x 2y 2z) (2x 2y 2z)
z
`u
`u `p
`u `q
`u `r
œ
œ (z y)# ; ` z œ ` p ` z ` q ` z ` r ` z
(2z 2y)#
rp
pq
ppq
2p
4y
y
"
œ q r (q r)# (q r)# œ q r (qr
œ 2q
r)#
(q r)# œ (2z 2y)# œ (z y)# ;
y
(z y) y(1)
y(1)
`u
`u
u œ pq qr œ 2z 2y
œ (z z y)# , and `` uz œ (z (zy)(0)
2y œ z y Ê ` x œ 0, ` y œ
(z y)#
y)#
œ (zyy)#
`u
`x
œ
(b) At ŠÈ3ß 2ß 1‹ :
12. (a)
œ
2
#
`w
`u
`u
`x
œ
`u
`y
`u
`z
`u
`x
œ 0,
œ
"
(1 2)#
œ 1, and
`u
`z
œ
2
(1 2)#
œ 2
œ yz if 1# x
eqr
È 1 p#
(cos x) areqr sin" pb (0) aqeqr sin" pb (0) œ
œ
eqr
È 1 p#
(0) areqr sin" pb Š zy ‹ aqeqr sin" pb (0) œ
œ
eqr
È 1 p#
(0) areqr sin" pb (2z ln y) aqeqr sin" pb ˆ z"# ‰ œ a2zreqr sin" pb (ln y)
#
œ (2z) ˆ "z ‰ ayz x ln yb
`u
`y
`u
`y
œ xzyz1 , and
(b) At ˆ 14 ß "# ß "# ‰ :
`u
`z
`u
`x
az# ln yb ayz b x
z#
z
eqr cos x
È 1 p#
œ
z# reqr sin " p
y
ez ln y cos x
È1 sin# x
œ
z# ˆ " ‰ y z x
z
y
1
#
;
œ xzyz1 ;
œ xyz ln y; u œ ez ln y sin" (sin x) œ xyz if 1# Ÿ x Ÿ
qeqr sin " p
z#
1
#
Ê
œ ˆ 14 ‰ ˆ "# ‰
"Î#
`u
`x
œ yz ,
œ œ xy ln y from direct calculations
œ ˆ "# ‰
"Î#
œ È2,
`u
`y
œ ˆ 14 ‰ ˆ "# ‰ ˆ "# ‰
Ð"Î#Ñ"
È
œ 14 2 ,
`u
`z
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
ln ˆ "# ‰ œ 1
È2 ln 2
4
Section 14.4 The Chain Rule
œ
` z dx
` x dt
` z dy
` y dt
` z du
` u dt
` z dv
` v dt
` x dw
` w dt
dz
dt
15.
`w
`u
œ
`w `x
`x `u
`w `y
`y `u
`w `z
`z `u
`w
`v
œ
`w `x
`x `v
`w `y
`y `v
`w `z
`z `v
16.
`w
`x
œ
`w `r
`r `x
`w `s
`s `x
`w `t
`t `x
`w
`y
œ
`w `r
`r `y
`w `s
`s `y
`w `t
`t `y
17.
`w
`u
œ
`w `x
`x `u
`w `y
`y `u
`w
`v
œ
`w `x
`x `v
`w `y
`y `v
14.
dz
dt
œ
13.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
819
820
Chapter 14 Partial Derivatives
18.
`w
`x
œ
`w `u
`u `x
19.
`z
`t
œ
`z `x
`x `t
20.
`y
`r
œ
dy ` u
du ` r
22.
`w
`p
œ
`w `x
`x `p
`w `y
`y `p
`w `z
`z `p
23.
`w
`r
œ
` w dx
` x dr
` w dy
` y dr
œ
` w dx
` x dr
since
`w `v
`v `x
`z `y
`y `t
21.
`w
`y
œ
`w `u
`u `y
`z
`s
œ
`z `x
`x `s
`w
`s
œ
dw ` u
du ` s
`w
`s
œ
` w dx
` x ds
`w `v
`v `y
`z `y
`y `s
`w
`t
œ
dw ` u
du ` t
`w `v
`v `p
dy
dr
œ0
` w dy
` y ds
œ
` w dy
` y ds
since
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
dx
ds
œ0
Section 14.4 The Chain Rule
24.
`w
`s
œ
`w `x
`x `s
25. Let F(xß y) œ x$ 2y# xy œ 0 Ê Jx (xß y) œ 3x# y
`w `y
`y `s
and Fy (xß y) œ 4y x Ê
Ê
dy
dx
(1ß 1) œ
dy
dx
dy
dx
#
œ FFxy œ (3x4yyx)
3
œ FFxy œ xy2y
dy
dx
(1ß 1) œ 2
27. Let F(xß y) œ x# xy y# 7 œ 0 Ê Fx (xß y) œ 2x y and Fy (xß y) œ x 2y Ê
Ê
dy
dx
4
3
26. Let F(xß y) œ xy y# 3x 3 œ 0 Ê Fx (xß y) œ y 3 and Fy (xß y) œ x 2y Ê
Ê
821
dy
dx
y
œ FFxy œ 2x
x 2y
(1ß 2) œ 45
28. Let F(xß y) œ xey sin xy y ln 2 œ 0 Ê Fx (xß y) œ ey y cos xy and Fy (xß y) œ xey x sin xy 1
Ê
dy
dx
œ FFxy œ xeye xysincosxyxy 1 Ê
y
dy
dx
(!ß ln 2) œ (2 ln 2)
29. Let F(xß yß z) œ z$ xy yz y$ 2 œ 0 Ê Fx (xß yß z) œ y, Fy (xß yß z) œ x z 3y# , Fz (xß yß z) œ 3z# y
Ê
Ê
Fx
`z
` x œ Fz
`z
` y (1ß 1ß 1)
30. Let F(xß yß z) œ
Ê
`z
`x
œ 3z# y y œ
y
3z# y
Ê
`z
`x
(1ß 1ß 1) œ
"
4
;
`z
`y
#
œ Fyz œ x3z#zy3y œ
F
x z 3y#
3z# y
œ 34
"
x
"
y
œ FFxz œ
"
z
1 œ 0 Ê Fx (xß yß z) œ x"# , Fy (xß yß z) œ y"# , Fz (xß yß z) œ z"#
Š x"# ‹
Š z"# ‹
#
œ xz# Ê
`z
`x
(2ß 3ß 6) œ 9;
`z
`y
F
œ Fyz œ
Š y"# ‹
Š z"# ‹
#
œ yz# Ê
`z
`y
(2ß 3ß 6) œ 4
31. Let F(xß yß z) œ sin (x y) sin (y z) sin (x z) œ 0 Ê Fx (xß yß z) œ cos (x y) cos (x z),
Fy (xß yß z) œ cos (x y) cos (y z), Fz (xß yß z) œ cos (y z) cos (x z) Ê `` xz œ FFxz
(x y) cos (x z)
œ cos
cos (y z) cos (x z) Ê
`z
`x
(1ß 1ß 1) œ 1;
`z
`y
(x y) cos (y z)
œ Fyz œ cos
cos (y z) cos (x z) Ê
F
`z
`y
(1 ß 1 ß 1 ) œ 1
32. Let F(xß yß z) œ xey yez 2 ln x 2 3 ln 2 œ 0 Ê Fx (xß yß z) œ ey 2x , Fy (xß yß z) œ xey ez , Fz (xß yß z) œ yez
Ê
33.
`w
`r
`z
`x
œ
œ FFxz œ
`w `x
`x `r
ˆey 2x ‰
yez
`w `y
`y `r
Ê
`w `z
`z `r
`z
`x
(1ß ln 2ß ln 3) œ 3 ln4 2 ;
`z
`y
œ Fyz œ xeyez e Ê
F
y
z
`z
`y
(1ß ln 2ß ln 3) œ 3 ln5 2
œ 2(x y z)(1) 2(x y z)[ sin (r s)] 2(x y z)[cos (r s)]
œ 2(x y z)[1 sin (r s) cos (r s)] œ 2[r s cos (r s) sin (r s)][1 sin (r s) cos (r s)]
Ê ``wr ¸ rœ1ßsœ1 œ 2(3)(2) œ 12
34.
`w
`v
œ
`w `x
`x `v
`w `y
`y `v
35.
`w
`v
œ
`w `x
`x `v
`w `y
`y `v
œ ˆ2x
`w ¸
` v uœ0ßvœ0
œ 7
Ê
`w `z
`z `v
‰
ˆ"‰
ˆ 2v ‰
œ y ˆ 2v
u x(1) z (0) œ (u v) u
y‰
x# (2)
ˆ "x ‰ (1) œ ’2(u 2v 1)
v#
u
Ê
`w ¸
` v uœ1ßvœ2
2u v 2
(u 2v 1)# “ (2)
œ (1) ˆ 41 ‰ ˆ 41 ‰ œ 8
"
u 2v 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
822
36.
Chapter 14 Partial Derivatives
`z
`u
œ
`z `x
`x `u
`z `y
`y `u
œ (y cos xy sin y)(2u) (x cos xy x cos y)(v)
$
œ cuv cos au v uv b sin uvd (2u) cau# v# b cos au$ v uv$ b au# v# b cos uvd (v)
Ê `` uz ¸ uœ0ßvœ1 œ 0 (cos 0 cos 0)(1) œ 2
37.
38.
$
`z
`u
œ
dz ` x
dx ` u
œ ˆ 1 5 x# ‰ eu œ ’ 1 aeu 5 ln vb# “ eu Ê
`z
`v
œ
dz ` x
dx ` v
œ ˆ 1 5 x# ‰ ˆ "v ‰ œ ’ 1 aeu 5 ln vb# “ ˆ "v ‰ Ê
`z
`u
œ
dz ` q
dq ` u
œ Š q" ‹ Š
`z
`v
œ
Èv 3
1 u# ‹
dz ` q
dq ` v
œ Š "q ‹ Š 2Èv 3 ‹ œ
`V
`I
41. V œ IR Ê
œ (600 ohms)
42. V œ abc Ê
¸
Ê dV
dt
ˆts2 ‰2
2
†
1
t
œ 2s4 t
s4 t
2
† ˆ ts2 ‰ œ s5
`V
`R
œ R and
œ
aœ1ßbœ2ßcœ3
œ
` V da
` a dt
œ ’ 1 5(2)# “ (1) œ 1
"
dw ` x
dx ` s
œ I;
dV
dt
œ
5s4 t ` w
2 ; `t
œ
s5
2
`z
` V db
` b dt
` V dI
` I dt
` V dc
` c dt
`w
`s
œ
œ f w axb † 3s2 œ 3s2 es t ,
3
`w `x
`x `s
œ
`w `x
`x `t
`w `y
`y `s
`w `y
`y `t
2
`w
`t
œ
œ
"
atan " 1b a1 1# b
dw ` x
dx ` t
œ
2
1
;
œ f w axb † 2t œ 2t es t
3
œ fx ax, yb † 2t s fy ax, yb †
œ fx ax, yb † s2 fy ax, yb †
1
t
s
t2
s5
2
œ
(0.04 amps)(0.5 ohms/sec)
dI
dt
dV
dt
`w
`s
Ê w œ fˆt s2 ß st ‰ œ faxß yb Ê
ˆts2 ‰2
2
œ at s2 bˆ st ‰ † 2t s
œ at s2 bˆ st ‰ † s2
s
t
`z ¸
` v uœln 2ßvœ1
¸
" u ‹ Š 1 u# ‹ œ atan " ub a1 u# b Ê ` u uœ1ßvœ2
"u
"
`z ¸
"
Š Èv 3"tan " u ‹ Š 2tan
Èv 3 ‹ œ #(v 3) Ê ` v uœ1ßvœ2 œ #
39. Let x œ s3 t2 Ê w œ fas3 t2 b œ faxb Ê
40. Let x œ t s2 and y œ
œ ’ 1 5(2)# “ (2) œ 2;
Èv 3
œ Š Èv 3"tan
tan " u
`z ¸
` u uœln 2ßvœ1
` V dR
dI
dR
` R dt œ R dt I dt Ê 0.01
Ê dI
dt œ 0.00005 amps/sec
volts/sec
db
dc
œ (bc) da
dt (ac) dt (ab) dt
œ (2 m)(3 m)(1 m/sec) (1 m)(3 m)(1 m/sec) (1 m)(2 m)(3 m/sec) œ 3 m$ /sec
and the volume is increasing; S œ 2ab 2ac 2bc Ê
db
dc
dS ¸
œ 2(b c) da
dt 2(a c) dt 2(a b) dt Ê dt
œ
dS
dt
` S da
` a dt
` S db
` b dt
` S dc
` c dt
aœ1ßbœ2ßcœ3
œ 2(5 m)(1 m/sec) 2(4 m)(1 m/sec) 2(3 m)(3 m/sec) œ 0 m# /sec and the surface area is not changing;
"
ˆa da b db c dc ‰ Ê dD ¸
D œ Èa# b# c# Ê dD œ ` D da ` D db ` D dc œ
dt
œ
"
Š È14
‹ [(1
m
` a dt
` b dt
` c dt
È a# b# c#
dt
m)(1 m/sec) (2 m)(1 m/sec) (3 m)(3 m/sec)] œ
dt
6
È14
dt
dt
aœ1ßbœ2ßcœ3
m/sec 0 Ê the diagonals are
decreasing in length
43.
`f
`x
`f
`y
`f
`z
44. (a)
(b)
œ
œ
œ
`f
`u
`f
`u
`f
`u
`w
`r
`w
`r
`u
`x
`u
`y
`u
`z
`f
`v
`f
`v
`f
`v
fy
Ê fy œ (sin ))
œ
`f
`w
`f
`w
`f
`w
`w
`x
`w
`y
`w
`z
œ
œ
œ
`f
`u
`f
`u
`f
`u
`f
` w (1)
(1) `` vf (1) ``wf (0)
(0) `` vf (1) ``wf (1)
(1)
`f
`v
(0)
œ
`f
`u
œ
œ
`f
`w
`f
`u
`f
`v
,
`f
`v ,
`f
`w
`y
`r
and
Ê
`f
`x
`f
`y
œ fx cos ) fy sin ) and ``w) œ fx (r sin )) fy (r cos )) Ê
sin ) œ fx sin ) cos ) fy sin# ) and ˆ cosr ) ‰ ``w) œ fx sin ) cos ) fy cos# )
œ fx
`x
`r
`v
`x
`v
`y
`v
`z
`w
`r
#
asin# )b
`w
`r
`w
`r
`f
`z
" `w
r `)
œ0
œ fx sin ) fy cos )
ˆ cosr ) ‰ ``w) ; then ``wr œ fx cos ) (sin )) ``wr ˆ cosr ) ‰ ``w) ‘ (sin )) Ê fx cos )
ˆ sin ) rcos ) ‰ ``w) œ a1 sin# )b ``wr ˆ sin ) rcos ) ‰ ``w) Ê fx œ (cos )) ``wr ˆ sinr ) ‰
#
`w ‰
`)
Š sinr# ) ‹ ˆ ``w) ‰ and
#
`w ‰
`)
Š cosr# ) ‹ ˆ ``w) ‰ Ê afx b# afy b# œ ˆ ``wr ‰
(c) afx b œ acos# )b ˆ ``wr ‰ ˆ 2 sin )r cos ) ‰ ˆ ``wr
afy b# œ asin# )b ˆ ``wr ‰ ˆ 2 sin )r cos ) ‰ ˆ ``wr
#
#
#
#
#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
r#
ˆ ``w) ‰#
`w
`)
2
Section 14.4 The Chain Rule
`w
`x
45. wx œ
œ
`w `u
`u `x
#
œ
`w
`u
x Š `` uw#
œ
`w
`u
x#
` #w
` u#
`w `v
`v `x
`u
`x
` #w ` v
` v` u ` x ‹
2xy
` #w
` v` u
#
Ê wyy œ ``wu y Š `` uw#
`w
`u
œx
#
`w
`v
#
y Š ``u`wv
` #w
` v#
y#
`u
`y
y
`u
`x
; wy œ
` #w ` v
` v` u ` y ‹
` #w ` v
` v# ` x ‹
`w
`y
œ
#
`w `u
`u `y
x Š ``u`wv
#
`w
`u
Ê wxx œ
#
`u
`y
x
œ
`w
`u
`
`x
ˆ ``wu ‰ y
wxx wyy œ ax# y# b
46.
`w
`x
`w
`y
` w
` u#
ax # y # b
#
` w
` v#
ˆ ``wv ‰
#
#
x Šx `` uw# y ``v`wu ‹ y Šx
`w `v
`v `y
œ y
`w
`u
x
` #w
` u` v
#
y `` vw# ‹
`w
`v
` #w ` v
` v# ` y ‹
#
œ ``wu y Šy `` uw# x ``v`wu ‹ x Šy ``u`wv x `` vw# ‹ œ ``wu y#
#
`
`x
823
` #w
` u#
2xy
` #w
` v` u
x#
` #w
` v#
; thus
œ ax# y# b (wuu wvv ) œ 0, since wuu wvv œ 0
œ f w (u)(1) gw (v)(1) œ f w (u) gw (v) Ê wxx œ f ww (u)(1) gww (v)(1) œ f ww (u) gww (v);
œ f w (u)(i) gw (v)(i) Ê wyy œ f ww (u) ai# b gww (v) ai# b œ f ww (u) gww (v) Ê wxx wyy œ 0
47. fx (xß yß z) œ cos t, fy (xß yß z) œ sin t, and fz (xß yß z) œ t# t 2 Ê
œ (cos t)( sin t) (sin t)(cos t) at# t 2b(1) œ t# t 2;
df
dt
df
dt
` f dx
` x dt
#
œ
` f dy
` y dt
` f dz
` z dt
œ 0 Ê t t 2 œ 0 Ê t œ 2
or t œ 1; t œ 2 Ê x œ cos (2), y œ sin (2), z œ 2 for the point (cos (2)ß sin (2)ß 2); t œ 1 Ê x œ cos 1,
y œ sin 1, z œ 1 for the point (cos 1ß sin 1ß 1)
48.
dw
dt
` w dx
` x dt
œ
` w dy
` y dt
` w dz
` z dt
" ‰
œ a2xe2y cos 3zb ( sin t) a2x# e2y cos 3zb ˆ t#
a3x# e2y sin 3zb (1)
2x# e2y cos 3z
3x# e2y
t#
2(1)# (4)(1)
0œ4
#
œ 2xe2y cos 3z sin t
Ê
49. (a)
dw ¸
dt Ð1ßln 2ß0Ñ
`T
`x
œ0
œ 8x 4y and
`T
`y
œ 8y 4x Ê
dT
dt
sin 3z; at the point on the curve z œ 0 Ê t œ z œ 0
œ
` T dx
` x dt
` T dy
` y dt
œ (8x 4y)( sin t) (8y 4x)(cos t)
œ (8 cos t 4 sin t)( sin t) (8 sin t 4 cos t)(cos t) œ 4 sin# t 4 cos# t Ê
dT
dt
d# T
dt#
œ 16 sin t cos t;
œ 0 Ê 4 sin t 4 cos t œ 0 Ê sin t œ cos t Ê sin t œ cos t or sin t œ cos t Ê t œ 14 ,
#
#
#
#
51 31 71
4 , 4 , 4
on
the interval 0 Ÿ t Ÿ 21;
d# T
dt# ¹ tœ 1
1
4
œ 16 sin
1
4
cos
0 Ê T has a minimum at (xß y) œ Š
4
È2
#
ß
È2
# ‹;
d# T
dt# ¹ tœ 31
œ 16 sin
31
4
cos
31
4
0 Ê T has a maximum at (xß y) œ Š
È2
#
ß
d# T
dt# ¹ tœ 51
œ 16 sin
51
4
cos
51
4
0 Ê T has a minimum at (xß y) œ Š
È2
#
ß
d# T
dt# ¹ tœ 71
œ 16 sin
71
4
cos
71
4
0 Ê T has a maximum at (xß y) œ Š
4
4
4
`T
`T
` x œ 8x 4y, and ` y œ 8y
È2 È2
È2
È2
# ß # ‹ œ TŠ # ß # ‹ œ
(b) T œ 4x# 4xy 4y# Ê
found in part (a): T Š
TŠ
50. (a)
`T
`x
È2
#
ß
È2
# ‹
œ y and
œ T Š
`T
`y
È2
#
œx Ê
ß
dT
dt
È2
# ‹
œ
È2
#
ß
È2
# ‹;
È2
# ‹;
È2
# ‹
4x so the extreme values occur at the four points
4 ˆ "# ‰ 4 ˆ "# ‰ 4 ˆ "# ‰ œ 6, the maximum and
œ 4 ˆ #" ‰ 4 ˆ #" ‰ 4 ˆ #" ‰ œ 2, the minimum
` T dx
` x dt
` T dy
` y dt
œ y Š2È2 sin t‹ x ŠÈ2 cos t‹
œ ŠÈ2 sin t‹ Š2È2 sin t‹ Š2È2 cos t‹ ŠÈ2 cos t‹ œ 4 sin# t 4 cos# t œ 4 sin# t 4 a1 sin# tb
œ 4 8 sin# t Ê
31 51 71
4 , 4 , 4
#
d T
dt# ¹ tœ 1
d# T
dt#
œ 16 sin t cost t;
dT
dt
œ 0 Ê 4 8 sin# t œ 0 Ê sin# t œ
"
#
Ê sin t œ „
on the interval 0 Ÿ t Ÿ 21;
œ 8 sin 2 ˆ 14 ‰ œ 8 Ê T has a maximum at (xß y) œ (2ß 1);
4
d# T
dt# ¹ tœ 31
œ 8 sin 2 ˆ 341 ‰ œ 8 Ê T has a minimum at (xß y) œ (2ß 1);
4
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
È2
Ê t œ 14 ,
824
Chapter 14 Partial Derivatives
d# T
dt# ¹ tœ 51
œ 8 sin 2 ˆ 541 ‰ œ 8 Ê T has a maximum at (xß y) œ (2ß 1);
d# T
dt# ¹ tœ 71
œ 8 sin 2 ˆ 741 ‰ œ 8 Ê T has a minimum at (xß y) œ (2ß 1)
4
4
(b) T œ xy 2 Ê
`T
`x
œ y and
`T
`y
œ x so the extreme values occur at the four points found in part (a):
T(2ß 1) œ T(2ß 1) œ 0, the maximum and T(2ß 1) œ T(2ß 1) œ 4, the minimum
51. G(uß x) œ 'a g(tß x) dt where u œ f(x) Ê
u
dG
dx
œ
` G du
` u dx
` G dx
` x dx
F(x) œ '0 Èt% x$ dt Ê Fw (x) œ Éax# b% x$ (2x) '0
x#
x#
`
`x
œ g(uß x)f w (x) 'a gx (tß x) dt; thus
u
Èt% x$ dt œ 2xÈx) x$ '
52. Using the result in Exercise 51, F(x) œ 'x# Èt$ x# dt œ '1 Èt$ x# dt Ê Fw (x)
x#
1
œ ’ Éax# b$ x# x# '
x#
`
1 `x
Èt$ x# dt “ œ x# Èx' x# ' # È $x # dt
x
t x
1
14.5 DIRECTIONAL DERIVATIVES AND GRADIENT VECTORS
1.
`f
`x
œ 1,
`f
`y
œ 1 Ê ™ f œ i j ; f(2ß 1) œ 1
Ê 1 œ y x is the level curve
2.
`f
`x
œ
Ê
2y
2x
`f
`f
x# y# Ê ` x ("ß ") œ 1; ` y œ x# y#
`f
` y ("ß ") œ 1 Ê ™ f œ i j ; f(1ß 1)
#
#
#
#
œ ln 2 Ê ln 2
œ ln ax y b Ê 2 œ x y is the level curve
3.
`g
`x
`g
` x a2ß 1b
œ y2 Ê
œ 1;
`g
`y
œ 2x y Ê
Ê ™ g œ i 4j ; ga2ß 1b œ 2 Ê x œ
`g
` x a2ß 1b œ 4;
2
y# is the level
curve
4.
`g
`x
œx Ê
Ê
`g
`y
`g
`x
ŠÈ2ß "‹ œ È2;
`g
`y
œ y
ŠÈ2ß "‹ œ 1 Ê ™ g œ È2 i j ;
g ŠÈ2ß "‹ œ
"
#
Ê
"
#
œ
x#
#
y#
#
x#
0
or 1 œ x# y# is the level
curve
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
3x#
2Èt% x$
dt
Section 14.5 Directional Derivatives and Gradient Vectors
5.
`f
`x
œ
1
È2x 3y
`f
`x
Ê
`f
`x
Ê
`f
`y
(1ß 2) œ 21 ;
œ
3
2È2x 3y
(1ß 2) œ 43 ; Ê ™ f œ 12 i 34 j ; f(1ß 2) œ 2
Ê 4 œ 2x 3y is the level curve
6.
`f
`x
œ
`f
`y
œ 2y2 x Ê
`f
`x
Ê
y
2y2 Èx 2x3Î2
Èx
`f
`y
1
a4ß 2b œ 16
;
1
a4ß 2b œ 14 Ê ™ f œ 16
i 41 j ;
f a4ß 2b œ 14 Ê y œ Èx is the level curve
7.
`f
`x
œ 2x
z
x
Ê
`f
`x
(1ß 1ß 1) œ 3;
`f
`y
`f
`y
œ 2y Ê
("ß "ß ") œ 2;
`f
`z
œ 4z ln x Ê
`f
`z
("ß "ß ") œ 4;
thus ™ f œ 3i 2j 4k
8.
`f
`x
œ 6xz
Ê
9.
10.
`f
`x
œ
`f
`z
œ
`f
`x
œ exy cos z
`f
`z
x
ax# y# z# b$Î#
z
ax# y# z# b$Î#
A
kAk
œ
4i 3j
È 4# 3#
`f
`y
(1ß "ß ") œ 11
# ;
`f
`y
œ 6yz Ê
("ß "ß ") œ 6;
`f
`z
œ 6z# 3 ax# y# b
x
x # z# 1
"
thus ™ f œ 11
# i 6j # k
"
x
Ê
`f
`x
(1ß 2ß 2) œ 26
27 ;
"
z
Ê
`f
`z
(1ß 2ß 2) œ 23
54 ; thus ™
y1
È 1 x#
`f
`z
œ exy sin z Ê
11. u œ
`f
`x
Ê
z
x # z# 1
`f
"
` z ("ß "ß ") œ # ;
œ
Ê
`f
`x
ˆ!ß !ß 16 ‰ œ
È3
#
1;
`f
`y
`f
`y
ˆ!ß !ß 16 ‰ œ #" ; thus ™ f œ
œ
y
y" Ê `` yf
ax# y# z# b$Î#
23
23
f œ 26
27 i 54 j 54 k
œ exy cos z sin" x Ê
È
Š 3#2 ‹ i
È3
#
(1ß 2ß 2) œ
`f
`y
ˆ0ß 0ß 16 ‰ œ
23
54
È3
#
;
;
j "# k
i 35 j ; fx (xß y) œ 2y Ê fx (5ß 5) œ 10; fy (xß y) œ 2x 6y Ê fy (5ß 5) œ 20
4
5
Ê ™ f œ 10i 20j Ê (Du f)P! œ ™ f † u œ 10 ˆ 45 ‰ 20 ˆ 35 ‰ œ 4
12. u œ
A
k Ak
œ
3i 4j
È3# (4)#
œ
3
5
i 45 j ; fx (xß y) œ 4x Ê fx (1ß 1) œ 4; fy (xß y) œ 2y Ê fy (1ß 1) œ 2
Ê ™ f œ 4i 2j Ê (Du f)P! œ ™ f † u œ 12
5
13. u œ
A
kAk
œ
12i 5j
È12# 5#
œ
12
13
i
5
13
A
kAk
œ
hy (xß y) œ
3i 2j
È3# (2)#
ˆ "x ‰
y
ˆ x ‰# 1
œ
3
È13
i
ˆ #x ‰ È3
x# y#
Ê1 Š 4 ‹
œ 4
y2 2
Ê gx a1ß 1b
axy 2b2
15
21
œ 36
13 13 œ 13
j ; gx axß yb œ
Ê ™ g œ 3i 3j Ê aDu gbP! œ ™ g † u
14. u œ
8
5
2
È13
j ; hx (xß y) œ
Ê hy (1ß 1) œ
3
#
Š x#y ‹
y
ˆ x ‰# 1
x 2
œ 3; gy axß yb œ axy
Ê gy a1ß 1b œ 3
2b2
ˆ #y ‰ È3
Ê1 Š
Ê ™hœ
"
#
x# y#
4 ‹
2
Ê hx (1ß 1) œ "# ;
i #3 j Ê (Du h)P! œ ™ h † u œ
œ 2È313
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
3
2È13
6
2È13
825
826
Chapter 14 Partial Derivatives
15. u œ
A
k Ak
œ
3 i 6 j #k
È3# 6# (2)#
œ
i 67 j 27 k ; fx (xß yß z) œ y z Ê fx (1ß 1ß 2) œ 1; fy (xß yß z) œ x z
3
7
Ê fy (1ß 1ß 2) œ 3; fz (xß yß z) œ y x Ê fz (1ß 1ß 2) œ 0 Ê ™ f œ i 3j Ê (Du f)P! œ ™ f † u œ
16. u œ
A
kAk
œ
ijk
È 1 # 1# 1#
œ
1
È3
i
1
È3
j
1
È3
3
7
18
7
œ3
k ; fx (xß yß z) œ 2x Ê fx (1ß 1ß 1) œ 2; fy (xß yß z) œ 4y
Ê fy (1ß 1ß 1) œ 4; fz (xß yß z) œ 6z Ê fz (1ß 1ß 1) œ 6 Ê ™ f œ 2i 4j 6k Ê (Du f)P! œ ™ f † u
œ 2 Š È"3 ‹ 4 Š È"3 ‹ 6 Š È"3 ‹ œ 0
17. u œ
A
k Ak
œ
2i j 2k
È2# 1# (2)#
œ
i 13 j 23 k ; gx (xß yß z) œ 3ex cos yz Ê gx (0ß 0ß 0) œ 3; gy (xß yß z) œ 3zex sin yz
2
3
Ê gy (0ß 0ß 0) œ 0; gz (xß yß z) œ 3yex sin yz Ê gz (0ß 0ß 0) œ 0 Ê ™ g œ 3i Ê (Du g)P! œ ™ g † u œ 2
18. u œ
A
k Ak
œ
i 2j 2k
È 1# 2# 2#
œ
1
3
i 23 j 23 k ; hx (xß yß z) œ y sin xy
"
x
Ê hx ˆ1ß 0ß "# ‰ œ 1;
hy (xß yß z) œ x sin xy zeyz Ê hy ˆ"ß !ß #" ‰ œ #" ; hz (xß yß z) œ yeyz
Ê (Du h)P! œ ™ h † u œ
"
3
"
3
4
3
Ê hz ˆ"ß !ß #" ‰ œ 2 Ê ™ h œ i #" j 2k
œ2
19. ™ f œ (2x y) i (x 2y) j Ê ™ f(1ß 1) œ i j Ê u œ
most rapidly in the direction u œ
"
z
"
È2
i
"
È2
™f
k™f k
œ
i j
È(1)# 1#
œ È" i
2
"
È2
j ; f increases
"
È2
j and decreases most rapidly in the direction u œ
i
"
È2
j;
(Du f)P! œ ™ f † u œ k ™ f k œ È2 and (Du f)P! œ È2
™f
k™ f k
20. ™ f œ a2xy yexy sin yb i ax# xexy sin y exy cos yb j Ê ™ f(1ß 0) œ 2j Ê u œ
œ j ; f increases most
rapidly in the direction u œ j and decreases most rapidly in the direction u œ j ; (Du f)P! œ ™ f † u œ k ™ f k
œ 2 and (Du f)P! œ 2
21. ™ f œ
"
y
i Š yx# z‹ j yk Ê ™ f(4ß "ß ") œ i 5j k Ê u œ
"
3È 3
f increases most rapidly in the direction of u œ
"
u œ 3È
i
3
5
3È 3
j
"
3È 3
i
5
3È 3
j
"
3È 3
™f
k™f k
œ
i 5j k
È1# (5)# (1)#
œ
"
3È 3
i
5
3È 3
j
"
3È 3
k and decreases most rapidly in the direction
k ; (Du f)P! œ ™ f † u œ k ™ f k œ 3È3 and (Du f)P! œ 3È3
22. ™ g œ ey i xey j 2zk Ê ™ g ˆ1ß ln 2ß "# ‰ œ 2i 2j k Ê u œ
g increases most rapidly in the direction u œ
2
3
™g
k™gk
œ
2i 2j k
È 2# 2# 1#
œ
2
3
i 32 j 3" k ;
i 23 j 3" k and decreases most rapidly in the direction
u œ 23 i 23 j 3" k ; (Du g)P! œ ™ g † u œ k ™ gk œ 3 and (Du g)P! œ 3
23. ™ f œ ˆ "x x" ‰ i Š y" y" ‹ j ˆ "z "z ‰ k Ê ™ f("ß "ß ") œ 2i 2j 2k Ê u œ
f increases most rapidly in the direction u œ
u œ È"3 i
"
È3
j
"
È3
"
È3
i
"
È3
j
"
È3
™f
k™f k
2
7
"
È3
2
7
™h
k™hk
œ
j
"
È3
k;
2i 3 j 6k
È 2# 3# 6#
i 37 j 67 k and decreases most rapidly in the
direction u œ i j k ; (Du h)P! œ ™ h † u œ k ™ hk œ 7 and (Du h)P! œ 7
3
7
"
È3
k; (Du f)P! œ ™ f † u œ k ™ f k œ 2È3 and (Du f)P! œ 2È3
i 37 j 67 k ; h increases most rapidly in the direction u œ
2
7
i
k and decreases most rapidly in the direction
2y
24. ™ h œ Š x# 2x
y# 1 ‹ i Š x# y# 1 1‹ j 6k Ê ™ h("ß "ß 0) œ 2i 3j 6k Ê u œ
œ
œ
6
7
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
k;
Section 14.5 Directional Derivatives and Gradient Vectors
827
25. ™ f œ 2xi 2yj Ê ™ f ŠÈ2ß È2‹ œ 2È2 i 2È2 j
Ê Tangent line: 2È2 Šx È2‹ 2È2 Šy È2‹ œ 0
Ê È2x È2y œ 4
26. ™ f œ 2xi j Ê ™ f ŠÈ2ß 1‹ œ 2È2 i j
Ê Tangent line: 2È2 Šx È2‹ (y 1) œ 0
Ê y œ 2È2x 3
27. ™ f œ yi xj Ê ™ f(2ß 2) œ 2i 2j
Ê Tangent line: 2(x 2) 2(y 2) œ 0
Ê yœx4
28. ™ f œ (2x y)i (2y x)j Ê ™ f(1ß 2) œ 4i 5j
Ê Tangent line: 4(x 1) 5(y 2) œ 0
Ê 4x 5y 14 œ 0
29. ™ f œ a2x ybi ax 2y 1bj
(a) ™ fa1, 1b œ 3i 4j Ê l ™ fa1, 1bl œ 5 Ê Du fa1, 1b œ 5 in the direction of u œ 35 i 45 j
(b) ™ fa1, 1b œ 3i 4j Ê l ™ fa1, 1bl œ 5 Ê Du fa1, 1b œ 5 in the direction of u œ 35 i 45 j
(c) Du fa1, 1b œ 0 in the direction of u œ 45 i 35 j or u œ 45 i 35 j
(d) Let u œ u1 i u2 j Ê lul œ Èu12 u22 œ 1 Ê u12 u22 œ 1; Du fa1, 1b œ ™ fa1, 1b † u œ a3i 4jb † au1 i u2 jb
2
œ 3u1 4u2 œ 4 Ê u2 œ 43 u1 1 Ê u12 ˆ 43 u1 1‰ œ 1 Ê
25 2
3
16 u1 2 u1
7
œ 24
25 i 25 j
œ 0 Ê u1 œ 0 or u1 œ
24
25 ;
7
u1 œ 0 Ê u2 œ 1 Ê u œ j, or u1 œ 24
25 Ê u2 œ 25 Ê u
(e) Let u œ u1 i u2 j Ê lul œ Èu12 u22 œ 1 Ê u12 u22 œ 1; Du fa1, 1b œ ™ fa1, 1b † u œ a3i 4jb † au1 i u2 jb
2
œ 3u1 4u2 œ 3 Ê u1 œ 43 u2 1 Ê ˆ 43 u2 1‰ u22 œ 1 Ê
u2 œ 0 Ê u1 œ 1 Ê u œ i, or u2 œ
24
25
Ê u2 œ
7
25
Êuœ
25 2
8
9 u2 3 u2
7
24
25 i 25 j
œ 0 Ê u2 œ 0 or u2 œ
24
25 ;
.
30. ™ f œ
2y
i
ax yb2
2x
j
a x y b2
(a) ™ fˆ 21 , 23 ‰ œ 3i j Ê l ™ fˆ 21 , 23 ‰l œ È10 Ê Du fˆ 21 , 23 ‰ œ È10 in the direction of u œ
3
È10 i
1
È10 j
(b) ™ fˆ 21 , 23 ‰ œ 3i j Ê l ™ fˆ 21 , 23 ‰l œ È10 Ê Du fa1, 1b œ È10 in the direction of u œ È310 i
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1
È10 j
828
Chapter 14 Partial Derivatives
(c) Du fˆ 12 , 23 ‰ œ 0 in the direction of u œ
1
È10 i
3
È10 j
or u œ È110 i
3
È10 j
(d) Let u œ u1 i u2 j Ê lul œ Èu12 u22 œ 1 Ê u12 u22 œ 1; Du fˆ 12 , 32 ‰ œ ™ fˆ 12 , 32 ‰ † u œ a3i jb † au1 i u2 jb
œ 3u1 u2 œ 2 Ê u2 œ 3u1 2 Ê u12 a3u1 2b2 œ 1 Ê 10u12 12u1 3 œ 0 Ê u1 œ
u1 œ
Êu
È
6 È 6
Ê u2 œ 2 103 6
10
È6
È
œ 6
i 2 103 6 j
10
6 È 6
i
10
Êuœ
2 3È 6
j,
10
or u1 œ
6 È 6
10
Ê u2 œ
6 „ È 6
10
2 3È 6
10
(e) Let u œ u1 i u2 j Ê lul œ Èu12 u22 œ 1 Ê u12 u22 œ 1; Du fˆ 12 , 32 ‰ œ ™ fˆ 12 , 32 ‰ † u œ a3i jb † au1 i u2 jb
œ 3u1 u2 œ 1 Ê u2 œ 1 3u1 Ê u12 a1 3u1 b2 œ 1 Ê 10u12 6u1 œ 0 Ê u1 œ 0 or u1 œ 35 ;
u1 œ 0 Ê u2 œ 1 Ê u œ j, or u1 œ
3
5
Ê u2 œ 45 Ê u œ 35 i 45 j
31. ™ f œ yi (x 2y)j Ê ™ f(3ß 2) œ 2i 7j ; a vector orthogonal to ™ f is v œ 7i 2j Ê u œ
œ
7
È53
32. ™ f œ
i
4xy#
a x # y # b#
Ê uœ
j and u œ È753 i
2
È53
v
kv k
2
È53
v
kvk
œ
7i 2j
È7# (2)#
j are the directions where the derivative is zero
4x# y
j Ê ™ f("ß ") œ i j ; a vector orthogonal to ™ f is v œ i j
a x # y # b#
ij
1
1
1
1
È1# 1# œ È2 i È2 j and u œ È2 i È2 j are the directions where the
i
œ
derivative is zero
33. ™ f œ (2x 3y)i (3x 8y)j Ê ™ f(1ß 2) œ 4i 13j Ê k ™ f(1ß 2)k œ È(4)# (13)# œ È185 ; no, the
maximum rate of change is È185 14
34. ™ T œ 2yi (2x z)j yk Ê ™ T(1ß 1ß 1) œ 2i j k Ê k ™ T(1ß 1ß 1)k œ È(2)# 1# 1# œ È6 ; no, the
minimum rate of change is È6 3
35. ™ f œ fx ("ß #)i fy ("ß #)j and u" œ
ij
È 1# 1#
œ
"
È2
i
"
È2
j Ê (Du" f)(1ß 2) œ fx (1ß 2) Š È"2 ‹ fy (1ß 2) Š È"2 ‹
œ 2È2 Ê fx (1ß 2) fy (1ß 2) œ 4; u# œ j Ê (Du# f)(1ß 2) œ fx (1ß 2)(0) fy (1ß 2)(1) œ 3 Ê fy (1ß 2) œ 3
Ê fy (1ß 2) œ 3; then fx (1ß 2) 3 œ 4 Ê fx (1ß 2) œ 1; thus ™ f(1ß 2) œ i 3j and u œ
œ È15 i
2
È5
j Ê (Du f)P! œ ™ f † u œ È"5
36. (a) (Du f)P œ 2È3 Ê k ™ f k œ 2È3; u œ
v
kvk
œ
v
kv k
œ
ij
È 1# 1#
œ
"
È2
i
"
È2
"
È3
i 2j
È(1)# (2)#
œ
œ È75
ijk
È1# 1# (1)#
Ê ™ f œ k ™ f k u Ê ™ f œ 2È3 Š È"3 i
(b) v œ i j Ê u œ
6
È5
v
kvk
j
"
È3
œ
1
È3
i
1
È3
j
"
È3
k; thus u œ
™f
k ™f k
k‹ œ 2i 2j 2k
j Ê (Du f)P! œ ™ f † u œ 2 Š È"2 ‹ 2 Š È"2 ‹ 2(0) œ 2È2
37. The directional derivative is the scalar component. With ™ f evaluated at P! , the scalar component of ™ f in the
direction of u is ™ f † u œ (Du f)P! .
38. Di f œ ™ f † i œ (fx i fy j fz k) † i œ fx ; similarly, Dj f œ ™ f † j œ fy and Dk f œ ™ f † k œ fz
39. If (xß y) is a point on the line, then T(xß y) œ (x x! )i (y y! )j is a vector parallel to the line Ê T † N œ 0
Ê A(x x! ) B(y y! ) œ 0, as claimed.
40. (a) ™ (kf) œ
` (kf)
`x
i
` (kf)
`y
j
` (kf)
`z
k œ k ˆ `` xf ‰ i k Š `` yf ‹ j k ˆ `` zf ‰ k œ k Š `` xf i
`f
`y
j
`f
`z
k‹ œ k ™ f
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.6 Tangent Planes and Differentials
` (f g)
`x
(b) ™ (f g) œ
œ
`f
`x
i
`g
`x
i
`f
`y
i
j
` (f g)
`y
`g
`y
j
j
`f
`z
` (f g)
`z
k
`g
`z
k œ Š `` xf
k œ Š `` xf i
`g
`x ‹ i
`f
`y
Š `` yf
j
`f
`z
`g
`y ‹ j
Š `` zf
k‹ Š `` gx i
`g
`y
829
`g
`z ‹ k
j
`g
`z
k‹ œ ™ f ™ g
f ‹ j Š `` zf g
`g
`z
f‹ k
(c) ™ (f g) œ ™ f ™ g (Substitute g for g in part (b) above)
` (fg)
`x
(d) ™ (fg) œ
i
` (fg)
`y
j
` (fg)
`z
`g
`x
k œ Š `` xf g
f ‹ i Š `` yf g
`g
`y
œ ˆ `` xf g‰ i Š `` xg f ‹ i Š `` yf g‹ j Š `` gy f ‹ j ˆ `` zf g‰ k Š `` gz f ‹ k
œ f Š `` gx i
(e) ™ Š gf ‹ œ
œŒ
œ
`g
`y
j
` Š gf ‹
`x
`g
`z
i
k‹ g Š `` xf i
` Š gf ‹
`y
g ``xf i g ``yf j g `` fz k
g#
g ™f
g#
f™g
g#
œ
j
Œ
` Š gf ‹
`z
`f
`y
j
kœŠ
`f
`z
k‹ œ f ™ g g ™ f
g ``xf f `` gx
‹i
g#
f `` gx i f `` gy j f ``gz k
g#
œ
Œ
g ``yf f `` gy
j
g#
g Š ``xf i ``yf j `` fz k‹
g#
Š
g `` zf f ``gz
‹k
g#
f Š `` gx i `` gy j ``gz k‹
g#
g™f f™g
g#
14.6 TANGENT PLANES AND DIFFERENTIALS
1. (a) ™ f œ 2xi 2yj 2zk Ê ™ f(1ß 1ß 1) œ 2i 2j 2k Ê Tangent plane: 2(x 1) 2(y 1) 2(z 1) œ 0
Ê x y z œ 3;
(b) Normal line: x œ 1 2t, y œ 1 2t, z œ 1 2t
2. (a) ™ f œ 2xi 2yj 2zk Ê ™ f(3ß 5ß 4) œ 6i 10j 8k Ê Tangent plane: 6(x 3) 10(y 5) 8(z 4) œ 0
Ê 3x 5y 4z œ 18;
(b) Normal line: x œ 3 6t, y œ 5 10t, z œ 4 8t
3. (a) ™ f œ 2xi 2k Ê ™ f(2ß 0ß 2) œ 4i 2k Ê Tangent plane: 4(x 2) 2(z 2) œ 0
Ê 4x 2z 4 œ 0 Ê 2x z 2 œ 0;
(b) Normal line: x œ 2 4t, y œ 0, z œ 2 2t
4. (a) ™ f œ (2x 2y)i (2x 2y)j 2zk Ê ™ f(1ß 1ß 3) œ 4j 6k Ê Tangent plane: 4(y 1) 6(z 3) œ 0
Ê 2y 3z œ 7;
(b) Normal line: x œ 1, y œ 1 4t, z œ 3 6t
5. (a) ™ f œ a1 sin 1x 2xy zexz b i ax# zb j axexz yb k Ê ™ f(0ß 1ß 2) œ 2i 2j k Ê Tangent plane:
2(x 0) 2(y 1) 1(z 2) œ 0 Ê 2x 2y z 4 œ 0;
(b) Normal line: x œ 2t, y œ 1 2t, z œ 2 t
6. (a) ™ f œ (2x y)i (x 2y)j k Ê ™ f(1ß 1ß 1) œ i 3j k Ê Tangent plane:
1(x 1) 3(y 1) 1(z 1) œ 0 Ê x 3y z œ 1;
(b) Normal line: x œ 1 t, y œ 1 3t, z œ 1 t
7. (a) ™ f œ i j k for all points Ê ™ f(0ß 1ß 0) œ i j k Ê Tangent plane: 1(x 0) 1(y 1) 1(z 0) œ 0
Ê x y z 1 œ 0;
(b) Normal line: x œ t, y œ 1 t, z œ t
8. (a) ™ f œ (2x 2y 1)i (2y 2x 3)j k Ê ™ f(2ß 3ß 18) œ 9i 7j k Ê Tangent plane:
9(x 2) 7(y 3) 1(z 18) œ 0 Ê 9x 7y z œ 21;
(b) Normal line: x œ 2 9t, y œ 3 7t, z œ 18 t
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
830
Chapter 14 Partial Derivatives
9. z œ f(xß y) œ ln ax# y# b Ê fx (xß y) œ
and fy (xß y) œ
2x
x# y#
2y
x# y#
Ê fx (1ß 0) œ 2 and fy (1ß 0) œ 0 Ê from
Eq. (4) the tangent plane at (1ß 0ß 0) is 2(x 1) z œ 0 or 2x z 2 œ 0
#
#
#
#
#
#
10. z œ f(xß y) œ e ax y b Ê fx (xß y) œ 2xe ax y b and fy (xß y) œ 2ye ax y b Ê fx (0ß 0) œ 0 and fy (!ß !) œ 0
Ê from Eq. (4) the tangent plane at (0ß 0ß 1) is z 1 œ 0 or z œ 1
11. z œ f(Bß y) œ Èy x Ê fx (xß y) œ "# (y x)"Î# and fy (xß y) œ
"
#
(y x)"Î# Ê fx (1ß 2) œ "# and fy ("ß #) œ
Ê from Eq. (4) the tangent plane at (1ß 2ß 1) is "# (x 1) "# (y 2) (z 1) œ 0 Ê x y 2z 1 œ 0
"
#
12. z œ f(Bß y) œ 4x# y# Ê fx (xß y) œ 8x and fy (xß y) œ #y Ê fx (1ß 1) œ 8 and fy ("ß 1) œ # Ê from Eq. (4) the
tangent plane at (1ß 1ß 5) is 8(x 1) 2(y 1) (z 5) œ 0 or 8x 2y z 5 œ 0
13. ™ f œ i 2yj 2k Ê ™ f(1ß 1ß 1) œ i 2j 2k and ™ g œ i for all points; v œ ™ f ‚ ™ g
â
â
â i j kâ
â
â
Ê v œ â " 2 2 â œ 2j 2k Ê Tangent line: x œ 1, y œ 1 2t, z œ 1 2t
â
â
â" 0 0â
14. ™ f œ yzi xzj xyk Ê ™ f(1ß 1ß 1) œ i j k; ™ g œ 2xi 4yj 6zk Ê ™ g(1ß 1ß 1) œ 2i 4j 6k ;
â
â
â i j kâ
â
â
Ê v œ ™ f ‚ ™ g Ê â " 1 1 â œ 2i 4j 2k Ê Tangent line: x œ 1 2t, y œ 1 4t, z œ 1 2t
â
â
â2 4 6â
15. ™ f œ 2xi 2j 2k Ê ™ f ˆ1ß 1ß "# ‰ œ 2i 2j 2k and ™ g œ j for all points; v œ ™ f ‚ ™ g
â
â
â i j kâ
â
â
Ê v œ â 2 2 2 â œ 2i 2k Ê Tangent line: x œ 1 2t, y œ 1, z œ "# 2t
â
â
â0 1 0â
16. ™ f œ i 2yj k Ê ™ f ˆ "# ß 1ß "# ‰ œ i 2j k and ™ g œ j for all points; v œ ™ f ‚ ™ g
â
â
â i j kâ
â
â
Ê v œ â 1 2 1 â œ i k Ê Tangent line: x œ "# t, y œ 1, z œ "# t
â
â
â0 1 0â
17. ™ f œ a3x# 6xy# 4yb i a6x# y 3y# 4xb j 2zk Ê ™ f(1ß 1ß 3) œ 13i 13j 6k ; ™ g œ 2xi 2yj 2zk
â
â
j
k â
â i
â
â
Ê ™ g("ß "ß $) œ 2i 2j 6k ; v œ ™ f ‚ ™ g Ê v œ â "3 13 6 â œ 90i 90j Ê Tangent line:
â
â
2
6 â
â 2
x œ 1 90t, y œ 1 90t, z œ 3
18. ™ f œ 2xi 2yj Ê ™ f ŠÈ2ß È2ß 4‹ œ 2È2 i 2È2 j ; ™ g œ 2xi 2yj k Ê ™ g ŠÈ2ß È2ß 4‹
â i
j
k ââ
â
â
â
œ 2È2i 2È2j k ; v œ ™ f ‚ ™ g Ê v œ â 2È2 2È2 0 â œ 2È2 i 2È2 j Ê Tangent line:
â
â
â 2È2 2È2 1 â
x œ È2 2È2 t, y œ È2 2È2 t, z œ 4
19. ™ f œ Š x# yx# z# ‹ i Š x# yy# z# ‹ j Š x# yz# z# ‹ k Ê ™ f(3ß 4ß 12) œ
uœ
v
kvk
œ
3i 6j 2k
È3# 6# (2)#
œ
3
7
i 76 j 27 k Ê ™ f † u œ
9
1183
3
169
i
4
169
j
12
169
k;
9 ‰
and df œ ( ™ f † u) ds œ ˆ 1183
(0.1) ¸ 0.0008
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.6 Tangent Planes and Differentials
20. ™ f œ aex cos yzb i azex sin yzb j ayex sin yzb k Ê ™ f(0ß 0ß 0) œ i ; u œ
œ
1
È3
i
1
È3
j
k Ê ™f†uœ
1
È3
1
È3
and df œ ( ™ f † u) ds œ
v
kvk
œ
831
2i 2j 2k
È2# 2# (2)#
(0.1) ¸ 0.0577
1
È3
Ä
21. ™ g œ (1 cos z)i (1 sin z)j (x sin z y cos z)k Ê ™ g(2ß 1ß 0) œ 2i j k; A œ P! P" œ 2i 2j 2k
Ê uœ
v
kvk
œ
2 i 2 j 2 k
È(2)# 2# 2#
œ È13 i
1
È3
j
1
È3
k Ê ™ g † u œ 0 and dg œ ( ™ g † u) ds œ (0)(0.2) œ 0
22. ™ h œ c1y sin (1xy) z# d i c1x sin (1xy)d j 2xzk Ê ™ h(1ß 1ß 1) œ (1 sin 1 1)i (1 sin 1)j 2k
Ä
k
œ i 2k ; v œ P! P" œ i j k where P" œ (!ß !ß !) Ê u œ kvvk œ È1i#j1#
œ È13 i È13 j È13 k
1#
Ê ™h†uœ
œ È3 and dh œ ( ™ h † u) ds œ È3(0.1) ¸ 0.1732
3
È3
23. (a) The unit tangent vector at Š "# ß
È3
# ‹
in the direction of motion is u œ
™ T œ (sin 2y)i (2x cos 2y)j Ê ™ T Š "# ß
È3
#
œ
sin È3
"
#
È3
# ‹
È3
#
i #" j ;
œ Šsin È3‹ i Šcos È3‹ j Ê Du T Š "# ß
œ ™T†vœŠ™T†
dT
dt
œŠ
È3
#
œ ™T†u
cos È3 ¸ 0.935° C/ft
` T dx
` x dt
` T dy
` y dt
we have u œ
È3
#
i #" j from part (a)
(b) r(t) œ (sin 2t)i (cos 2t)j Ê v(t) œ (2 cos 2t)i (2 sin 2t)j and kvk œ 2;
Ê
È3
# ‹
v
kvk ‹
sin È3
"
#
kvk œ (Du T) kvk , where u œ
v
kv k
; at Š "# ß
È3
# ‹
dT
dt
œ
cos È3‹ † 2 œ È3 sin È3 cos È3 ¸ 1.87° C/sec
24. (a) ™ T œ (4x yz)i xzj xyk Ê ™ T(8ß 6ß 4) œ 56i 32j 48k ; r(t) œ 2t# i 3tj t# k Ê the particle is
at the point P()ß 6ß 4) when t œ 2; v(t) œ 4ti 3j 2tk Ê v(2) œ 8i 3j 4k Ê u œ kvvk
(b)
œ
8
È89
dT
dt
œ
i
` T dx
` x dt
3
È89
j
` T dy
` y dt
4
È89
k Ê Du T(8ß 6ß 4) œ ™ T † u œ
"
È89
œ ™ T † v œ ( ™ T † u) kvk Ê at t œ 2,
[56 † 8 32 † 3 48 † (4)] œ
dT
dt
736
È89
° C/m
736
œ Du T¸ tœ2 v(2) œ Š È
‹ È89 œ 736° C/sec
89
25. (a) f(!ß 0) œ 1, fx (xß y) œ 2x Ê fx (0ß 0) œ 0, fy (xß y) œ 2y Ê fy (0ß 0) œ 0 Ê L(xß y) œ 1 0(x 0) 0(y 0) œ 1
(b) f(1ß 1) œ 3, fx (1ß 1) œ 2, fy (1ß 1) œ 2 Ê L(xß y) œ 3 2(x 1) 2(y 1) œ 2x 2y 1
26. (a) f(!ß 0) œ 4, fx (xß y) œ 2(x y 2) Ê fx (0ß 0) œ 4, fy (xß y) œ 2(x y 2) Ê fy (0ß 0) œ 4
Ê L(xß y) œ 4 4(x 0) 4(y 0) œ 4x 4y 4
(b) f(1ß 2) œ 25, fx (1ß 2) œ 10, fy (1ß 2) œ 10 Ê L(xß y) œ 25 10(x 1) 10(y 2) œ 10x 10y 5
27. (a) f(0ß 0) œ 5, fx (xß y) œ 3 for all (xß y), fy (xß y) œ 4 for all (xß y) Ê L(xß y) œ 5 3(x 0) 4(y 0) œ 3x 4y 5
(b) f(1ß 1) œ 4, fx (1ß 1) œ 3, fy (1ß 1) œ 4 Ê L(xß y) œ 4 3(x 1) 4(y 1) œ 3x 4y 5
28. (a) f(1ß 1) œ 1, fx (xß y) œ 3x# y% Ê fx (1ß 1) œ 3, fy (xß y) œ 4x$ y$ Ê fy (1ß 1) œ 4
Ê L(xß y) œ 1 3(x 1) 4(y 1) œ 3x 4y 6
(b) f(0ß 0) œ 0, fx (!ß 0) œ 0, fy (0ß 0) œ 0 Ê L(xß y) œ 0
29. (a) f(0ß 0) œ 1, fx (xß y) œ ex cos y Ê fx (0ß 0) œ 1, fy (xß y) œ ex sin y Ê fy (0ß 0) œ 0
Ê L(xß y) œ 1 1(x 0) 0(y 0) œ x 1
(b) f ˆ0ß 1# ‰ œ 0, fx ˆ0ß 1# ‰ œ 0, fy ˆ0ß 1# ‰ œ 1 Ê L(xß y) œ 0 0(x 0) 1 ˆy 1# ‰ œ y
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1
#
832
Chapter 14 Partial Derivatives
30. (a) f(0ß 0) œ 1, fx (xß y) œ e2yx Ê fx (!ß !) œ 1, fy (xß y) œ 2e2yx Ê fy (0ß 0) œ 2
Ê L(xß y) œ 1 1(x 0) 2(y 0) œ x 2y 1
(b) f(1ß 2) œ e$ , fx (1ß 2) œ e$ , fy (1ß 2) œ 2e$ Ê L(xß y) œ e$ e$ (x 1) 2e$ (y 2) œ e$ x 2e$ y 2e$
31. (a) Wa20, 25b œ 11‰ F; Wa30, 10b œ 39‰ F; Wa15, 15b œ 0‰ F
(b) Wa10, 40b œ 65.5‰ F; Wa50, 40b œ 88‰ F; Wa60, 30b œ 10.2‰ F;
5.72
0.0684t
`W
(c) Wa25, 5b œ 17.4088‰ F; ``W
V œ v0.84 v0.84 Ê ` V a25, 5b œ 0.36;
Ê
`W
` T a25,
`W
`T
œ 0.6215 0.4275v0.16
5b œ 1.3370 Ê LaV, Tb œ 17.4088 0.36aV 25b 1.337aT 5b œ 1.337T 0.36V 15.0938
(d) i) Wa24, 6b ¸ La24, 6b œ 15.7118 ¸ 15.7‰ F
ii) Wa27, 2b ¸ La27, 2b œ 22.1398 ¸ 22.1‰ F
ii) Wa5, 10b ¸ La5, 10b œ 30.2638 ¸ 30.2‰ F This value is very different because the point a5, 10b is not
close to the point a25, 5b.
32. Wa50, 20b œ 59.5298‰ F;
Ê
`W
` T a50,
`W
`V
œ v5.72
0.84
0.0684t
v0.84
Ê
`W
` V a50,
20b œ 0.2651;
`W
`T
œ 0.6215 0.4275v0.16
20b œ 1.4209 Ê LaV, Tb œ 59.5298 0.2651aV 50b 1.4209aT 20b
œ 1.4209T 0.2651V 17.8568
(a) Wa49, 22b ¸ La49, 22b œ 62.1065 ¸ 62.1‰ F
(b) Wa53, 19b ¸ La53, 19b œ 58.9042 ¸ 58.9‰ F
(c) Wa60, 30b ¸ La60, 30b œ 76.3898 ¸ 76.4‰ F
33. f(2ß 1) œ 3, fx (xß y) œ 2x 3y Ê fx (2ß 1) œ 1, fy (xß y) œ 3x Ê fy (2ß 1) œ 6 Ê L(xß y) œ 3 1(x 2) 6(y 1)
œ 7 x 6y; fxx (xß y) œ 2, fyy (xß y) œ 0, fxy (xß y) œ 3 Ê M œ 3; thus kE(xß y)k Ÿ ˆ "# ‰ (3) akx 2k ky 1kb#
Ÿ ˆ 3# ‰ (0.1 0.1)# œ 0.06
34. f(2ß 2) œ 11, fx (xß y) œ x y 3 Ê fx (2ß 2) œ 7, fy (xß y) œ x
y
#
3 Ê fy (2ß 2) œ 0
Ê L(xß y) œ 11 7(x 2) 0(y 2) œ 7x 3; fxx (xß y) œ 1, fyy (xß y) œ "# , fxy (xß y) œ 1
Ê M œ 1; thus kE(xß y)k Ÿ ˆ "# ‰ (1) akx 2k ky 2kb# Ÿ ˆ #1 ‰ (0.1 0.1)# œ 0.02
35. f(0ß 0) œ 1, fx (xß y) œ cos y Ê fx (0ß 0) œ 1, fy (xß y) œ 1 x sin y Ê fy (0ß 0) œ 1
Ê L(xß y) œ 1 1(x 0) 1(y 0) œ x y 1; fxx (xß y) œ 0, fyy (xß y) œ x cos y, fxy (xß y) œ sin y Ê Q œ 1;
thus kE(xß y)k Ÿ ˆ "# ‰ (1) akxk kykb# Ÿ ˆ #1 ‰ (0.2 0.2)# œ 0.08
36. f("ß #) œ 6, fx (xß y) œ y# y sin (x 1) Ê fx (1ß 2) œ 4, fy (xß y) œ 2xy cos (x 1) Ê fy (1ß 2) œ 5
Ê L(xß y) œ 6 4(x 1) 5(y 2) œ 4x 5y 8; fxx (xß y) œ y cos (x 1), fyy (xß y) œ 2x,
fxy (xß y) œ 2y sin (x 1); kx 1k Ÿ 0.1 Ê 0.9 Ÿ x Ÿ 1.1 and ky 2k Ÿ 0.1 Ê 1.9 Ÿ y Ÿ 2.1; thus the max of
kfxx (xß y)k on R is 2.1, the max of kfyy (xß y)k on R is 2.2, and the max of kfxy (xß y)k on R is 2(2.1) sin (0.9 1)
Ÿ 4.3 Ê M œ 4.3; thus kE(xß y)k Ÿ ˆ "# ‰ (4.3) akx 1k ky 2kb# Ÿ (2.15)(0.1 0.1)# œ 0.086
37. f(0ß 0) œ 1, fx (xß y) œ ex cos y Ê fx (0ß 0) œ 1, fy (xß y) œ ex sin y Ê fy (0ß 0) œ 0
Ê L(xß y) œ 1 1(x 0) 0(y 0) œ 1 x; fxx (xß y) œ ex cos y, fyy (xß y) œ ex cos y, fxy (xß y) œ ex sin y;
kxk Ÿ 0.1 Ê 0.1 Ÿ x Ÿ 0.1 and kyk Ÿ 0.1 Ê 0.1 Ÿ y Ÿ 0.1; thus the max of kfxx (xß y)k on R is e0Þ1 cos (0.1)
Ÿ 1.11, the max of kfyy (xß y)k on R is e0Þ1 cos (0.1) Ÿ 1.11, and the max of kfxy (xß y)k on R is e0Þ1 sin (0.1)
Ÿ 0.12 Ê M œ 1.11; thus kE(xß y)k Ÿ ˆ "# ‰ (1.11) akxk kykb# Ÿ (0.555)(0.1 0.1)# œ 0.0222
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.6 Tangent Planes and Differentials
38. f(1ß 1) œ 0, fx (xß y) œ
"
x
Ê fx (1ß 1) œ 1, fy (xß y) œ
œ x y 2; fxx (xß y) œ
"
(0.98)#
kfxx (xß y)k on R is
"
(0.98)#
"
x#
, fyy (xß y) œ
"
y#
"
y
833
Ê fy (1ß 1) œ 1 Ê L(xß y) œ 0 1(x 1) 1(y 1)
, fxy (xß y) œ 0; kx 1k Ÿ 0.2 Ê 0.98 Ÿ x Ÿ 1.2 so the max of
Ÿ 1.04; ky 1k Ÿ 0.2 Ê 0.98 Ÿ y Ÿ 1.2 so the max of kfyy (xß y)k on R is
Ÿ 1.04 Ê M œ 1.04; thus kE(xß y)k Ÿ ˆ #" ‰ (1.04) akx 1k ky 1kb# Ÿ (0.52)(0.2 0.2)# œ 0.0832
39. (a) f("ß "ß ") œ 3, fx (1ß 1ß 1) œ y zkÐ1ß1ß1Ñ œ 2, fy (1ß 1ß 1) œ x zkÐ1ß1ß1Ñ œ 2, fz (1ß 1ß 1) œ y xkÐ1ß1ß1Ñ œ 2
Ê L(xß yß z) œ 3 2(x 1) 2(y 1) 2(z 1) œ 2x 2y 2z 3
(b) f(1ß 0ß 0) œ 0, fx (1ß 0ß 0) œ 0, fy (1ß 0ß 0) œ 1, fz (1ß 0ß 0) œ 1 Ê L(xß yß z) œ 0 0(x 1) (y 0) (z 0) œ y z
(c) f(0ß 0ß 0) œ 0, fx (0ß 0ß 0) œ 0, fy (0ß 0ß 0) œ 0, fz (0ß 0ß 0) œ 0 Ê L(xß yß z) œ 0
40. (a) f(1ß 1ß 1) œ 3, fx (1ß 1ß 1) œ 2xkÐ"ß"ß"Ñ œ 2, fy (1ß 1ß 1) œ 2ykÐ"ß"ß"Ñ œ 2, fz (1ß 1ß 1) œ 2zkÐ"ß"ß"Ñ œ 2
Ê L(xß yß z) œ 3 2(x 1) 2(y 1) 2(z 1) œ 2x 2y 2z 3
(b) f(0ß 1ß 0) œ 1, fx (0ß 1ß 0) œ 0, fy (!ß 1ß 0) œ 2, fz (0ß 1ß 0) œ 0 Ê L(xß yß z) œ 1 0(x 0) 2(y 1) 0(z 0)
œ 2y 1
(c) f(1ß 0ß 0) œ 1, fx (1ß 0ß 0) œ 2, fy (1ß 0ß 0) œ 0, fz (1ß 0ß 0) œ 0 Ê L(xß yß z) œ 1 2(x 1) 0(y 0) 0(z 0)
œ 2x 1
41. (a) f(1ß 0ß 0) œ 1, fx (1ß 0ß 0) œ
fz (1ß 0ß 0) œ
z
È x # y# z# ¹
x
È x # y # z# ¹
Ð1ß0ß0Ñ
(b) f(1ß 1ß 0) œ È2, fx (1ß 1ß 0) œ
Ê L(xß yß z) œ È2
"
È2
Ð1ß0ß0Ñ
œ 1, fy (1ß 0ß 0) œ
"
3
Ð1 ß0 ß0 Ñ
œ 0,
œ 0 Ê L(xß yß z) œ 1 1(x 1) 0(y 0) 0(z 0) œ x
"
È2
, fy (1ß 1ß 0) œ
(x 1)
"
È2
"
È2
, fz (1ß 1ß 0) œ 0
(y 1) 0(z 0) œ
(c) f(1ß 2ß 2) œ 3, fx (1ß 2ß 2) œ "3 , fy (1ß 2ß 2) œ 23 , fz (1ß 2ß 2) œ
œ
y
È x # y # z# ¹
2
3
"
È2
x
"
È2
y
Ê L(xß yß z) œ 3 "3 (x 1) 23 (y 2) 23 (z 2)
x 32 y 32 z
42. (a) f ˆ 12 ß 1ß 1‰ œ 1, fx ˆ 1# ß 1ß 1‰ œ
fz ˆ 1# ß 1ß 1‰ œ
sin xy
z# ¹ ˆ 1 ß"ß"‰
y cos xy ¸
ˆ 1# ß"ß"‰
z
œ 0, fy ˆ 1# ß 1ß 1‰ œ
x cos xy ¸
ˆ 1# ß"ß"‰
z
œ 0,
œ 1 Ê L(xß yß z) œ 1 0 ˆx 1# ‰ 0(y 1) 1(z 1) œ 2 z
#
(b) f(2ß 0ß 1) œ 0, fx (2ß 0ß 1) œ 0, fy (2ß 0ß 1) œ 2, fz (2ß 0ß 1) œ 0 Ê L(xß yß z) œ 0 0(x 2) 2(y 0) 0(z 1) œ 2y
43. (a) f(0ß 0ß 0) œ 2, fx (0ß 0ß 0) œ ex k Ð!ß!ß!Ñ œ 1, fy (0ß 0ß 0) œ sin (y z)k Ð!ß!ß!Ñ œ 0,
fz (0ß 0ß 0) œ sin (y z)k Ð!ß!ß!Ñ œ 0 Ê L(xß yß z) œ 2 1(x 0) 0(y 0) 0(z 0) œ 2 x
(b) f ˆ0ß 1# ß 0‰ œ 1, fx ˆ0ß 1# ß 0‰ œ 1, fy ˆ0ß 1# ß 0‰ œ 1, fz ˆ0ß 1# ß 0‰ œ 1 Ê L(xß yß z)
œ 1 1(x 0) 1 ˆy 12 ‰ 1(z 0) œ x y z 1# 1
(c) f ˆ0ß 14 ß 14 ‰ œ 1, fx ˆ0ß 14 ß 14 ‰ œ 1, fy ˆ0ß 14 ß 14 ‰ œ 1, fz ˆ0ß 14 ß 14 ‰ œ 1 Ê L(xß yß z)
œ 1 1(x 0) 1 ˆy 14 ‰ 1 ˆz 14 ‰ œ x y z 1# 1
44. (a) f(1ß 0ß 0) œ 0, fx (1ß 0ß 0) œ
fz (1ß 0ß 0) œ
xy
(xyz)# 1 ¹ Ð"ß!ß!Ñ
yz
(xyz)# 1 ¹ Ð"ß!ß!Ñ
œ 0, fy (1ß 0ß 0) œ
xz
(xyz)# 1 ¹ Ð"ß!ß!Ñ
œ 0,
œ 0 Ê L(xß yß z) œ 0
(b) f(1ß 1ß 0) œ 0, fx (1ß 1ß 0) œ 0, fy (1ß 1ß 0) œ 0, fz (1ß 1ß 0) œ 1 Ê L(xß yß z) œ 0 0(x 1) 0(y 1) 1(z 0) œ z
(c) f(1ß 1ß 1) œ 14 , fx (1ß 1ß 1) œ "# , fy (1ß 1ß 1) œ "# , fz (1ß 1ß 1) œ "# Ê L(xß yß z) œ 14 "# (x 1) "# (y 1) "# (z 1)
œ
"
#
x "# y "# z
1
4
3
#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
834
Chapter 14 Partial Derivatives
45. f(xß yß z) œ xz 3yz 2 at P! (1ß 1ß 2) Ê f(1ß 1ß 2) œ 2; fx œ z, fy œ 3z, fz œ x 3y Ê L(xß yß z)
œ 2 2(x 1) 6(y 1) 2(z 2) œ 2x 6y 2z 6; fxx œ 0, fyy œ 0, fzz œ 0, fxy œ 0, fyz œ 3
Ê M œ 3; thus, kE(xß yß z)k Ÿ ˆ "# ‰ (3)(0.01 0.01 0.02)# œ 0.0024
46. f(xß yß z) œ x# xy yz "4 z# at P! (1ß 1ß 2) Ê f(1ß 1ß 2) œ 5; fx œ 2x y, fy œ x z, fz œ y "# z
Ê L(xß yß z) œ 5 3(x 1) 3(y 1) 2(z 2) œ 3x 3y 2z 5; fxx œ 2, fyy œ 0, fzz œ "# , fxy œ 1, fxz œ 0,
fyz œ 1 Ê M œ 2; thus kE(xß yß z)k Ÿ ˆ "# ‰ (2)(0.01 0.01 0.08)# œ 0.01
47. f(xß yß z) œ xy 2yz 3xz at P! (1ß 1ß 0) Ê f(1ß 1ß 0) œ 1; fx œ y 3z, fy œ x 2z, fz œ 2y 3x
Ê L(xß yß z) œ 1 (x 1) (y 1) (z 0) œ x y z 1; fxx œ 0, fyy œ 0, fzz œ 0, fxy œ 1, fxz œ 3,
fyz œ 2 Ê M œ 3; thus kE(xß yß z)k Ÿ ˆ "# ‰ (3)(0.01 0.01 0.01)# œ 0.00135
48. f(xß yß z) œ È2 cos x sin (y z) at P! ˆ0ß 0ß 14 ‰ Ê f ˆ0ß 0ß 14 ‰ œ 1; fx œ È2 sin x sin (y z),
fy œ È2 cos x cos (y z), fz œ È2 cos x cos (y z) Ê L(xß yß z) œ 1 0(x 0) (y 0) ˆz 14 ‰
œ y z 14 1; fxx œ È2 cos x sin (y z), fyy œ È2 cos x sin (y z), fzz œ È2 cos x sin (y z),
fxy œ È2 sin x cos (y z), fxz œ È2 sin x cos (y z), fyz œ È2 cos x sin (y z). The absolute value of
each of these second partial derivatives is bounded above by È2 Ê M œ È2; thus kE(xß yß z)k
Ÿ ˆ " ‰ ŠÈ2‹ (0.01 0.01 0.01)# œ 0.000636.
#
49. Tx (xß y) œ ey ey and Ty (xß y) œ x aey ey b Ê dT œ Tx (xß y) dx Ty (xß y) dy
œ aey ey b dx x aey ey b dy Ê dTkÐ#ßln 2Ñ œ 2.5 dx 3.0 dy. If kdxk Ÿ 0.1 and kdyk Ÿ 0.02, then the
maximum possible error in the computed value of T is (2.5)(0.1) (3.0)(0.02) œ 0.31 in magnitude.
#
21rh dr 1r dh
50. Vr œ 21rh and Vh œ 1r# Ê dV œ Vr dr Vh dh Ê dV
œ 2r dr "h dh; now ¸ drr † 100¸ Ÿ 1 and
1 r# h
V œ
¸ dh
¸
¸ dV
¸ ¸ˆ2 drr ‰ (100) ˆ dh
‰
¸
¸ dr
¸ ¸ dh
¸
h † 100 Ÿ 1 Ê
V † 100 Ÿ
h (100) Ÿ 2 r † 100 h † 100 Ÿ 2(1) 1 œ 3 Ê 3%
51.
dx
x
Ÿ 0.02,
dy
y
Ÿ 0.03
dy
2
(a) S œ 2x2 4xy Ê dS œ a4x 4ybdx 4x dy œ a4x2 4xyb dx
x 4xy y Ÿ a4x 4xyba0.02b a4xyba0.03b
œ 0.04a2x2 b 0.05a4xyb Ÿ 0.05a2x2 b 0.05a4xyb œ a0.05ba2x2 4xyb œ 0.05S
2 dy
2
2
2
(b) V œ x2 y Ê dV œ 2xy dx x2 dy œ 2x2 y dx
x x y y Ÿ a2x yba0.02b ax yba0.03b œ 0.07ax yb=0.07V
52. V œ
41 3
3 r
1 r2 h Ê dV œ a41 r2 21 rhbdr 1 r2 dh; r œ 10, h œ 15, dr œ
1
2
and dh œ 0 Ê
dV œ Š41a10b2 21 a10ba15b‹ˆ 12 ‰ 1 a10b2 a0b œ 3501 cm3
53. Vr œ 21rh and Vh œ 1r# Ê dV œ Vr dr Vh dh Ê dV œ 21rh dr 1r# dh Ê dVkÐ5ß12Ñ œ 1201 dr 251 dh;
kdrk Ÿ 0.1 cm and kdhk Ÿ 0.1 cm Ê dV Ÿ (1201)(0.1) (251)(0.1) œ 14.51 cm$ ; V(5ß 12) œ 3001 cm$
1
Ê maximum percentage error is „ 14.5
3001 ‚ 100 œ „ 4.83%
54. (a)
"
R
œ
"
R"
"
R#
Ê R"# dR œ R"# dR"
"
"
R##
#
"
(b) dR œ R# ’Š R"# ‹ dR" Š R"# ‹ dR# “ Ê dRk Ð100 400Ñ œ R# ’ (100)
# dR"
"
ß
#
sensitive to a variation in R" since
"
(100)#
#
dR# Ê dR œ Š RR" ‹ dR" Š RR# ‹ dR#
"
(400)#
dR# “ Ê R will be more
"
(400)#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.6 Tangent Planes and Differentials
#
835
#
(c) From part (a), dR œ Š RR" ‹ dR" Š RR# ‹ dR# so that R" changing from 20 to 20.1 ohms Ê dR" œ 0.1 ohm
and R# changing from 25 to 24.9 ohms Ê dR# œ 0.1 ohms;
Ê dRk Ð20 25Ñ œ
ß
œ
0.011
ˆ 100
‰
9
ˆ 100
‰#
9
(20)#
(0.1)
ˆ 100
‰#
9
(25)#
"
R
œ
"
R"
"
R#
Ê Rœ
(0.1) ¸ 0.011 ohms Ê percentage change is
100
9
ohms
dR ¸
R Ð20ß25Ñ
‚ 100
‚ 100 ¸ 0.1%
55. A œ xy Ê dA œ x dy y dx; if x y then a 1-unit change in y gives a greater change in dA than a 1-unit change in x.
Thus, pay more attention to y which is the smaller of the two dimensions.
56. (a) fx (xß y) œ 2x(y 1) Ê fx (1ß 0) œ 2 and fy (xß y) œ x# Ê fy (1ß 0) œ 1 Ê df œ 2 dx 1 dy Ê df is more
sensitive to changes in x
dx
"
(b) df œ 0 Ê 2 dx dy œ 0 Ê 2 dx
dy 1 œ 0 Ê dy œ #
57. (a) r# œ x# y# Ê 2r dr œ 2x dx 2y dy Ê dr œ
œ „
œ
0.07
5
y
y# x#
œ „ 0.014 Ê ¸ drr ‚ 100¸ œ ¸ „
dx
x
y# x#
0.014
5
x
r
dx
y
r
dy Ê dr|Ð$ß%Ñ œ ˆ 35 ‰ a „ 0.01b ˆ 45 ‰ a „ 0.01b
‚ 100¸ œ 0.28%; d) œ
3 ‰
dy Ê d)|Ð$ß%Ñ œ ˆ 254 ‰ a „ 0.01b ˆ 25
a „ 0.01b œ
y
‹
x#
#
y
ˆ ‰ 1
x
Š
…0.04
25
dx
Š x" ‹
y
ˆ ‰# 1
x
dy
„0.03
#5
Ê maximum change in d) occurs when dx and dy have opposite signs (dx œ 0.01 and dy œ 0.01 or vice
„0.0028
" ˆ 4 ‰
¸ d)) ‚ 100¸ œ ¸ 0.927255218
versa) Ê d) œ „#0.07
‚ 100¸
5 ¸ „ 0.0028; ) œ tan
3 ¸ 0.927255218 Ê
¸ 0.30%
(b) the radius r is more sensitive to changes in y, and the angle ) is more sensitive to changes in x
58. (a) V œ 1r# h Ê dV œ 21rh dr 1r# dh Ê at r œ 1 and h œ 5 we have dV œ 101 dr 1 dh Ê the volume is
about 10 times more sensitive to a change in r
"
(b) dV œ 0 Ê 0 œ 21rh dr 1r# dh œ 2h dr r dh œ 10 dr dh Ê dr œ 10
dh; choose dh œ 1.5
Ê dr œ 0.15 Ê h œ 6.5 in. and r œ 0.85 in. is one solution for ?V ¸ dV œ 0
59. f(aß bß cß d) œ º
a b
œ ad bc Ê fa œ d, fb œ c, fc œ b, fd œ a Ê df œ d da c db b dc a dd; since
c dº
kak is much greater than kbk , kck , and kdk , the function f is most sensitive to a change in d.
60. ux œ ey , uy œ xey sin z, uz œ y cos z Ê du œ ey dx axey sin zb dy (y cos z) dz
Ê duk ˆ2ßln 3ß 12 ‰ œ 3 dx 7 dy 0 dz œ 3 dx 7 dy Ê magnitude of the maximum possible error
Ÿ 3(0.2) 7(0.6) œ 4.8
61. QK œ
"
#
ˆ 2KM
‰"Î# ˆ 2M
‰ , QM œ
h
h
"
#
ˆ 2KM
‰"Î# ˆ 2K
‰
h
h , and Qh œ
"
#
ˆ 2KM
‰"Î# ˆ 2KM
‰
h
h#
" ˆ 2KM ‰"Î# ˆ 2M ‰
‰"Î# ˆ 2K
‰ dM "# ˆ 2KM
‰"Î# ˆ 2KM
‰ dh
dK "# ˆ 2KM
#
h
h
h
h
h
h#
"Î#
2K
2KM
ˆ 2KM
‰
2M
‘
h
h dK h dM h# dh Ê dQk Ð2ß20ß0Þ0.05Ñ
"Î#
(2)(2)
(2)(2)(20)
’ (2)(2)(20)
’ (2)(20)
0.05 “
0.05 dK 0.05 dM (0.05)# dh“ œ (0.0125)(800 dK 80 dM
Ê dQ œ
œ
"
#
œ
"
#
32,000 dh)
Ê Q is most sensitive to changes in h
ab sin C Ê Aa œ "# b sin C, Ab œ "# a sin C, Ac œ "# ab cos C
Ê dA œ ˆ "# b sin C‰ da ˆ "# a sin C‰ db ˆ "# ab cos C‰ dC; dC œ k2°k œ k0.0349k radians, da œ k0.5k ft,
62. A œ
"
#
db œ k0.5k ft; at a œ 150 ft, b œ 200 ft, and C œ 60°, we see that the change is approximately
dA œ "# (200)(sin 60°) k0.5k "# (150)(sin 60°) k0.5k "# (200)(150)(cos 60°) k0.0349k œ „ 338 ft#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
836
Chapter 14 Partial Derivatives
63. z œ f(xß y) Ê g(xß yß z) œ f(xß y) z œ 0 Ê gx (xß yß z) œ fx (xß y), gy (xß yß z) œ fy (xß y) and gz (xß yß z) œ 1
Ê gx (x! ß y! ß f(x! ß y! )) œ fx (x! ß y! ), gy (x! ß y! ß f(x! ß y! )) œ fy (x! ß y! ) and gz (x! ß y! ß f(x! ß y! )) œ 1 Ê the tangent
plane at the point P! is fx (x! ß y! )(x x! ) fy (x! ß y! )(y y! ) [z f(x! ß y! )] œ 0 or
z œ fx (x! ß y! )(x x! ) fy (x! ß y! )(y y! ) f(x! ß y! )
64. ™ f œ 2xi 2yj œ 2(cos t t sin t)i 2(sin t t cos t)j and v œ (t cos t)i (t sin t)j Ê u œ
œ
(t cos t)i (t sin t)j
È(t cos t)# (t sin t)#
v
kvk
œ (cos t)i (sin t)j since t 0 Ê (Du f)P! œ ™ f † u
œ 2(cos t t sin t)(cos t) 2(sin t t cos t)(sin t) œ 2
65. ™ f œ 2xi 2yj 2zk œ (2 cos t)i (2 sin t)j 2tk and v œ ( sin t)i (cos t)j k Ê u œ
œ
( sin t)i (cos t)j k
È(sin t)# (cos t)# 1#
t
œ Š Èsin t ‹ i Š cos
È ‹j
2
2
"
È2
k Ê (Du f)P! œ ™ f † u
t
"
œ (2 cos t) Š Èsin2 t ‹ (2 sin t) Š cos
È2 ‹ (2t) Š È2 ‹ œ
(Du f) ˆ 14 ‰ œ
"
#
" "Î#
i "# t"Î# j
# t
#
#
" "Î#
i
# t
#
#
67. r œ Èti Ètj (2t 1)k Ê v œ
v(1) œ
Ê (Du f) ˆ 41 ‰ œ
1
2È 2
, (Du f)(0) œ 0 and
4" k ; t œ 1 Ê x œ 1, y œ 1, z œ 1 Ê P! œ (1ß 1ß 1)
i "# j "4 k ; f(xß yß z) œ x y z 3 œ 0 Ê ™ f œ 2xi 2yj k
Ê ™ f(1ß 1ß 1) œ 2i 2j k ; therefore v œ
"
#
2t
È2
1
2È 2
66. r œ Èti Ètj 4" (t 3)k Ê v œ
and v(1) œ
v
kvk
"
#
"
4
( ™ f) Ê the curve is normal to the surface
"# t"Î# j 2k ; t œ 1 Ê x œ 1, y œ 1, z œ 1 Ê P! œ (1ß 1ß 1) and
i j 2k ; f(xß yß z) œ x y z 1 œ 0 Ê ™ f œ 2xi 2yj k Ê ™ f(1ß 1ß 1) œ 2i 2j k ;
now va1b † ™ fa1ß 1ß 1b œ 0, thus the curve is tangent to the surface when t œ 1
14.7 EXTREME VALUES AND SADDLE POINTS
1. fx (xß y) œ 2x y 3 œ 0 and fy (xß y) œ x 2y 3 œ 0 Ê x œ 3 and y œ 3 Ê critical point is (3ß 3);
#
œ 3 0 and fxx 0 Ê local minimum of
fxx (3ß 3) œ 2, fyy (3ß 3) œ 2, fxy (3ß 3) œ 1 Ê fxx fyy fxy
f(3ß 3) œ 5
2. fx (xß y) œ 2y 10x 4 œ 0 and fy (xß y) œ 2x 4y 4 œ 0 Ê x œ 23 and y œ 43 Ê critical point is ˆ 23 ß 43 ‰ ;
#
œ 36 0 and fxx 0 Ê local maximum of
fxx ˆ 23 ß 43 ‰ œ 10, fyy ˆ 23 ß 43 ‰ œ 4, fxy ˆ 23 ß 43 ‰ œ 2 Ê fxx fyy fxy
f ˆ 23 ß 43 ‰ œ 0
3. fx (xß y) œ 2x y 3 œ 0 and fy (xß y) œ x 2 œ 0 Ê x œ 2 and y œ 1 Ê critical point is (2ß 1);
#
œ 1 0 Ê saddle point
fxx (2ß 1) œ 2, fyy (2ß 1) œ 0, fxy (2ß 1) œ 1 Ê fxx fyy fxy
ˆ 6 69 ‰
4. fx (xß y) œ 5y 14x 3 œ 0 and fy (xß y) œ 5x 6 œ 0 Ê x œ 65 and y œ 69
#5 Ê critical point is 5 ß 25 ;
#
‰
ˆ 6 69 ‰
ˆ 6 69 ‰
fxx ˆ 65 ß 69
25 œ 14, fyy 5 ß 25 œ 0, fxy 5 ß 25 œ 5 Ê fxx fyy fxy œ 25 0 Ê saddle point
5. fx (xß y) œ 2y 2x 3 œ 0 and fy (xß y) œ 2x 4y œ 0 Ê x œ 3 and y œ 3# Ê critical point is ˆ3ß 32 ‰ ;
#
œ 4 0 and fxx 0 Ê local maximum of
fxx ˆ3ß 32 ‰ œ 2, fyy ˆ3ß 32 ‰ œ 4, fxy ˆ3ß 32 ‰ œ 2 Ê fxx fyy fxy
f ˆ3ß 3# ‰ œ
17
#
6. fx (xß y) œ 2x 4y œ 0 and fy (xß y) œ 4x 2y 6 œ 0 Ê x œ 2 and y œ 1 Ê critical point is (2ß 1);
#
œ 12 0 Ê saddle point
fxx (2ß 1) œ 2, fyy (2ß 1) œ 2, fxy (2ß 1) œ 4 Ê fxx fyy fxy
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.7 Extreme Values and Saddle Points
837
7. fx (xß y) œ 4x 3y 5 œ 0 and fy (xß y) œ 3x 8y 2 œ 0 Ê x œ 2 and y œ 1 Ê critical point is (2ß 1);
#
œ 23 0 and fxx 0 Ê local minimum of f(2ß 1) œ 6
fxx (2ß 1) œ 4, fyy (2ß 1) œ 8, fxy (2ß 1) œ 3 Ê fxx fyy fxy
8. fx (xß y) œ 2x 2y 2 œ 0 and fy (xß y) œ 2x 4y 2 œ 0 Ê x œ 1 and y œ 0 Ê critical point is (1ß 0);
#
œ 4 0 and fxx 0 Ê local minimum of f(1ß 0) œ 0
fxx (1ß 0) œ 2, fyy (1ß 0) œ 4, fxy (1ß 0) œ 2 Ê fxx fyy fxy
9. fx (xß y) œ 2x 2 œ 0 and fy (xß y) œ 2y 4 œ 0 Ê x œ 1 and y œ 2 Ê critical point is (1ß 2); fxx (1ß 2) œ 2,
#
œ 4 0 Ê saddle point
fyy (1ß 2) œ 2, fxy (1ß 2) œ 0 Ê fxx fyy fxy
10. fx (xß y) œ 2x 2y œ 0 and fy (xß y) œ 2x œ 0 Ê x œ 0 and y œ 0 Ê critical point is (0ß 0); fxx (0ß 0) œ 2,
#
œ 4 0 Ê saddle point
fyy (0ß 0) œ 0, fxy (0ß 0) œ 2 Ê fxx fyy fxy
11. fx axß yb œ
112x 8x
È56x2 8y2 16x 31
8 œ 0 and fy axß yb œ
8y
È56x2 8y2 16x 31
8
8
#
‰
ˆ 16 ‰
ˆ 16 ‰
fxx ˆ 16
7 ß 0 œ 15 , fyy 7 ß 0 œ 15 , fxy 7 ß 0 œ 0 Ê fxx fyy fxy œ
16
‰
fˆ 16
7 ß0 œ 7
12. fx axß yb œ
2x
3ax2 y2 b2Î3
œ 0 and fy axß yb œ
2y
3ax2 y2 b2Î3
‰
œ 0 Ê critical point is ˆ 16
7 ß0 ;
64
225
0 and fxx 0 Ê local maximum of
œ 0 Ê there are no solutions to the system fx axß yb œ 0 and
fy axß yb œ 0, however, we must also consider where the partials are undefined, and this occurs when x œ 0 and y œ 0
Ê critical point is a0ß 0b. Note that the partial derivatives are defined at every other point other than a0ß 0b. We cannot use
the second derivative test, but this is the only possible local maximum, local minimum, or saddle point. faxß yb has a local
3
maximum of fa0ß 0b œ 1 at a0ß 0b since faxß yb œ 1 È
x2 y2 Ÿ 1 for all axß yb other than a0ß 0b.
13. fx (xß y) œ 3x# 2y œ 0 and fy (xß y) œ 3y# 2x œ 0 Ê x œ 0 and y œ 0, or x œ 23 and y œ 23 Ê critical points
are (0ß 0) and ˆ 23 ß 23 ‰ ; for (0ß 0): fxx (0ß 0) œ 6xk Ð0ß0Ñ œ 0, fyy (0ß 0) œ 6yk Ð0ß0Ñ œ 0, fxy (0ß 0) œ 2
#
Ê fxx fyy fxy
œ 4 0 Ê saddle point; for ˆ 32 ß 32 ‰ : fxx ˆ 32 ß 32 ‰ œ 4, fyy ˆ 32 ß 32 ‰ œ 4, fxy ˆ 32 ß 32 ‰ œ 2
#
Ê fxx fyy fxy
œ 12 0 and fxx 0 Ê local maximum of f ˆ 23 ß 32 ‰ œ 170
27
14. fx (xß y) œ 3x# 3y œ 0 and fy (xß y) œ 3x 3y# œ 0 Ê x œ 0 and y œ 0, or x œ 1 and y œ 1 Ê critical points
#
are (0ß 0) and (1ß 1); for (!ß !): fxx (0ß 0) œ 6xk Ð0ß0Ñ œ 0, fyy (0ß 0) œ 6yk Ð0ß0Ñ œ 0, fxy (0ß 0) œ 3 Ê fxx fyy fxy
#
œ 9 0 Ê saddle point; for (1ß 1): fxx (1ß 1) œ 6, fyy (1ß 1) œ 6, fxy (1ß 1) œ 3 Ê fxx fyy fxy
œ 27 0 and fxx 0 Ê local maximum of f(1ß 1) œ 1
15. fx (xß y) œ 12x 6x# 6y œ 0 and fy (xß y) œ 6y 6x œ 0 Ê x œ 0 and y œ 0, or x œ 1 and y œ 1 Ê critical
#
points are (0ß 0) and (1ß 1); for (!ß !): fxx (0ß 0) œ 12 12xk Ð0ß0Ñ œ 12, fyy (0ß 0) œ 6, fxy (0ß 0) œ 6 Ê fxx fyy fxy
œ 36 0 and fxx 0 Ê local minimum of f(0ß 0) œ 0; for (1ß 1): fxx (1ß 1) œ 0, fyy (1ß 1) œ 6,
#
fxy (1ß 1) œ 6 Ê fxx fyy fxy
œ 36 0 Ê saddle point
16. fx (xß y) œ 3x# 6x œ 0 Ê x œ 0 or x œ 2; fy (xß y) œ 3y# 6y œ 0 Ê y œ 0 or y œ 2 Ê the critical points are
(0ß 0), (0ß 2), (2ß 0), and (2ß 2); for (!ß !): fxx (0ß 0) œ 6x 6k Ð0ß0Ñ œ 6, fyy (0ß 0) œ 6y 6k Ð0ß0Ñ œ 6,
#
fxy (0ß 0) œ 0 Ê fxx fyy fxy
œ 36 0 Ê saddle point; for (0ß 2): fxx (0ß 2) œ 6, fyy (0ß 2) œ 6, fxy (0ß 2) œ 0
#
Ê fxx fyy fxy
œ 36 0 and fxx 0 Ê local minimum of f(0ß 2) œ 12; for (2ß 0): fxx (2ß 0) œ 6,
#
fyy (2ß 0) œ 6, fxy (2ß 0) œ 0 Ê fxx fyy fxy
œ 36 0 and fxx 0 Ê local maximum of f(2ß 0) œ 4;
#
for (2ß 2): fxx (2ß 2) œ 6, fyy (2ß 2) œ 6, fxy (2ß 2) œ 0 Ê fxx fyy fxy
œ 36 0 Ê saddle point
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
838
Chapter 14 Partial Derivatives
17. fx axß yb œ 3x2 3y2 15 œ 0 and fy axß yb œ 6x y 3y2 15 œ 0 Ê critical points are a2ß 1b, a2ß 1b, Š0ß È5‹, and
Š0ß È5‹; for a2ß 1b: fxx a2ß 1b œ 6xk a2ß1b œ 12, fyy a2ß 1b œ a6x 6ybk a2ß1b œ 18, fxy a2ß 1b œ 6yk a2ß1b œ 6
#
Ê fxx fyy fxy
œ 180 0 and fxx 0 Ê local minimum of fa2ß 1b œ 30; for a2ß 1b: fxx a2ß 1b œ 6xk a2ß1b
#
œ 12, fyy a2ß 1b œ a6x 6ybk a2ß1b œ 18, fxy a2ß 1b œ 6yk a2ß1b œ 6 Ê fxx fyy fxy
œ 180 0 and
fxx 0 Ê local maximum of fa2ß 1b œ 30; for Š0ß È5‹: fxx Š0ß È5‹ œ 6x¹
œ a6x 6ybk Š0ßÈ5‹ œ 6È5, fxy Š0ß È5‹ œ 6y¹
for Š0ß È5‹: fxx Š0ß È5‹ œ 6x¹
fxy Š0ß È5‹ œ 6y¹
Š0ßÈ5‹
Š0ßÈ5‹
Š0ßÈ5‹
Š0ßÈ5‹
œ 0, fyy Š0ß È5‹
#
œ 6È5 Ê fxx fyy fxy
œ 180 0 Ê saddle pointà
œ 0, fyy Š0ß È5‹ œ a6x 6ybk Š0ßÈ5‹ œ 6È5,
#
œ 6È5 Ê fxx fyy fxy
œ 180 0 Ê saddle point.
18. fx (xß y) œ 6x# 18x œ 0 Ê 6x(x 3) œ 0 Ê x œ 0 or x œ 3; fy (xß y) œ 6y# 6y 12 œ 0 Ê 6(y 2)(y 1) œ 0
Ê y œ 2 or y œ 1 Ê the critical points are (0ß 2), (0ß 1), (3ß 2), and (3ß 1); fxx (xß y) œ 12x 18,
fyy (xß y) œ 12y 6, and fxy (xß y) œ 0; for (!ß 2): fxx (0ß 2) œ 18, fyy (0ß 2) œ 18, fxy (0ß 2) œ 0
#
Ê fxx fyy fxy
œ 324 0 and fxx 0 Ê local maximum of f(0ß 2) œ 20; for (0ß 1): fxx (0ß 1) œ 18,
#
fyy (0ß 1) œ 18, fxy (0ß 1) œ 0 Ê fxx fyy fxy
œ 324 0 Ê saddle point; for (3ß 2): fxx (3ß 2) œ 18,
#
fyy (3ß 2) œ 18, fxy (3ß 2) œ 0 Ê fxx fyy fxy
œ 324 0 Ê saddle point; for (3ß 1): fxx (3ß 1) œ 18,
#
fyy (3ß 1) œ 18, fxy (3ß 1) œ 0 Ê fxx fyy fxy
œ 324 0 and fxx 0 Ê local minimum of f(3ß 1) œ 34
19. fx (xß y) œ 4y 4x$ œ 0 and fy (xß y) œ 4x 4y$ œ 0 Ê x œ y Ê x a1 x# b œ 0 Ê x œ 0, 1, 1 Ê the critical
points are (0ß 0), (1ß 1), and (1ß 1); for (!ß !): fxx (0ß 0) œ 12x# k Ð0ß0Ñ œ 0, fyy (0ß 0) œ 12y# k Ð0ß0Ñ œ 0,
#
fxy (0ß 0) œ 4 Ê fxx fyy fxy
œ 16 0 Ê saddle point; for (1ß 1): fxx (1ß 1) œ 12, fyy (1ß 1) œ 12, fxy (1ß 1) œ 4
#
Ê fxx fyy fxy
œ 128 0 and fxx 0 Ê local maximum of f(1ß 1) œ 2; for (1ß 1): fxx (1ß 1) œ 12,
#
fyy (1ß 1) œ 12, fxy (1ß 1) œ 4 Ê fxx fyy fxy
œ 128 0 and fxx 0 Ê local maximum of f(1ß 1) œ 2
20. fx (xß y) œ 4x$ 4y œ 0 and fy (xß y) œ 4y$ 4x œ 0 Ê x œ y Ê x$ x œ 0 Ê x a1 x# b œ 0 Ê x œ 0, 1, 1
Ê the critical points are (0ß 0), (1ß 1), and (1ß 1); fxx (xß y) œ 12x# , fyy (xß y) œ 12y# , and fxy (xß y) œ 4;
#
for (!ß 0): fxx (0ß 0) œ 0, fyy (0ß 0) œ 0, fxy (0ß 0) œ 4 Ê fxx fyy fxy
œ 16 0 Ê saddle point; for (1ß 1):
#
fxx (1ß 1) œ 12, fyy (1ß 1) œ 12, fxy (1ß 1) œ 4 Ê fxx fyy fxy
œ 128 0 and fxx 0 Ê local minimum of
#
f("ß 1) œ 2; for (1ß 1): fxx (1ß 1) œ 12, fyy (1ß 1) œ 12, fxy (1ß 1) œ 4 Ê fxx fyy fxy
œ 128 0 and
fxx 0 Ê local minimum of f(1ß 1) œ 2
21. fx (xß y) œ
2x
ax# y# 1b#
œ 0 and fy (xß y) œ
2y
ax# y# 1b#
œ 0 Ê x œ 0 and y œ 0 Ê the critical point is (!ß 0);
#
#
4x# 2y# 2
, fyy œ ax2x# y#4y 1b$2 , fxy œ ax# 8xy
; fxx (!ß !) œ 2, fyy (0ß 0)
y# 1b$
ax# y# 1b$
#
fxx fyy fxy œ 4 0 and fxx 0 Ê local maximum of f(0ß 0) œ 1
fxx œ
Ê
22. fx (xß y) œ x1# y œ 0 and fy (xß y) œ x
1
y#
œ 2, fxy (0ß 0) œ 0
œ 0 Ê x œ 1 and y œ 1 Ê the critical point is (1ß 1); fxx œ
fxy œ 1; fxx (1ß 1) œ 2, fyy (1ß 1) œ 2, fxy (1ß 1) œ 1 Ê fxx fyy
#
fxy
2
x$
, fyy œ
2
y$
œ 3 0 and fxx 2 Ê local minimum of f(1ß 1) œ 3
23. fx (xß y) œ y cos x œ 0 and fy (xß y) œ sin x œ 0 Ê x œ n1, n an integer, and y œ 0 Ê the critical points are
(n1ß 0), n an integer (Note: cos x and sin x cannot both be 0 for the same x, so sin x must be 0 and y œ 0);
fxx œ y sin x, fyy œ 0, fxy œ cos x; fxx (n1ß 0) œ 0, fyy (n1ß 0) œ 0, fxy (n1ß 0) œ 1 if n is even and fxy (n1ß 0) œ 1
#
if n is odd Ê fxx fyy fxy
œ 1 0 Ê saddle point.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
,
Section 14.7 Extreme Values and Saddle Points
839
24. fx (xß y) œ 2e2x cos y œ 0 and fy (xß y) œ e2x sin y œ 0 Ê no solution since e2x Á 0 for any x and the functions
cos y and sin y cannot equal 0 for the same y Ê no critical points Ê no extrema and no saddle points
25. fx axß yb œ a2x 4bex
fyy a2ß 0b œ
2
e4
2
y2 4x
Ê fxx fyy
#
fxy
œ 0 and fy axß yb œ 2yex
œ
4
e8
2
y2 4x
œ 0 Ê critical point is a2ß 0b; fxx a2ß 0b œ
0 and fxx 0 Ê local mimimum of fa2ß 0b œ
2
e4 , fxy a2ß 0b
œ 0,
1
e4
26. fx axß yb œ yex œ 0 and fy axß yb œ ey ex œ 0 Ê critical point is a0ß 0b; fxx a2ß 0b œ 0, fxy a2ß 0b œ 1, fyy a2ß 0b œ 1
#
Ê fxx fyy fxy
œ 1 0 Ê saddle point
27. fx axß yb œ 2xey œ 0 and fy axß yb œ 2yey ey ax2 y2 b œ 0 Ê critical points are a0ß 0b and a0ß 2b; for a0ß 0b:
fxx a0ß 0b œ 2ey k a0ß0b œ 2, fyy a0ß 0b œ a2ey 4yey ey ax2 y2 bbk a0ß0b œ 2, fxy a0ß 0b œ 2xey k a0ß0b œ 0
#
Ê fxx fyy fxy
œ 4 0 and fxx 0 Ê local mimimum of fa0ß 0b œ 0; for a0ß 2b: fxx a0ß 2b œ 2ey k a0ß2b œ
#
fyy a0ß 2b œ a2ey 4yey ey ax2 y2 bbk a0ß2b œ e22 , fxy a0ß 2b œ 2xey k a0ß2b œ 0 Ê fxx fyy fxy
œ
2
e2 ,
e44
0
Ê saddle point
28. fx axß yb œ ex ax2 2x y2 b œ 0 and fy axß yb œ 2yex œ 0 Ê critical points are a0ß 0b and a2ß 0b; for a0ß 0b:
fxx a0ß 0b œ ex ax2 4x 2 y2 bk a0ß0b œ 2, fyy a0ß 0b œ 2ex k a0ß0b œ 2, fxy a0ß 0b œ 2yex k a0ß0b œ 0
#
Ê fxx fyy fxy
œ 4 0 and fxx 0 Ê saddle point; for a2ß 0b: fxx a2ß 0b œ ex ax2 4x 2 y2 bk a2ß0b œ e22 ,
#
fyy a2ß 0b œ 2ex k a2ß0b œ e22 , fxy a2ß 0b œ 2yex k a2ß0b œ 0 Ê fxx fyy fxy
œ
of fa2ß 0b œ
4
e4
0 and fxx 0 Ê local maximum
4
e2
29. fx axß yb œ 4
2
x
œ 0 and fy axß yb œ 1
1
y
œ 0 Ê critical point is ˆ 21 , 1‰ ; fxx ˆ 21 , 1‰ œ 8, fyy ˆ 12 , 1‰ œ 1,
#
œ 8 0 and fxx 0 Ê local maximum of fˆ 12 , 1‰ œ 3 2ln 2
fxy ˆ 12 , 1‰ œ 0 Ê fxx fyy fxy
30. fx axß yb œ 2x
1
xy
œ 0 and fy axß yb œ 1
1
xy
œ 0 Ê critical point is ˆ 21 , 23 ‰ ; fxx ˆ 21 , 23 ‰ œ 1, fyy ˆ 12 , 32 ‰ œ 1,
#
œ 2 0 Ê saddle point
fxy ˆ 12 , 32 ‰ œ 1 Ê fxx fyy fxy
On OA, f(xß y) œ f(0ß y) œ y# 4y 1 on 0 Ÿ y Ÿ 2;
f w (0ß y) œ 2y 4 œ 0 Ê y œ 2;
f(0ß 0) œ 1 and f(!ß #) œ 3
(ii) On AB, f(xß y) œ f(xß 2) œ 2x# 4x 3 on 0 Ÿ x Ÿ 1;
f w (xß 2) œ 4x 4 œ 0 Ê x œ 1;
f(0ß 2) œ 3 and f(1ß #) œ 5
(iii) On OB, f(xß y) œ f(xß 2x) œ 6x# 12x 1 on
0 Ÿ x Ÿ 1; endpoint values have been found above;
f w (xß 2x) œ 12x 12 œ 0 Ê x œ 1 and y œ 2, but ("ß #) is not an interior point of OB
(iv) For interior points of the triangular region, fx (xß y) œ 4x 4 œ 0 and fy (xß y) œ 2y 4 œ 0
Ê x œ 1 and y œ 2, but (1ß 2) is not an interior point of the region. Therefore, the absolute maximum is
1 at (0ß 0) and the absolute minimum is 5 at ("ß #).
31. (i)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
840
Chapter 14 Partial Derivatives
On OA, D(xß y) œ D(0ß y) œ y# 1 on 0 Ÿ y Ÿ 4;
Dw (0ß y) œ 2y œ 0 Ê y œ 0; D(!ß !) œ 1 and
D(!ß %) œ 17
(ii) On AB, D(xß y) œ D(xß 4) œ x# 4x 17 on
0 Ÿ x Ÿ 4; Dw (xß 4) œ 2x 4 œ 0 Ê x œ 2 and (2ß 4)
is an interior point of AB; D(#ß %) œ 13 and
D(%ß %) œ D(!ß %) œ 17
(iii) On OB, D(xß y) œ D(xß x) œ x# 1 on 0 Ÿ x Ÿ 4;
Dw (xß x) œ 2x œ 0 Ê x œ 0 and y œ 0, which is not an interior point of OB; endpoint values have been found
above
(iv) For interior points of the triangular region, fx (xß y) œ 2x y œ 0 and fy (xß y) œ x 2y œ 0 Ê x œ 0 and y œ 0,
which is not an interior point of the region. Therefore, the absolute maximum is 17 at (!ß %) and (%ß %), and the
absolute minimum is 1 at (0ß 0).
32. (i)
On OA, f(xß y) œ f(!ß y) œ y# on 0 Ÿ y Ÿ 2;
f w (0ß y) œ 2y œ 0 Ê y œ 0 and x œ 0; f(0ß 0) œ 0 and
f(0ß #) œ 4
(ii) On OB, f(xß y) œ f(xß 0) œ x# on 0 Ÿ x Ÿ 1;
f w (xß 0) œ 2x œ 0 Ê x œ 0 and y œ 0; f(0ß 0) œ 0 and
f(1ß 0) œ 1
(iii) On AB, f(xß y) œ f(xß 2x 2) œ 5x# 8x 4 on
0 Ÿ x Ÿ 1; f w (xß 2x 2) œ 10x 8 œ 0 Ê x œ 45
and y œ 25 ; f ˆ 45 ß 25 ‰ œ 45 ; endpoint values have been found above.
33. (i)
(iv) For interior points of the triangular region, fx (xß y) œ 2x œ 0 and fy (xß y) œ 2y œ 0 Ê x œ 0 and y œ 0, but (!ß 0) is
not an interior point of the region. Therefore the absolute maximum is 4 at (0ß 2) and the absolute minimum is 0 at
(0ß 0).
34. (i)
(ii)
On AB, T(xß y) œ T(!ß y) œ y# on 3 Ÿ y Ÿ 3;
Tw (0ß y) œ 2y œ 0 Ê y œ 0 and x œ 0; T(0ß 0) œ 0,
T(!ß 3) œ 9, and T(!ß 3) œ 9
On BC, T(xß y) œ T(xß 3) œ x# 3x 9 on 0 Ÿ x Ÿ 5;
Tw (xß 3) œ 2x 3 œ 0 Ê x œ 3# and y œ 3;
T ˆ 3# ß 3‰ œ 27
4 and T(5ß 3) œ 19
(iii) On CD, T(xß y) œ T(5ß y) œ y# 5y 5 on
3 Ÿ y Ÿ 3;Tw (5ß y) œ 2y 5 œ 0 Ê y œ 5# and
x œ 5;T ˆ5ß 5# ‰ œ 45
4 , T(&ß 3) œ 11 and T(5ß 3) œ 19
(iv) On AD, T(xß y) œ T(xß 3) œ x# 9x 9 on 0 Ÿ x Ÿ 5; Tw (xß 3) œ 2x 9 œ 0 Ê x œ
T ˆ 9# ß 3‰ œ 45
4 , T(!ß 3) œ 9 and T(&ß 3) œ 11
(v)
9
#
and y œ 3;
For interior points of the rectangular region, Tx (xß y) œ 2x y 6 œ 0 and Ty (xß y) œ x 2y œ 0 Ê x œ 4
and y œ 2 Ê (4ß 2) is an interior critical point with T(4ß 2) œ 12. Therefore the absolute maximum
is 19 at (5ß 3) and the absolute minimum is 12 at (4ß 2).
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.7 Extreme Values and Saddle Points
35. (i)
(ii)
841
On OC, T(xß y) œ T(xß 0) œ x# 6x 2 on
0 Ÿ x Ÿ 5; Tw (xß 0) œ 2x 6 œ 0 Ê x œ 3 and
y œ 0; T(3ß 0) œ 7, T(0ß 0) œ 2, and T(5ß 0) œ 3
On CB, T(xß y) œ T(5ß y) œ y# 5y 3 on
3 Ÿ y Ÿ 0; Tw (5ß y) œ 2y 5 œ 0 Ê y œ 5# and
x œ 5; T ˆ5ß 5# ‰ œ 37
4 and T(5ß 3) œ 9
(iii) On AB, T(xß y) œ T(xß 3) œ x# 9x 11 on
0 Ÿ x Ÿ 5; Tw (xß 3) œ 2x 9 œ 0 Ê x œ 9# and
y œ 3; T ˆ 9# ß 3‰ œ 37
4 and T(!ß 3) œ 11
(iv) On AO, T(xß y) œ T(!ß y) œ y# 2 on 3 Ÿ y Ÿ 0; Tw (0ß y) œ 2y œ 0 Ê y œ 0 and x œ 0, but (0ß 0) is
not an interior point of AO
(v) For interior points of the rectangular region, Tx (xß y) œ 2x y 6 œ 0 and Ty (xß y) œ x 2y œ 0 Ê x œ 4
and y œ 2, an interior critical point with T(%ß 2) œ 10. Therefore the absolute maximum is 11 at
(!ß 3) and the absolute minimum is 10 at (4ß 2).
36. (i)
(ii)
On OA, f(xß y) œ f(!ß y) œ 24y# on 0 Ÿ y Ÿ 1;
f w (0ß y) œ 48y œ 0 Ê y œ 0 and x œ 0, but (0ß 0) is
not an interior point of OA; f(!ß 0) œ 0 and
f(!ß 1) œ 24
On AB, f(xß y) œ f(xß 1) œ 48x 32x$ 24 on
0 Ÿ x Ÿ 1; f w (xß 1) œ 48 96x# œ 0 Ê x œ È"2 and
y œ 1, or x œ È"2 and y œ 1, but Š È"2 ß 1‹ is not in
the interior of AB; f Š È"2 ß 1‹ œ 16È2 24 and f(1ß 1) œ 8
(iii) On BC, f(xß y) œ f("ß y) œ 48y 32 24y# on 0 Ÿ y Ÿ 1; f w ("ß y) œ 48 48y œ 0 Ê y œ 1 and x œ 1, but
("ß ") is not an interior point of BC; f("ß 0) œ 32 and f("ß ") œ 8
(iv) On OC, f(xß y) œ f(xß 0) œ 32x$ on 0 Ÿ x Ÿ 1; f w (xß 0) œ 96x# œ 0 Ê x œ 0 and y œ 0, but (0ß 0) is not an
interior point of OC; f(!ß 0) œ 0 and f("ß 0) œ 32
(v) For interior points of the rectangular region, fx (xß y) œ 48y 96x# œ 0 and fy (xß y) œ 48x 48y œ 0
Ê x œ 0 and y œ 0, or x œ "# and y œ "# , but (0ß 0) is not an interior point of the region; f ˆ "# ß "# ‰ œ 2.
Therefore the absolute maximum is 2 at ˆ "# ß "# ‰ and the absolute minimum is 32 at (1ß 0).
37. (i)
On AB, f(xß y) œ f(1ß y) œ 3 cos y on 14 Ÿ y Ÿ
w
1
4
;
1
4
;
f (1ß y) œ 3 sin y œ 0 Ê y œ 0 and x œ 1;
f("ß 0) œ 3, f ˆ1ß 14 ‰ œ
(ii)
3È 2
#
, and f ˆ1ß 14 ‰ œ
3È 2
#
On CD, f(xß y) œ f($ß y) œ 3 cos y on 14 Ÿ y Ÿ
f w (3ß y) œ 3 sin y œ 0 Ê y œ 0 and x œ 3;
È
3È 2
ˆ 1‰ 3 2
# and f 3ß 4 œ #
È2
1‰
#
4 œ # a4x x b on
f(3ß 0) œ 3, f ˆ3ß 14 ‰ œ
(iii) On BC, f(xß y) œ f ˆxß
1 Ÿ x Ÿ 3; f w ˆxß 14 ‰ œ È2(2 x) œ 0 Ê x œ 2 and y œ
f ˆ3ß 14 ‰ œ
3È 2
#
; f ˆ2ß 14 ‰ œ 2È2, f ˆ1ß 14 ‰ œ
È2
#
w
# a4x x b on 1 Ÿ x Ÿ 3; f
È
È
œ 3 # 2 , and f ˆ3ß 14 ‰ œ 3 # 2
(iv) On AD, f(xß y) œ f ˆxß 14 ‰ œ
f ˆ2ß 14 ‰ œ 2È2, f ˆ1ß 14 ‰
1
4
3È 2
#
, and
ˆxß 14 ‰ œ È2(2 x) œ 0 Ê x œ 2 and y œ 14 ;
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
842
Chapter 14 Partial Derivatives
For interior points of the region, fx (xß y) œ (4 2x) cos y œ 0 and fy (xß y) œ a4x x# b sin y œ 0 Ê x œ 2
and y œ 0, which is an interior critical point with f(2ß 0) œ 4. Therefore the absolute maximum is 4 at
(v)
(2ß 0) and the absolute minimum is
3È 2
#
at ˆ3ß 14 ‰ , ˆ3ß 14 ‰ , ˆ1ß 14 ‰ , and ˆ1ß 14 ‰ .
On OA, f(xß y) œ f(!ß y) œ 2y 1 on 0 Ÿ y Ÿ 1;
f w (0ß y) œ 2 Ê no interior critical points; f(0ß 0) œ 1
and f(0ß 1) œ 3
(ii) On OB, f(xß y) œ f(xß 0) œ 4x 1 on 0 Ÿ x Ÿ 1;
f w (xß 0) œ 4 Ê no interior critical points; f(1ß 0) œ 5
(iii) On AB, f(xß y) œ f(xß x 1) œ 8x# 6x 3 on
0 Ÿ x Ÿ 1; f w (xß x 1) œ 16x 6 œ 0 Ê x œ 38
and y œ 58 ; f ˆ 38 ß 58 ‰ œ 15
8 , f(0ß 1) œ 3, and f("ß 0) œ 5
38. (i)
(iv) For interior points of the triangular region, fx (xß y) œ 4 8y œ 0 and fy (xß y) œ 8x 2 œ 0
Ê y œ "# and x œ 4" which is an interior critical point with f ˆ 4" ß #" ‰ œ 2. Therefore the absolute maximum is 5 at
(1ß 0) and the absolute minimum is 1 at (0ß 0).
39. Let F(aß b) œ 'a a6 x x# b dx where a Ÿ b. The boundary of the domain of F is the line a œ b in the ab-plane, and
b
F(aß a) œ 0, so F is identically 0 on the boundary of its domain. For interior critical points we have:
`F
`F
#
#
` a œ a6 a a b œ 0 Ê a œ 3, 2 and ` b œ a6 b b b œ 0 Ê b œ 3, 2. Since a Ÿ b, there is only one
interior critical point (3ß 2) and F(3ß 2) œ 'c3 a6 x x# b dx gives the area under the parabola y œ 6 x x# that is
2
above the x-axis. Therefore, a œ 3 and b œ 2.
40. Let F(aß b) œ 'a a24 2x x# b
b
"Î$
dx where a Ÿ b. The boundary of the domain of F is the line a œ b and on this line F is
identically 0. For interior critical points we have:
`F
`b
# "Î$
œ a24 2b b b
`F
`a
œ a24 2a a# b
"Î$
œ 0 Ê a œ 4, 6 and
œ 0 Ê b œ 4, 6. Since a Ÿ b, there is only one critical point (6ß 4) and
F(6ß 4) œ 'c6 a24 2x x# b dx gives the area under the curve y œ a24 2x x# b
4
"Î$
that is above the x-axis.
Therefore, a œ 6 and b œ 4.
41. Tx (xß y) œ 2x 1 œ 0 and Ty (xß y) œ 4y œ 0 Ê x œ
"
#
and y œ 0 with T ˆ "# ß 0‰ œ 4" ; on the boundary
x# y# œ 1: T(xß y) œ x# x 2 for 1 Ÿ x Ÿ 1 Ê Tw (xß y) œ 2x 1 œ 0 Ê x œ "# and y œ „
T Š
" È3
#ß # ‹
Š "# ß
œ
È3
# ‹;
9
4
, T Š
œ 2 ln
"
#
œ
9
4
"
4
, T(1ß 0) œ 2, and T("ß 0) œ 0 Ê the hottest is 2 ° at Š
" È3
#ß # ‹
2
x
"
y# ¹ ˆ 1 ß2‰
œ 0 and fy (xß y) œ x
œ
2
"
4
"
y
œ0 Ê xœ
"
#
and y œ 2; fxx ˆ "# ß 2‰ œ
2¸
x# ˆ 12 ß2‰
œ 8,
#
, fxy ˆ "# ß 2‰ œ 1 Ê fxx fyy fxy
œ 1 0 and fxx 0 Ê a local minimum of f ˆ "# ß 2‰
œ 2 ln 2
43. (a) fx (xß y) œ 2x 4y œ 0 and fy (xß y) œ 2y 4x œ 0 Ê x œ 0 and y œ 0; fxx (0ß 0) œ 2, fyy (0ß 0) œ 2,
#
œ 12 0 Ê saddle point at (0ß 0)
fxy (0ß 0) œ 4 Ê fxx fyy fxy
(b) fx (xß y) œ 2x 2 œ 0 and fy (xß y) œ 2y 4 œ 0 Ê x œ 1 and y œ 2; fxx (1ß 2) œ 2, fyy (1ß 2) œ 2,
#
œ 4 0 and fxx 0 Ê local minimum at ("ß #)
fxy (1ß 2) œ 0 Ê fxx fyy fxy
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
;
and
the coldest is "4 ° at ˆ "# ß 0‰ .
42. fx (xß y) œ y 2
fyy ˆ #" ß 2‰ œ
È3
"
#ß # ‹
È3
#
Section 14.7 Extreme Values and Saddle Points
843
(c) fx (xß y) œ 9x# 9 œ 0 and fy (xß y) œ 2y 4 œ 0 Ê x œ „ 1 and y œ 2; fxx (1ß 2) œ 18xk Ð1ß2Ñ œ 18,
#
œ 36 0 and fxx 0 Ê local minimum at ("ß #);
fyy (1ß 2) œ 2, fxy (1ß 2) œ 0 Ê fxx fyy fxy
#
fxx (1ß 2) œ 18, fyy ("ß 2) œ 2, fxy ("ß 2) œ 0 Ê fxx fyy fxy
œ 36 0 Ê saddle point at ("ß 2)
44. (a)
(b)
(c)
(d)
(e)
(f)
Minimum at (0ß 0) since f(xß y) 0 for all other (xß y)
Maximum of 1 at (!ß !) since f(xß y) 1 for all other (xß y)
Neither since f(xß y) 0 for x 0 and f(xß y) 0 for x 0
Neither since f(xß y) 0 for x 0 and f(xß y) 0 for x 0
Neither since f(xß y) 0 for x 0 and y 0, but f(xß y) 0 for x 0 and y 0
Minimum at (0ß 0) since f(xß y) 0 for all other (xß y)
45. If k œ 0, then f(xß y) œ x# y# Ê fx (xß y) œ 2x œ 0 and fy (xß y) œ 2y œ 0 Ê x œ 0 and y œ 0 Ê (0ß 0) is the only
critical point. If k Á 0, fx (xß y) œ 2x ky œ 0 Ê y œ 2k x; fy (xß y) œ kx 2y œ 0 Ê kx 2 ˆ 2k x‰ œ 0
4‰
ˆ
ˆ 2‰
Ê kx 4x
k œ 0 Ê k k x œ 0 Ê x œ 0 or k œ „ 2 Ê y œ k (0) œ 0 or y œ „ x; in any case (0ß 0) is a
critical point.
#
46. (See Exercise 45 above): fxx (xß y) œ 2, fyy (xß y) œ 2, and fxy (xß y) œ k Ê fxx fyy fxy
œ 4 k# ; f will have a saddle point
at (0ß 0) if 4 k# 0 Ê k 2 or k 2; f will have a local minimum at (0ß 0) if 4 k# 0 Ê 2 k 2; the test is
inconclusive if 4 k# œ 0 Ê k œ „ 2.
47. No; for example f(xß y) œ xy has a saddle point at (aß b) œ (0ß 0) where fx œ fy œ 0.
#
48. If fxx (aß b) and fyy (aß b) differ in sign, then fxx (aß b) fyy (aß b) 0 so fxx fyy fxy
0. The surface must therefore have a
saddle point at (aß b) by the second derivative test.
49. We want the point on z œ 10 x# y# where the tangent plane is parallel to the plane x 2y 3z œ 0. To find a normal
vector to z œ 10 x# y# let w œ z x# y# 10. Then ™ w œ 2xi 2yj k is normal to z œ 10 x# y# at
(xß y). The vector ™ w is parallel to i 2j 3k which is normal to the plane x 2y 3z œ 0 if
"
‰
6xi 6yj 3k œ i 2j 3k or x œ "6 and y œ "3 . Thus the point is ˆ "6 ß "3 ß 10 36
9" ‰ or ˆ 6" ß 3" ß 355
36 .
50. We want the point on z œ x# y# 10 where the tangent plane is parallel to the plane x 2y z œ 0. Let
w œ z x# y# 10, then ™ w œ 2xi 2yj k is normal to z œ x# y# 10 at (xß y). The vector ™ w is parallel
‰
to i 2j k which is normal to the plane if x œ "# and y œ 1. Thus the point ˆ "# ß 1ß 4" 1 10‰ or ˆ #" ß 1ß 45
4 is the point
on the surface z œ x# y# 10 nearest the plane x 2y z œ 0.
51. daxß yß zb œ Éax 0b2 ay 0b2 az 0b2 Ê we can minimize daxß yß zb by minimizing Daxß yß zb œ x2 y2 z2 ;
3x 2y z œ 6 Ê z œ 6 3x 2y Ê Daxß yb œ x2 y2 a6 3x 2yb2 Ê Dx axß yb œ 2x 6a6 3x 2yb œ 0
and Dy axß yb œ 2y 4a6 3x 2yb œ 0 Ê critical point is ˆ 97 , 67 ‰ Ê z œ 37 ; Dxx ˆ 97 , 67 ‰ œ 20, Dyy ˆ 12 , 1‰ œ 10,
Dxy ˆ 12 , 1‰ œ 12 Ê Dxx Dyy D#xy œ 56 0 and Dxx 0 Ê local minimum of dˆ 97 , 67 , 37 ‰ œ
3È14
7
52. daxß yß zb œ Éax 2b2 ay 1b2 az 1b2 Ê we can minimize daxß yß zb by minimizing
Daxß yß zb œ ax 2b2 ay 1b2 az 1b2 ; x y z œ 2 Ê z œ x y 2
Ê Daxß yb œ ax 2b2 ay 1b2 ax y 3b2 Ê Dx axß yb œ 2ax 2b 2ax y 3b œ 0
and Dy axß yb œ 2ay 1b 2ax y 3b œ 0 Ê critical point is ˆ 83 , 13 ‰ Ê z œ 13 ; Dxx ˆ 83 , 13 ‰ œ 4, Dyy ˆ 83 , 13 ‰ œ 4,
Dxy ˆ 83 , 13 ‰ œ 2 Ê Dxx Dyy D#xy œ 12 0 and Dxx 0 Ê local minimum of dˆ 83 , 13 , 13 ‰ œ È2
3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
844
Chapter 14 Partial Derivatives
53. saxß yß zb œ x2 y2 z2 ; x y z œ 9 Ê z œ 9 x y Ê saxß yb œ x2 y2 a9 x yb2
Ê sx axß yb œ 2x 2a9 x yb œ 0 and sy axß yb œ 2y 2a9 x yb œ 0 Ê critical point is a3, 3b Ê z œ 3;
sxx a3, 3b œ 4, syy a3, 3b œ 4, sxy a3, 3b œ 2 Ê sxx syy s#xy œ 12 0 and sxx 0 Ê local minimum of sa3, 3, 3b œ 27
54. paxß yß zb œ xyz; x y z œ 3 Ê z œ 3 x y Ê paxß yb œ x ya3 x yb œ 3x y x2 y x y2
Ê px axß yb œ 3y 2xy y2 œ 0 and py axß yb œ 3x x2 2xy œ 0 Ê critical points are a0, 0b, a0, 3b, a3, 0b, and
a1, 1b; for a0, 0b Ê z œ 3; pxx a0, 0b œ 0, pyy a0, 0b œ 0, pxy a0, 0b œ 3 Ê pxx pyy p#xy œ 9 0 Ê saddle point;
for a0, 3b Ê z œ 0; pxx a0, 3b œ 6, pyy a0, 3b œ 0, pxy a0, 3b œ 3 Ê pxx pyy p#xy œ 9 0 Ê saddle point;
for a3, 0b Ê z œ 0; pxx a3, 0b œ 0, pyy a3, 0b œ 6, pxy a3, 0b œ 3 Ê pxx pyy p#xy œ 9 0 Ê saddle point;
for a1, 1b Ê z œ 1; pxx a1, 1b œ 2, pyy a1, 1b œ 2, pxy a1, 1b œ 1 Ê pxx pyy p#xy œ 3 0 and pxx 0 Ê local
maximum of pa1, 1, 1b œ 1
55. saxß yß zb œ xy yz xz; x y z œ 6 Ê z œ 6 x y Ê saxß yb œ xy ya6 x yb xa6 x yb
œ 6x 6y xy x2 y2 Ê sx axß yb œ 6 2x y œ 0 and sy axß yb œ 6 x 2y œ 0 Ê critical point is a2, 2b
Ê z œ 2; sxx a2, 2b œ 2, syy a2, 2b œ 2, sxy a2, 2b œ 1 Ê sxx syy s#xy œ 3 0 and sxx 0 Ê local maximum of
sa2, 2, 2b œ 12
56. daxß yß zb œ Éax 6b2 ay 4b2 az 0b2 Ê we can minimize daxß yß zb by minimizing
Daxß yß zb œ ax 6b2 ay 4b2 z2 ; z œ Èx2 y2 Ê Daxß yb œ ax 6b2 ay 4b2 x2 y2
œ 2x2 2y2 12x 8y 52 Ê Dx axß yb œ 4x 12 œ 0 and Dy axß yb œ 4y 8 œ 0 Ê critical point is a3, 2b
Ê z œ È13; Dxx a3, 2b œ 4, Dyy a3, 2b œ 4, Dxy a3, 2b œ 0 Ê Dxx Dyy D# œ 16 0 and Dxx 0 Ê local
xy
minimum of dŠ3, 2, È13‹ œ È26
57. Vaxß yß zb œ a2xba2yba2zb œ 8xyz; x2 y2 z2 œ 4 Ê z œ È4 x2 y2 Ê Vaxß yb œ 8xyÈ4 x2 y2 ,
x
0 and y
a0, 0b, Š È#3 ,
0 Ê Vx axß yb œ
#
È3 ‹,
32y 16x2 y 8y3
È 4 x2 y2
Š È#3 , È#3 ‹, Š È#3 ,
Va0ß 0b œ 0 and VŠ È#3 ,
#
È3 ‹
œ
64
;
3È 3
#
È3 ‹,
œ 0 and Vy axß yb œ
32x 16x y2 8x3
È 4 x2 y2
œ 0 Ê critical points are
and Š È#3 , È#3 ‹. Only a0, 0b and Š È#3 ,
#
È3 ‹
satisfy x
0 and y
0
On x œ 0, 0 Ÿ y Ÿ 2 Ê Va0ß yb œ 8a0byÈ4 02 y2 œ 0, no critical points,
Va0ß 0b œ 0, Va0ß 2b œ 0; On y œ 0, 0 Ÿ x Ÿ 2 Ê Vaxß 0b œ 8xa0bÈ4 x2 02 œ 0, no critical points, Va0ß 0b œ 0,
2
Va0ß 2b œ 0; On y œ È4 x2 , 0 Ÿ x Ÿ 2 Ê VŠxß È4 x2 ‹ œ 8xÈ4 x2 Ê4 x2 ŠÈ4 x2 ‹ œ 0
no critical points, Va0ß 2b œ 0, Va2ß 0b œ 0. Thus, there is a maximum volume of
58. Saxß yß zb œ 2xy 2yz 2xz; xyz œ 27 Ê z œ
y 0; Sx axß yb œ 2y
54
x2
27
xy
Syy a3, 3b œ 4, Dxy a3, 3b œ 2 Ê Dxx Dyy
if the box is
#
È3
‚
27
27
Ê Saxß yß zb œ 2xy 2yŠ xy
‹ 2xŠ xy
‹ œ 2xy
œ 0 and Sy axß yb œ 2x
D#xy
64
3È 3
54
y2
#
È3
54
x
‚
54
y ,
œ 0 Ê Critical point is a3, 3b Ê z œ 3; Sxx a3, 3b œ 4,
œ 12 0 and Dxx 0 Ê local minimum of Sa3ß 3ß 3b œ 54
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
#
È3 .
x 0,
Section 14.7 Extreme Values and Saddle Points
845
59. Let x œ height of the box, y œ width, and z œ length, cut out
squares of length x from corner of the material See diagram
at right. Fold along the dashed lines to form the box. From
the diagram we see that the length of the material is 2x y
and the width is 2x z. Thus a2x yba2x zb œ 12
2ˆ6 2 x2 xy‰
. Since Vax, y, zb œ x y z
2x y
ˆ
2x y 6 2 x2 xy‰
Vax, yb œ
, where x 0, y
2x y
2
3
2 2
ˆ
4 3y 4x y 4x y xy3 ‰
Êzœ
Ê
Vx ax, yb œ
Vy ax, yb œ
œ 0 and
a2x yb2
2ˆ12x 4 x 4x y x y
a2x yb2
2
4
0.
2 2‰
3
œ 0 Ê critical points are ŠÈ3, 0‹, ŠÈ3, 0‹, Š È13 ,
and Š È13 , È43 ‹. Only ŠÈ3, 0‹ and Š È13 ,
4
È3 ‹
4
È3 ‹,
satisfy x 0 and y 0. For ŠÈ3, 0‹: z œ 0; Vxx ŠÈ3, 0‹ œ 0,
#
Vyy ŠÈ3, 0‹ œ 2È3, Vxy ŠÈ3, 0‹ œ 4È3 Ê Vxx Vyy Vxy
œ 48 0 Ê saddle point. For Š È13 ,
Vxx Š È13 ,
4
È3 ‹
1
œ 380
È3 , Vyy Š È3 ,
4
È3 ‹
Vxx 0 Ê local maximum of VŠ È13 ,
2
œ 3È
, Vxy Š È13 ,
3
4
4
È3 , È3 ‹
œ
4
È3 ‹
4
#
œ 3È
Ê Vxx Vyy Vxy
œ
3
16
3
4
È3 ‹:
zœ
0 and
16
3È 3
60. (a) (i) On x œ 0, f(xß y) œ f(0ß y) œ y# y 1 for 0 Ÿ y Ÿ 1; f w (0ß y) œ 2y 1 œ 0 Ê y œ
f ˆ0ß "# ‰ œ 34 , f(0ß 0) œ 1, and f(0ß 1) œ 1
"
#
and x œ 0;
On y œ 1, f(xß y) œ f(xß 1) œ x# x 1 for 0 Ÿ x Ÿ 1; f w (xß 1) œ 2x 1 œ 0 Ê x œ "# and y œ 1, but
ˆ "# ß 1‰ is outside the domain; f(0ß 1) œ 1 and f("ß ") œ 3
(ii)
(iii) On x œ 1, f(xß y) œ f("ß y) œ y# y 1 for 0 Ÿ y Ÿ 1; f w (1ß y) œ 2y 1 œ 0 Ê y œ "# and x œ 1, but
ˆ1ß "# ‰ is outside the domain; f(1ß 0) œ 1 and f("ß ") œ 3
(iv) On y œ 0, f(xß y) œ f(xß 0) œ x# x 1 for 0 Ÿ x Ÿ 1; f w (xß 0) œ 2x 1 œ 0 Ê x œ
f ˆ "# ß 0‰ œ 34 ; f(0ß 0) œ 1, and f("ß 0) œ 1
"
#
and y œ 0;
On the interior of the square, fx (xß y) œ 2x 2y 1 œ 0 and fy (xß y) œ 2y 2x 1 œ 0 Ê 2x 2y œ 1
Ê (x y) œ "# . Then f(xß y) œ x# y# 2xy x y 1 œ (x y)# (x y) 1 œ 34 is the absolute
(v)
minimum value when 2x 2y œ 1.
(b) The absolute maximum is f("ß ") œ 3.
61. (a)
df
dt
œ
` f dx
` x dt
` f dy
` y dt
œ
dx
dt
dy
dt
œ 2 sin t 2 cos t œ 0 Ê cos t œ sin t Ê x œ y
On the semicircle x# y# œ 4, y
(i)
0, we have t œ
1
4
and x œ y œ È2 Ê f ŠÈ2ß È2‹ œ 2È2. At the
endpoints, f(2ß 0) œ 2 and f(#ß !) œ 2. Therefore the absolute minimum is f(2ß 0) œ 2 when t œ 1;
the absolute maximum is f ŠÈ2ß È2‹ œ 2È2 when t œ 1 .
4
On the quartercircle x# y# œ 4, x 0 and y 0, the endpoints give f(!ß 2) œ 2 and f(#ß 0) œ 2.
Therefore the absolute minimum is f(2ß 0) œ 2 and f(!ß 2) œ 2 when t œ 0, 1# respectively; the absolute
(ii)
maximum is f ŠÈ2ß È2‹ œ 2È2 when t œ
(b)
(i)
dg
dt
œ
` g dx
` x dt
` g dy
` y dt
œy
dx
dt
x
dy
dt
31
4
.
œ 4 sin# t 4 cos# t œ 0 Ê cos t œ „ sin t Ê x œ „ y.
On the semicircle x# y# œ 4, y
tœ
1
4
0, we obtain x œ y œ È2 at t œ
1
4
and x œ È2, y œ È2 at
. Then g ŠÈ2ß È2‹ œ 2 and g ŠÈ2ß È2‹ œ 2. At the endpoints, g(2ß 0) œ g(#ß 0) œ 0.
Therefore the absolute minimum is g ŠÈ2ß È2‹ œ 2 when t œ
g ŠÈ2 ß È2‹ œ 2 when t œ
1
4
31
4
; the absolute maximum is
.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
4
È3 ;
846
Chapter 14 Partial Derivatives
On the quartercircle x# y# œ 4, x 0 and y 0, the endpoints give g(!ß 2) œ 0 and g(#ß 0) œ 0.
Therefore the absolute minimum is g(2ß 0) œ 0 and g(!ß 2) œ 0 when t œ 0, 1# respectively; the absolute
(ii)
maximum is g ŠÈ2ß È2‹ œ 2 when t œ
dh
dt
(c)
œ
` h dx
` x dt
` h dy
` y dt
1
4
.
dy
œ 4x dx
dt 2y dt œ (8 cos t)(2 sin t) (4 sin t)(2 cos t) œ 8 cos t sin t œ 0
Ê t œ 0, 1# , 1 yielding the points (2ß 0), (0ß 2) for 0 Ÿ t Ÿ 1.
On the semicircle x# y# œ 4, y 0 we have h(2ß 0) œ 8, h(0ß 2) œ 4, and h(2ß 0) œ 8. Therefore,
the absolute minimum is h(!ß 2) œ 4 when t œ 1# ; the absolute maximum is h(2ß 0) œ 8 and h(2ß 0) œ 8
(i)
when t œ 0, 1 respectively.
On the quartercircle x# y# œ 4, x
(ii)
0 and y
0 the absolute minimum is h(0ß 2) œ 4 when t œ
absolute maximum is h(2ß 0) œ 8 when t œ 0.
df
dt
62. (a)
(i)
œ
` f dx
` x dt
` f dy
` y dt
1
4
x#
9
y#
4
œ 1, y
0, f(xß y) œ 2x 3y œ 6 cos t 6 sin t œ
È
1
4
.
On the quarter ellipse, at the endpoints f(0ß 2) œ 6 and f(3ß 0) œ 6. The absolute minimum is f(3ß 0) œ 6
È
and f(0ß 2) œ 6 when t œ 0, 1 respectively; the absolute maximum is f Š 3 2 ß È2‹ œ 6È2 when t œ 1 .
(ii)
#
` g dy
dx
œ `` gx dx
dt ` y dt œ y dt
Ê t œ 14 , 341 for 0 Ÿ t Ÿ
dg
dt
x
dy
dt
#
1.
È
31
4
. At the endpoints, g(3ß 0) œ g($ß 0) œ 0. The absolute minimum is
È
31
4
; the absolute maximum is g Š 3 # 2 ß È2‹ œ 3 when t œ
#
dh
dt
œ
` h dx
` x dt
Ê t œ 0,
(i)
(ii)
œ
(ii)
, and
È
1
4
.
On the quarter ellipse, at the endpoints g(!ß 2) œ 0 and g($ß 0) œ 0. The absolute minimum is g(3ß 0) œ 0
È
and g(0ß 2) œ 0 at t œ 0, 1 respectively; the absolute maximum is g Š 3 2 ß È2‹ œ 3 when t œ 1 .
(ii)
(i)
1
4
È
g Š 3 # 2 ß È2‹ œ 3 when t œ
df
dt
#
œ (2 sin t)(3 sin t) (3 cos t)(2 cos t) œ 6 acos t sin tb œ 6 cos 2t œ 0
g Š 3 # 2 ß È2‹ œ 3 when t œ
63.
4
#
On the semi-ellipse, g(xß y) œ xy œ 6 sin t cos t. Then g Š 3 # 2 ß È2‹ œ 3 when t œ
(i)
(c)
œ 6È 2
. At the endpoints, f(3ß 0) œ 6 and f(3ß 0) œ 6. The absolute minimum is f(3ß 0) œ 6 when
t œ 1; the absolute maximum is f Š 3 # 2 ß È2‹ œ 6È2 when t œ
(b)
1
4 for 0 Ÿ t Ÿ 1.
È
È
6 Š #2 ‹ 6 Š #2 ‹
; the
dy
œ 2 dx
dt 3 dt œ 6 sin t 6 cos t œ 0 Ê sin t œ cos t Ê t œ
On the semi-ellipse,
at t œ
1
#
1
#
` h dy
` y dt
œ 2x
dx
dt
6y
#
dy
dt
4
œ (6 cos t)(3 sin t) (12 sin t)(2 cos t) œ 6 sin t cos t œ 0
, 1 for 0 Ÿ t Ÿ 1, yielding the points (3ß 0), (0ß 2), and (3ß 0).
On the semi-ellipse, y 0 so that h(3ß 0) œ 9, h(0ß 2) œ 12, and h(3ß 0) œ 9. The absolute minimum is
h(3ß 0) œ 9 and h(3ß 0) œ 9 when t œ 0, 1 respectively; the absolute maximum is h(!ß 2) œ 12 when t œ
On the quarter ellipse, the absolute minimum is h(3ß 0) œ 9 when t œ 0; the absolute maximum is
h(!ß 2) œ 12 when t œ 1# .
` f dx
` x dt
` f dy
` y dt
1
#
dy
œ y dx
dt x dt
"
"
x œ 2t and y œ t 1 Ê df
dt œ (t 1)(2) (2t)(1) œ 4t 2 œ 0 Ê t œ # Ê x œ 1 and y œ # with
f ˆ1ß "# ‰ œ "# . The absolute minimum is f ˆ1ß "# ‰ œ "# when t œ "# ; there is no absolute maximum.
For the endpoints: t œ 1 Ê x œ 2 and y œ 0 with f(2ß 0) œ 0; t œ 0 Ê x œ 0 and y œ 1 with
f(!ß 1) œ 0. The absolute minimum is f ˆ1ß "# ‰ œ "# when t œ "# ; the absolute maximum is f(0ß 1) œ 0
and f(#ß 0) œ 0 when t œ 1, 0 respectively.
(iii) There are no interior critical points. For the endpoints: t œ 0 Ê x œ 0 and y œ 1 with f(0ß 1) œ 0;
t œ 1 Ê x œ 2 and y œ 2 with f(2ß 2) œ 4. The absolute minimum is f(0ß 1) œ 0 when t œ 0; the absolute
maximum is f(2ß 2) œ 4 when t œ 1.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
.
Section 14.7 Extreme Values and Saddle Points
df
dt
64. (a)
` f dx
` x dt
œ
` f dy
` y dt
dy
œ 2x dx
dt 2y dt
4
4
x œ t and y œ 2 2t Ê df
dt œ (2t)(1) 2(2 2t)(2) œ 10t 8 œ 0 Ê t œ 5 Ê x œ 5 and y œ
4
f ˆ 45 ß 25 ‰ œ "#65 25
œ 45 . The absolute minimum is f ˆ 45 ß 25 ‰ œ 45 when t œ 45 ; there is no absolute
(i)
2
5
with
maximum along the line.
For the endpoints: t œ 0 Ê x œ 0 and y œ 2 with f(0ß 2) œ 4; t œ 1 Ê x œ 1 and y œ 0 with f(1ß 0) œ 1.
The absolute minimum is f ˆ 45 ß 25 ‰ œ 45 at the interior critical point when t œ 45 ; the absolute maximum is
(ii)
f(0ß 2) œ 4 at the endpoint when t œ 0.
œ
dg
dt
(b)
` g dx
` x dt
` g dy
` y dt
œ ’ ax#2xy# b# “
’ ax#2yy# b# “
dx
dt
dy
dt
x œ t and y œ 2 2t Ê x# y# œ 5t# 8t 4 Ê
(i)
#
œ a5t# 8t 4b (10t 8) œ 0 Ê t œ
maximum is g ˆ 45 ß 25 ‰ œ
5
4
when t œ
4
5
4
5
dg
dt
#
œ a5t# 8t 4b [(2t)(1) (2)(2 2t)(2)]
Ê xœ
4
5
and y œ
"
4
The absolute minimum is g(0ß 2) œ
with g ˆ 45 ß 25 ‰ œ
"
ˆ 45 ‰
œ
5
4
. The absolute
; there is no absolute minimum along the line since x and y can be
as large as we please.
For the endpoints: t œ 0 Ê x œ 0 and y œ 2 with g(0ß 2) œ
(ii)
2
5
"
4
; t œ 1 Ê x œ 1 and y œ 0 with g(1ß 0) œ 1.
when t œ 0; the absolute maximum is g ˆ 45 ß 52 ‰ œ 45 when t œ 54 .
65. w œ am x1 b y1 b2 am x2 b y2 b2 â am xn b yn b2
w
Ê `` m
œ 2am x1 b y1 bax1 b 2am x2 b y2 bax2 b â 2am xn b yn baxn b
Ê
`w
`m
`w
`b
œ 2am x1 b y1 ba1b 2am x2 b y2 ba1b â 2am xn b yn ba1b
œ 0 Ê 2am x1 b y1 bax1 b am x2 b y2 bax2 b â am xn b yn baxn b‘ œ 0
Ê m x21 b x1 x1 y1 m x#2 b x2 x2 y2 â m xn2 b xn xn yn œ 0
Ê max21 x2# â xn2 b bax1 x2 â xn b ax1 y1 x2 y2 â xn yn b œ 0
n
n
n
k œ1
k œ1
k œ1
Ê m! ax2k b b! xk ! axk yk b œ 0
`w
`b
œ 0 Ê 2am x1 b y1 b am x2 b y2 b â am xn b yn b‘ œ 0
Ê m x1 b y1 m x2 b y2 â m xn b yn œ 0
Ê max1 x2 â xn b ab b â bb ay1 y2 â yn b œ 0
n
n
n
n
n
k œ1
k œ1
k œ1
k œ1
k œ1
n
n
kœ1
n
kœ 1
n
n
n
k œ1
k œ1
k œ1
Ê m ! xk b ! 1 ! yk œ 0 Ê m ! xk bn ! yk œ 0 Ê b œ 1n Œ ! yk m! xk .
Substituting for b in the equation obtained for
`w
`m
n
we get m ! ax2k b 1n Œ ! yk m! xk ! xk ! axk yk b œ 0.
n
k œ1
n
n
k œ1
n
n
k œ1
kœ1
Multiply both sides by n to obtain m n ! ax2k b Œ ! yk m! xk ! xk n ! axk yk b œ 0
k œ1
n
n
k œ1
k œ1
k œ1
n
2
n
k œ1
n
Ê m n ! ax2k b Œ ! xk Œ ! yk mŒ ! xk n ! axk yk b œ 0
n
k œ1
2
n
k œ1
kœ1
n
n
n
k œ1
k œ1
k œ1
n
n
n
Ê m n ! ax2k b mŒ ! xk œ n ! axk yk b Œ ! xk Œ ! yk
kœ1
kœ1
n
2
n
Ê m–n! ax2k b Œ ! xk — œ n ! axk yk b Œ ! xk Œ ! yk
k œ1
k œ1
n
Êmœ
k œ1
n
n
n ! axk yk bΠ! xk Π! yk
kœ1
kœ1
n
n
kœ1
kœ1
kœ1
2
n! ax2k bΠ! xk
k œ1
n
œ
n
kœ1
n
Π! xk Π! yk n ! axk yk b
kœ1
kœ1
n
kœ1
2
n
2
Π! x k n ! ax k b
kœ1
kœ1
To show that these values for m and b minimize the sum of the squares of the distances, use second derivative test.
` 2w
` m2
n
œ 2 x21 2 x#2 â 2 x2n œ 2 ! ax2k b;
k œ1
` 2w
`m `b
n
œ 2 x1 2 x2 â 2 xn œ 2! xk ;
k œ1
` 2w
` b2
œ 2 2 â 2 œ 2n
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
847
848
Chapter 14 Partial Derivatives
2
n
n
kœ1
kœ1
#
n
n
kœ1
k œ1
2
The discriminant is: Š `` mw2 ‹Š `` bw2 ‹ Š ``m w` b ‹ œ ”2 ! ax2k b•a2 nb ”2 ! xk • œ 4–n ! ax2k b Œ ! xk —.
2
n
2
2
2
n
Now, n ! ax2k b Œ ! xk œ nax12 x#2 â x#n b ax1 x2 â xn bax1 x2 â xn b
k œ1
kœ1
2
œ
n x# â n x#n x21 x1 x2 â x1 xn x2 x1 x2# â x2 xn xn x1 xn x2 â x#n
œ an 1b x21 an 1b x2# â an 1b x#n 2 x1 x2 2 x1 x3 â 2 x1 xn 2 x2 x3 â 2 x2 xn â 2 xn1 xn
œ a x21 2 x1 x2 x#2 b a x12 2 x1 x3 x23 b â ax21 2 x1 xn xn# b ax2# 2 x2 x3 x23 b â a x#2 2 x2 xn xn# b
â ax2n1 2 xn1 xn x#n b
œ ax1 x2 b2 ax1 x3 b2 â ax1 xn b2 ax2 x3 b2 â ax2 xn b2 â axn1 xn b2 0.
2
n
n
2
2
2
2
Thus we have : Š `` mw2 ‹Š `` bw2 ‹ Š ``m w` b ‹ œ 4–n ! ax2k b Œ ! xk — 4a0b œ 0. If x1 œ x2 œ â œ xn then
kœ1
k œ1
n x21
2
Š `` mw2 ‹Š `` bw2 ‹ Š ``m w` b ‹ œ 0. Also,
2
2
2
` 2w
` m2
n
œ 2 ! ax2k b
k œ1
` 2w
` m2
0. If x1 œ x2 œ â œ xn œ 0, then
œ 0.
2
Provided that at least one xi is nonzero and different from the rest of xj , j Á i, then Š `` mw2 ‹Š `` bw2 ‹ Š ``m w` b ‹ 0 and
2
` 2w
` m2
bœ
Ê
67. m œ
bœ
Ê
68. m œ
Ê
2
0 Ê the values given above for m and b minimize w.
66. m œ
bœ
2
(0)(5) 3(6)
3
(0)# 3(8) œ 4 and
"
3
‘ 5
3 5 4 (0) œ 3
y œ 34 x 53 ; y¸ xœ4
œ
14
3
(2)(1) 3("4)
œ 20
(2)# 3(10)
13 and
"
20
9
ˆ
‰ ‘
3 1 13 (2) œ 13
9
¸
y œ 20
13 x 13 ; y xœ4 œ
(3)(5) 3(8)
3
(3)# 3(5) œ 2 and
"
3
‘ 1
3 5 2 (3) œ 6
y œ 32 x 16 ; y¸ xœ4
œ
37
6
71
13
k
1
2
3
D
xk
2
0
2
0
yk
0
2
3
5
x#k
4
0
4
8
xk yk
0
0
6
6
k
1
2
3
D
xk
1
0
3
2
yk
2
1
4
1
x#k
1
0
9
10
xk yk
2
0
12
14
k
1
2
3
D
xk
0
1
2
3
yk
0
2
3
5
x#k
0
1
4
5
xk yk
0
2
6
8
69-74. Example CAS commands:
Maple:
f := (x,y) -> x^2+y^3-3*x*y;
x0,x1 := -5,5;
y0,y1 := -5,5;
plot3d( f(x,y), x=x0..x1, y=y0..y1, axes=boxed, shading=zhue, title="#69(a) (Section 14.7)" );
plot3d( f(x,y), x=x0..x1, y=y0..y1, grid=[40,40], axes=boxed, shading=zhue, style=patchcontour, title="#69(b)
(Section 14.7)" );
fx := D[1](f);
# (c)
fy := D[2](f);
crit_pts := solve( {fx(x,y)=0,fy(x,y)=0}, {x,y} );
fxx := D[1](fx);
# (d)
fxy := D[2](fx);
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.8 Lagrange Multipliers
849
fyy := D[2](fy);
discr := unapply( fxx(x,y)*fyy(x,y)-fxy(x,y)^2, (x,y) );
for CP in {crit_pts} do
# (e)
eval( [x,y,fxx(x,y),discr(x,y)], CP );
end do;
# (0,0) is a saddle point
# ( 9/4, 3/2) is a local minimum
Mathematica: (assigned functions and bounds will vary)
Clear[x,y,f]
f[x_,y_]:= x2 y3 3x y
xmin= 5; xmax= 5; ymin= 5; ymax= 5;
Plot3D[f[x,y], {x, xmin, xmax}, {y, ymin, ymax}, AxesLabel Ä {x, y, z}]
ContourPlot[f[x,y], {x, xmin, xmax}, {y, ymin, ymax}, ContourShading Ä False, Contours Ä 40]
fx= D[f[x,y], x];
fy= D[f[x,y], y];
critical=Solve[{fx==0, fy==0},{x, y}]
fxx= D[fx, x];
fxy= D[fx, y];
fyy= D[fy, y];
discriminant= fxx fyy fxy2
{{x, y}, f[x, y], discriminant, fxx} /.critical
14.8 LAGRANGE MULTIPLIERS
1.
™ f œ yi xj and ™ g œ 2xi 4yj so that ™ f œ - ™ g Ê yi xj œ -(2xi 4yj) Ê y œ 2x- and x œ 4yÊ x œ 8x-# Ê - œ „
È2
4
or x œ 0.
CASE 1: If x œ 0, then y œ 0. But (0ß 0) is not on the ellipse so x Á 0.
CASE 2: x Á 0 Ê - œ „
È2
4
Therefore f takes on its extreme values at Š „
are „
2.
È2
#
#
Ê x œ „ È2y Ê Š „ È2y‹ 2y# œ 1 Ê y œ „ "# .
È2 "
2 ß #‹
and Š „
È2
"
2 ß #‹ .
The extreme values of f on the ellipse
.
™ f œ yi xj and ™ g œ 2xi 2yj so that ™ f œ - ™ g Ê yi xj œ -(2xi 2yj) Ê y œ 2x- and x œ 2yÊ x œ 4x-# Ê x œ 0 or - œ „ 12 .
CASE 1: If x œ 0, then y œ 0. But (0ß 0) is not on the circle x# y# 10 œ 0 so x Á 0.
CASE 2: x Á 0 Ê - œ „ 12 Ê y œ 2x ˆ „ "# ‰ œ „ x Ê x# a „ xb# 10 œ 0 Ê x œ „ È5 Ê y œ „ È5.
Therefore f takes on its extreme values at Š „ È5ß È5‹ and Š „ È5ß È5‹ . The extreme values of f on the
circle are 5 and 5.
3.
™ f œ 2xi 2yj and ™ g œ i 3j so that ™ f œ - ™ g Ê 2xi 2yj œ -(i 3j) Ê x œ -# and y œ 3#Ê ˆ -# ‰ 3 ˆ 3#- ‰ œ 10 Ê - œ 2 Ê x œ 1 and y œ 3 Ê f takes on its extreme value at (1ß 3) on the line.
The extreme value is f("ß $) œ 49 1 9 œ 39.
4.
™ f œ 2xyi x# j and ™ g œ i j so that ™ f œ - ™ g Ê 2xyi x# j œ -(i j) Ê 2xy œ - and x# œ Ê 2xy œ x# Ê x œ 0 or 2y œ x.
CASE 1: If x œ 0, then x y œ 3 Ê y œ 3.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
850
Chapter 14 Partial Derivatives
CASE 2: If x Á 0, then 2y œ x so that x y œ 3 Ê 2y y œ 3 Ê y œ 1 Ê x œ 2.
Therefore f takes on its extreme values at (!ß 3) and (2ß "). The extreme values of f are f(0ß 3) œ 0 and f(2ß 1) œ 4.
5. We optimize f(xß y) œ x# y# , the square of the distance to the origin, subject to the constraint
g(xß y) œ xy# 54 œ 0. Thus ™ f œ 2xi 2yj and ™ g œ y# i 2xyj so that ™ f œ - ™ g Ê 2xi 2yj
œ - ay# i 2xyjb Ê 2x œ -y# and 2y œ 2-xy.
CASE 1: If y œ 0, then x œ 0. But (0ß 0) does not satisfy the constraint xy# œ 54 so y Á 0.
CASE 2: If y Á 0, then 2 œ 2-x Ê x œ -" Ê 2 ˆ -" ‰ œ -y# Ê y# œ -2# . Then xy# œ 54 Ê ˆ -" ‰ ˆ -2# ‰ œ 54
Ê -$ œ " Ê - œ " Ê x œ 3 and y# œ 18 Ê x œ 3 and y œ „ 3È2.
27
3
Therefore Š$ß „ 3È2‹ are the points on the curve xy# œ 54 nearest the origin (since xy# œ 54 has points increasingly
far away as y gets close to 0, no points are farthest away).
6. We optimize f(xß y) œ x# y# , the square of the distance to the origin subject to the constraint g(xß y) œ x# y 2 œ 0.
Thus ™ f œ 2xi 2yj and ™ g œ 2xyi x# j so that ™ f œ - ™ g Ê 2x œ 2xy- and 2y œ x# - Ê - œ 2y
x# , since
2y
x œ 0 Ê y œ 0 (but g(0ß 0) Á 0). Thus x Á 0 and 2x œ 2xy ˆ x# ‰ Ê x# œ 2y# Ê a2y# b y 2 œ 0 Ê y œ 1 (since
y 0) Ê x œ „ È2 . Therefore Š „ È2ß 1‹ are the points on the curve x# y œ 2 nearest the origin (since x# y œ 2 has
points increasingly far away as x gets close to 0, no points are farthest away).
7. (a) ™ f œ i j and ™ g œ yi xj so that ™ f œ - ™ g Ê i j œ -(yi xj) Ê 1 œ -y and 1 œ -x Ê y œ
xœ
"
-
Ê
"
-#
œ 16 Ê - œ „
"
4.
Use - œ
"
4
"
-
and
since x 0 and y 0. Then x œ 4 and y œ 4 Ê the minimum value is 8
at the point (4ß 4). Now, xy œ 16, x 0, y 0 is a branch of a hyperbola in the first quadrant with the x-and y-axes
as asymptotes. The equations x y œ c give a family of parallel lines with m œ 1. As these lines move away from
the origin, the number c increases. Thus the minimum value of c occurs where x y œ c is tangent to the hyperbola's
branch.
(b) ™ f œ yi xj and ™ g œ i j so that ™ f œ - ™ g Ê yi xj œ -(i j) Ê y œ - œ x y y œ 16 Ê y œ 8
Ê x œ 8 Ê f()ß )) œ 64 is the maximum value. The equations xy œ c (x 0 and y 0 or x 0 and y 0
to get a maximum value) give a family of hyperbolas in the first and third quadrants with the x- and y-axes as
asymptotes. The maximum value of c occurs where the hyperbola xy œ c is tangent to the line x y œ 16.
8. Let f(xß y) œ x# y# be the square of the distance from the origin. Then ™ f œ 2xi 2yj and
™ g œ (2x y)i (2y x)j so that ™ f œ - ™ g Ê 2x œ -(2x y) and 2y œ -(2y x) Ê
2y
2yx
œ-
Ê 2x œ Š 2y2yx ‹ (2x y) Ê x(2y x) œ y(2x y) Ê x# œ y# Ê y œ „ x.
CASE 1: y œ x Ê x# x(x) x# 1 œ 0 Ê x œ „
"
È3
and y œ x.
CASE 2: y œ x Ê x# x(x) (x)# 1 œ 0 Ê x œ „ 1 and y œ x. Thus f Š È"3 ß È"3 ‹ œ
2
3
œ f Š È"3 ß È"3 ‹ and f(1ß 1) œ 2 œ f(1ß 1).
Therefore the points (1ß 1) and (1ß 1) are the farthest away; Š È"3 ß È"3 ‹ and Š È"3 ß È"3 ‹ are the closest
points to the origin.
9. V œ 1r# h Ê 161 œ 1r# h Ê 16 œ r# h Ê g(rß h) œ r# h 16; S œ 21rh 21r# Ê ™ S œ (21h 41r)i 21rj and
™ g œ 2rhi r# j so that ™ S œ - ™ g Ê (21rh 41r)i 21rj œ - a2rhi r# jb Ê 21rh 41r œ 2rh- and 21r œ -r#
Ê r œ 0 or - œ 2r1 . But r œ 0 gives no physical can, so r Á 0 Ê - œ 2r1 Ê 21h 41r œ 2rh ˆ 2r1 ‰ Ê 2r œ h
Ê 16 œ r# (2r) Ê r œ 2 Ê h œ 4; thus r œ 2 cm and h œ 4 cm give the only extreme surface area of 241 cm# . Since
r œ 4 cm and h œ 1 cm Ê V œ 161 cm$ and S œ 401 cm# , which is a larger surface area, then 241 cm# must be the
minimum surface area.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.8 Lagrange Multipliers
851
10. For a cylinder of radius r and height h we want to maximize the surface area S œ 21rh subject to the constraint
#
g(rß h) œ r# ˆ h# ‰ a# œ 0. Thus ™ S œ 21hi 21rj and ™ g œ 2ri h# j so that ™ S œ - ™ g Ê 21h œ 2-r and
21r œ
-h
#
Ê
1h
r
4r#
4
œ - and 21r œ ˆ 1rh ‰ ˆ #h ‰ Ê 4r# œ h# Ê h œ 2r Ê r#
œ a# Ê 2r# œ a# Ê r œ
a
È2
Ê h œ aÈ2 Ê S œ 21 Š Èa2 ‹ ŠaÈ2‹ œ 21a# .
#
#
x
11. A œ (2x)(2y) œ 4xy subject to g(xß y) œ 16
y9 1 œ 0; ™ A œ 4yi 4xj and ™ g œ x8 i 2y
9 j so that ™ A
2y
2y
32y
x
x
‰ ˆ 32y
‰
œ - ™ g Ê 4yi 4xj œ - ˆ 8 i 9 j‰ Ê 4y œ ˆ 8 ‰ - and 4x œ ˆ 9 ‰ - Ê - œ x and 4x œ ˆ 2y
9
x
Ê y œ „ 34 x Ê
Then y œ
3
4
x#
16
Š2È2‹ œ
ˆ „43 x‰#
œ 1 Ê x#
9
3È 2
# , so the length is
12. P œ 4x 4y subject to g(xß y) œ
x#
a#
y#
b#
and height œ 2y œ
2b#
È a# b#
2x œ 4È2 and the width is 2y œ 3È2.
1 œ 0; ™ P œ 4i 4j and ™ g œ
‰
ˆ 2y ‰
Ê 4 œ ˆ 2x
a# - and 4 œ b# - Ê - œ
œ 1 Ê aa# b# b x# œ a% Ê x œ
œ 8 Ê x œ „ 2È2 . We use x œ 2È2 since x represents distance.
2a#
x
a#
È a# b#
#
2x
a#
i
#
b
‰ 2a
and 4 œ ˆ 2y
b# Š x ‹ Ê y œ Š a# ‹ x Ê
#
, since x 0 Ê y œ Š ba# ‹ x œ
Ê perimeter is P œ 4x 4y œ
4a# 4b#
È a# b#
b#
È a# b#
2y
b#
x#
a#
j so that ™ P œ - ™ g
#
#
Š ba# ‹ x#
b#
œ1 Ê
Ê width œ 2x œ
x#
a#
b# x#
a%
2a#
È a# b#
œ 4Èa# b#
13. ™ f œ 2xi 2yj and ™ g œ (2x 2)i (2y 4)j so that ™ f œ - ™ g œ 2xi 2yj œ -[(2x 2)i (2y 4)j]
2#
#
Ê 2x œ -(2x 2) and 2y œ -(2y 4) Ê x œ -
1 and y œ -1 , - Á 1 Ê y œ 2x Ê x 2x (2x) 4(2x) œ 0
Ê x œ 0 and y œ 0, or x œ 2 and y œ 4. Therefore f(0ß 0) œ 0 is the minimum value and f(2ß 4) œ 20 is the maximum
value. (Note that - œ 1 gives 2x œ 2x 2 or ! œ 2, which is impossible.)
14. ™ f œ 3i j and ™ g œ 2xi 2yj so that ™ f œ - ™ g Ê 3 œ 2-x and 1 œ 2-y Ê - œ
#
Ê y œ x3 Ê x# ˆ x3 ‰ œ 4 Ê 10x# œ 36 Ê x œ „
yœ
2
È10 .
Therefore f Š È610 ß È210 ‹ œ
20
È10
6
È10
Ê xœ
6
È10
3
2x
3 ‰
and 1 œ 2 ˆ 2x
y
and y œ È210 , or x œ È610 and
6 œ 2È10 6 ¸ 12.325 is the maximum value, and f Š È610 ß È210 ‹
œ 2È10 6 ¸ 0.325 is the minimum value.
15. ™ T œ (8x 4y)i (4x 2y)j and g(xß y) œ x# y# 25 œ 0 Ê ™ g œ 2xi 2yj so that ™ T œ - ™ g
Ê (8x 4y)i (4x 2y)j œ -(2xi 2yj) Ê 8x 4y œ 2-x and 4x 2y œ 2-y Ê y œ -2x1 , - Á 1
Ê 8x 4 ˆ -2x1 ‰ œ 2-x Ê x œ 0, or - œ 0, or - œ 5.
CASE 1: x œ 0 Ê y œ 0; but (0ß 0) is not on x# y# œ 25 so x Á 0.
CASE 2: - œ 0 Ê y œ 2x Ê x# (2x)# œ 25 Ê x œ „ È5 and y œ 2x.
CASE 3: - œ 5 Ê y œ
and y œ È5 .
2x
4
#
œ #x Ê x# ˆ #x ‰ œ 25 Ê x œ „ 2È5 Ê x œ 2È5 and y œ È5, or x œ 2È5
Therefore T ŠÈ5ß 2È5‹ œ 0° œ T ŠÈ5ß 2È5‹ is the minimum value and T Š2È5ß È5‹ œ 125°
œ T Š2È5ß È5‹ is the maximum value. (Note: - œ 1 Ê x œ 0 from the equation 4x 2y œ 2-y; but we
found x Á 0 in CASE 1.)
16. The surface area is given by S œ 41r# 21rh subject to the constraint V(rß h) œ
#
4
3
1r$ 1r# h œ 8000. Thus
#
™ S œ (81r 21h)i 21rj and ™ V œ a41r 21rhb i 1r j so that ™ S œ - ™ V œ (81r 21h)i 21rj
œ - ca41r# 21rhb i 1r# jd Ê 81r 21h œ - a41r# 21rhb and 21r œ -1r# Ê r œ 0 or 2 œ r-. But r Á 0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
852
Chapter 14 Partial Derivatives
so 2 œ r- Ê - œ
4
3
2
r
Ê 4r h œ
1r$ œ 8000 Ê r œ 10 ˆ 16 ‰
"Î$
2
r
a2r# rhb Ê h œ 0 Ê the tank is a sphere (there is no cylindrical part) and
.
17. Let f(xß yß z) œ (x 1)# (y 1)# (z 1)# be the square of the distance from (1ß 1ß 1). Then
™ f œ 2(x 1)i 2(y 1)j 2(z 1)k and ™ g œ i 2j 3k so that ™ f œ - ™ g
Ê 2(x 1)i 2(y 1)j 2(z 1)k œ -(i 2j 3k) Ê 2(x 1) œ -, 2(y 1) œ 2-, 2(z 1) œ 3Ê 2(y 1) œ 2[2(x 1)] and 2(z 1) œ 3[2(x 1)] Ê x œ y # 1 Ê z 2 œ 3 ˆ y # 1 ‰ or z œ 3y # 1 ; thus
y1
ˆ 3y # 1 ‰ 13 œ 0 Ê y œ 2 Ê x œ 3# and z œ #5 . Therefore the point ˆ #3 ß 2ß 5# ‰ is closest (since no
# 2y 3
point on the plane is farthest from the point (1ß 1ß 1)).
18. Let f(xß yß z) œ (x 1)# (y 1)# (z 1)# be the square of the distance from (1ß 1ß 1). Then
™ f œ 2(x 1)i 2(y 1)j 2(z 1)k and ™ g œ 2xi 2yj 2zk so that ™ f œ - ™ g Ê x 1 œ -x, y 1 œ -y
#
‰# ˆ 1 " - ‰# œ 4
and z 1 œ -z Ê x œ 1 " - , y œ 1 " - , and z œ 1" - for - Á 1 Ê ˆ 1 " - ‰ ˆ 1"
Ê
"
"-
œ „
2
È3
Ê xœ
2
È3
, y œ È23 , z œ
2
È3
or x œ È23 , y œ
2
È3
, z œ È23 . The largest value of f
occurs where x 0, y 0, and z 0 or at the point Š È23 ß È23 ß È23 ‹ on the sphere.
19. Let f(xß yß z) œ x# y# z# be the square of the distance from the origin. Then ™ f œ 2xi 2yj 2zk and
™ g œ 2xi 2yj 2zk so that ™ f œ - ™ g Ê 2xi 2yj 2zk œ -(2xi 2yj 2zk) Ê 2x œ 2x-, 2y œ 2y-,
and 2z œ 2z- Ê x œ 0 or - œ 1.
CASE 1: - œ 1 Ê 2y œ 2y Ê y œ 0; 2z œ 2z Ê z œ 0 Ê x# 1 œ 0 Ê x# 1 œ 0 Ê x œ „ 1 and y œ z œ 0.
CASE 2: x œ 0 Ê y# z# œ 1, which has no solution.
Therefore the points on the unit circle x# y# œ 1, are the points on the surface x# y# z# œ 1 closest to the originÞ
The minimum distance is 1.
20. Let f(xß yß z) œ x# y# z# be the square of the distance to the origin. Then ™ f œ 2xi 2yj 2zk and
™ g œ yi xj k so that ™ f œ - ™ g Ê 2xi 2yj 2zk œ -(yi xj k) Ê 2x œ -y, 2y œ -x, and 2z œ Ê xœ
-y
#
Ê 2y œ - Š -#y ‹ Ê y œ 0 or - œ „ 2.
CASE 1: y œ 0 Ê x œ 0 Ê z 1 œ 0 Ê z œ 1.
CASE 2: - œ 2 Ê x œ y and z œ 1 Ê x# (1) 1 œ 0 Ê x# 2 œ 0, so no solution.
CASE 3: - œ 2 Ê x œ y and z œ 1 Ê (y)y 1 1 œ 0 Ê y œ 0, again.
Therefore (0ß 0ß 1) is the point on the surface closest to the origin since this point gives the only extreme value
and there is no maximum distance from the surface to the origin.
21. Let f(xß yß z) œ x# y# z# be the square of the distance to the origin. Then ™ f œ 2xi 2yj 2zk and
™ g œ yi xj 2zk so that ™ f œ - ™ g Ê 2xi 2yj 2zk œ -(yi xj 2zk) Ê 2x œ y-, 2y œ x-, and
2z œ 2z- Ê - œ 1 or z œ 0.
CASE 1: - œ 1 Ê 2x œ y and 2y œ x Ê y œ 0 and x œ 0 Ê z# 4 œ 0 Ê z œ „ 2 and x œ y œ 0.
CASE 2: z œ 0 Ê xy 4 œ 0 Ê y œ 4x . Then 2x œ
4
x
- Ê -œ
x#
#
#
, and x8 œ x- Ê x8 œ x Š x# ‹
Ê x% œ 16 Ê x œ „ 2. Thus, x œ 2 and y œ 2, or x = 2 and y œ 2.
Therefore we get four points: (#ß 2ß 0), (2ß 2ß 0), (0ß 0ß 2) and (!ß 0ß 2). But the points (!ß 0ß 2) and (!ß !ß 2)
are closest to the origin since they are 2 units away and the others are 2È2 units away.
22. Let f(xß yß z) œ x# y# z# be the square of the distance to the origin. Then ™ f œ 2xi 2yj 2zk and
™ g œ yzi xzj xyk so that ™ f œ - ™ g Ê 2x œ -yz, 2y œ -xz, and 2z œ -xy Ê 2x# œ -xyz and 2y# œ -yxz
Ê x# œ y# Ê y œ „ x Ê z œ „ x Ê x a „ xb a „ xb œ 1 Ê x œ „ 1 Ê the points are (1ß 1ß 1), ("ß 1ß 1),
("ß "ß "), and (1ß 1, 1).
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.8 Lagrange Multipliers
853
23. ™ f œ i 2j 5k and ™ g œ 2xi 2yj 2zk so that ™ f œ - ™ g Ê i 2j 5k œ -(2xi 2yj 2zk) Ê 1 œ 2x-,
2 œ 2y-, and 5 œ 2z- Ê x œ #"- , y œ -" œ 2x, and z œ #5- œ 5x Ê x# (2x)# (5x)# œ 30 Ê x œ „ 1.
Thus, x œ 1, y œ 2, z œ 5 or x œ 1, y œ 2, z œ 5. Therefore f(1ß 2ß 5) œ 30 is the maximum value and
f(1ß 2ß 5) œ 30 is the minimum value.
24. ™ f œ i 2j 3k and ™ g œ 2xi 2yj 2zk so that ™ f œ - ™ g Ê i 2j 3k œ -(2xi 2yj 2zk) Ê 1 œ 2x-,
2 œ 2y-, and 3 œ 2z- Ê x œ #"- , y œ -" œ 2x, and z œ #3- œ 3x Ê x# (2x)# (3x)# œ 25 Ê x œ „ È514 .
Thus, x œ
5
È14
,yœ
10
È14
,zœ
15
È14
or x œ È514 , y œ È1014 , z œ È1514 . Therefore f Š È514 ß È1014 ß È1514 ‹
œ 5È14 is the maximum value and f Š È514 ß È1014 , È1514 ‹ œ 5È14 is the minimum value.
25. f(xß yß z) œ x# y# z# and g(xß yß z) œ x y z 9 œ 0 Ê ™ f œ 2xi 2yj 2zk and ™ g œ i j k so that
™ f œ - ™ g Ê 2xi 2yj 2zk œ -(i j k) Ê 2x œ -, 2y œ -, and 2z œ - Ê x œ y œ z Ê x x x 9 œ 0
Ê x œ 3, y œ 3, and z œ 3.
26. f(xß yß z) œ xyz and g(xß yß z) œ x y z# 16 œ 0 Ê ™ f œ yzi xzj xyk and ™ g œ i j 2zk so that
™ f œ - ™ g Ê yzi xzj xyk œ -(i j 2zk) Ê yz œ -, xz œ -, and xy œ 2z- Ê yz œ xz Ê z œ 0 or y œ x.
But z 0 so that y œ x Ê x# œ 2z- and xz œ -. Then x# œ 2z(xz) Ê x œ 0 or x œ 2z# . But x 0 so that
32
x œ 2z# Ê y œ 2z# Ê 2z# 2z# z# œ 16 Ê z œ „ È45 . We use z œ È45 since z 0. Then x œ 32
5 and y œ 5
32
4
which yields f Š 32
5 ß 5 ß È5 ‹ œ
4096
25È5
.
27. V œ xyz and g(xß yß z) œ x# y# z# 1 œ 0 Ê ™ V œ yzi xzj xyk and ™ g œ 2xi 2yj 2zk so that
™ V œ - ™ g Ê yz œ -x, xz œ -y, and xy œ -z Ê xyz œ -x# and xyz œ -y# Ê y œ „ x Ê z œ „ x
Ê x# x# x# œ 1 Ê x œ È"3 since x 0 Ê the dimensions of the box are È13 by È13 by È13 for maximum
volume. (Note that there is no minimum volume since the box could be made arbitrarily thin.)
28. V œ xyz with xß yß z all positive and
x
a
y
b
z
c
œ 1; thus V œ xyz and g(xß yß z) œ bcx acy abz abc œ 0
Ê ™ V œ yzi xzj xyk and ™ g œ bci acj abk so that ™ V œ - ™ g Ê yz œ -bc, xz œ -ac, and xy œ -ab
Ê xyz œ -bcx, xyz œ -acy, and xyz œ -abz Ê - Á 0. Also, -bcx œ -acy œ -abz Ê bx œ ay, cy œ bz, and
a
cx œ az Ê y œ ba x and z œ ac x. Then xa by zc œ 1 Ê xa b" ˆ ba x‰ "c ˆ ca x‰ œ 1 Ê 3x
a œ 1 Ê xœ 3
Ê y œ ˆ ba ‰ ˆ 3a ‰ œ b3 and z œ ˆ ca ‰ ˆ 3a ‰ œ 3c Ê V œ xyz œ ˆ 3a ‰ ˆ b3 ‰ ˆ 3c ‰ œ abc
27 is the maximum volume. (Note that
there is no minimum volume since the box could be made arbitrarily thin.)
29. ™ T œ 16xi 4zj (4y 16)k and ™ g œ 8xi 2yj 8zk so that ™ T œ - ™ g Ê 16xi 4zj (4y 16)k
œ -(8xi 2yj 8zk) Ê 16x œ 8x-, 4z œ 2y-, and 4y 16 œ 8z- Ê - œ 2 or x œ 0.
CASE 1: - œ 2 Ê 4z œ 2y(2) Ê z œ y. Then 4z 16 œ 16z Ê z œ 43 Ê y œ 43 . Then
#
#
4x# ˆ 43 ‰ 4 ˆ 43 ‰ œ 16 Ê x œ „ 43 .
CASE 2: x œ 0 Ê - œ
2z
y
#
#
#
#
#
Ê 4y 16 œ 8z Š 2z
y ‹ Ê y 4y œ 4z Ê 4(0) y ay 4yb 16 œ 0
Ê y# 2y 8 œ 0 Ê (y 4)(y 2) œ 0 Ê y œ 4 or y œ 2. Now y œ 4 Ê 4z# œ 4# 4(4)
Ê z œ 0 and y œ 2 Ê 4z# œ (2)# 4(2) Ê z œ „ È3.
°
°
The temperatures are T ˆ „ 43 ß 43 ß 43 ‰ œ 642 23 , T(0ß 4ß 0) œ 600°, T Š0ß 2ß È3‹ œ Š600 24È3‹ , and
°
T Š0ß 2ß È3‹ œ Š600 24È3‹ ¸ 641.6°. Therefore ˆ „ 43 ß 43 ß 43 ‰ are the hottest points on the space probe.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
854
Chapter 14 Partial Derivatives
30. ™ T œ 400yz# i 400xz# j 800xyzk and ™ g œ 2xi 2yj 2zk so that ™ T œ - ™ g
Ê 400yz# i 400xz# j 800xyzk œ -(2xi 2yj 2zk) Ê 400yz# œ 2x-, 400xz# œ 2y-, and 800xyz œ 2z-.
Solving this system yields the points a!ß „ 1ß 0b , a „ 1ß 0ß 0b , and Š „ "# ß „ "# ß „
temperatures are T a!ß „ 1ß 0b œ 0, T a „ 1ß 0ß 0b œ 0, and T Š „ "# ß „ "# ß „
È2
# ‹
maximum temperature at Š "# ß "# ß „
Š "# ß "# ß „
È2
# ‹
and Š #" ß #" ß „
and Š "# ß "# ß „
È2
# ‹;
È2
# ‹
È2
# ‹.
The corresponding
œ „ 50. Therefore 50 is the
50 is the minimum temperature at
È2
# ‹.
31. ™ U œ (y 2)i xj and ™ g œ 2i j so that ™ U œ - ™ g Ê (y 2)i xj œ -(2i j) Ê y # œ 2- and
x œ - Ê y 2 œ 2x Ê y œ 2x 2 Ê 2x (2x 2) œ 30 Ê x œ 8 and y œ 14. Therefore U(8ß 14) œ $128
is the maximum value of U under the constraint.
32. ™ M œ (6 z)i 2yj xk and ™ g œ 2xi 2yj 2zk so that ™ M œ - ™ g Ê (6 z)i 2yj xk
œ -(2xi 2yj 2zk) Ê 6 z œ 2x-, 2y œ 2y-, x œ 2z- Ê - œ 1 or y œ 0.
CASE 1: - œ 1 Ê 6 z œ 2x and x œ 2z Ê 6 z œ 2(2z) Ê z œ 2 and x œ 4. Then
(4)# y# 2# 36 œ 0 Ê y œ „ 4.
x
x ‰
CASE 2: y œ 0, 6 z œ 2x-, and x œ 2z- Ê - œ 2z
Ê 6 z œ 2x ˆ 2z
Ê 6z z# œ x#
Ê a6z z# b 0# z# œ 36 Ê z œ 6 or z œ 3. Now z œ 6 Ê x# œ 0 Ê x œ 0; z œ 3
Ê x# œ 27 Ê x œ „ 3È3.
Therefore we have the points Š „ 3È3ß 0ß 3‹ , (0ß 0ß 6), and a4ß „ 4ß 2b . Then M Š3È3ß 0ß 3‹ œ 27È3 60
¸ 106.8, M Š3È3ß 0ß 3‹ œ 60 27È3 ¸ 13.2, M(0ß 0ß 6) œ 60, and M(4ß 4ß 2) œ 12 œ M(4ß 4ß 2). Therefore,
the weakest field is at a4ß „ 4ß 2b .
33. Let g" (xß yß z) œ 2x y œ 0 and g# (xß yß z) œ y z œ 0 Ê ™ g" œ 2i j , ™ g# œ j k , and ™ f œ 2xi 2j 2zk
so that ™ f œ - ™ g" . ™ g# Ê 2xi 2j 2zk œ -(2i j) .(j k) Ê 2xi 2j 2zk œ 2-i (. -)j .k
Ê 2x œ 2-, 2 œ . -, and 2z œ . Ê x œ -. Then 2 œ 2z x Ê x œ 2z 2 so that 2x y œ 0
Ê 2(2z 2) y œ 0 Ê 4z 4 y œ 0. This equation coupled with y z œ 0 implies z œ 43 and y œ 43 . Then
xœ
2
3
#
#
so that ˆ 23 ß 43 ß 43 ‰ is the point that gives the maximum value f ˆ 23 ß 43 ß 43 ‰ œ ˆ 23 ‰ 2 ˆ 43 ‰ ˆ 43 ‰ œ
4
3
.
34. Let g" (xß yß z) œ x 2y 3z 6 œ 0 and g# (xß yß z) œ x 3y 9z 9 œ 0 Ê ™ g" œ i 2j 3k ,
™ g# œ i 3j 9k , and ™ f œ 2xi 2yj 2zk so that ™ f œ - ™ g" . ™ g# Ê 2xi 2yj 2zk
œ -(i 2j 3k) .(i 3j 9k) Ê 2x œ - ., 2y œ 2- 3., and 2z œ 3- 9.. Then 0 œ x 2y 3z 6
‰
œ "# (- .) (2- 3.) ˆ 9# - 27
# . 6 Ê 7- 17. œ 6; 0 œ x 3y 9z 9
"
9
27
81
Ê # (- .) ˆ3- # .‰ ˆ # - # .‰ 9 Ê 34- 91. œ 18. Solving these two equations for - and . gives
-.
2- 3.
3- 9.
78
81
9
- œ 240
œ 123
œ 59
. The minimum value is
59 and . œ 59 Ê x œ # œ 59 , y œ
#
59 , and z œ
#
21,771
81
123
9
369
f ˆ 59 ß 59 ß 59 ‰ œ 59# œ 59 . (Note that there is no maximum value of f subject to the constraints because
at least one of the variables x, y, or z can be made arbitrary and assume a value as large as we please.)
35. Let f(xß yß z) œ x# y# z# be the square of the distance from the origin. We want to minimize f(xß yß z) subject to the
constraints g" (xß yß z) œ y 2z 12 œ 0 and g# (xß yß z) œ x y 6 œ 0. Thus ™ f œ 2xi 2yj 2zk , ™ g" œ j 2k,
and ™ g# œ i j so that ™ f œ - ™ g" . ™ g# Ê 2x œ ., 2y œ - ., and 2z œ 2-. Then 0 œ y 2z 12
œ ˆ -# .# ‰ 2- 12 Ê #5 - "# . œ 12 Ê 5- . œ 24; 0 œ x y 6 œ .# ˆ -# .# ‰ 6 Ê "# - . œ 6
Ê - #. œ 12. Solving these two equations for - and . gives - œ 4 and . œ 4 Ê x œ
.
#
œ 2, y œ
-.
#
œ 4, and
z œ - œ 4. The point (2ß 4ß 4) on the line of intersection is closest to the origin. (There is no maximum distance from the
origin since points on the line can be arbitrarily far away.)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.8 Lagrange Multipliers
36. The maximum value is f ˆ 23 ß 43 ß 43 ‰ œ
4
3
855
from Exercise 33 above.
37. Let g" (xß yß z) œ z 1 œ 0 and g# (xß yß z) œ x# y# z# 10 œ 0 Ê ™ g" œ k , ™ g# œ 2xi 2yj 2zk , and
™ f œ 2xyzi x# zj x# yk so that ™ f œ - ™ g" . ™ g# Ê 2xyzi x# zj x# yk œ -(k) .(2xi 2yj 2zk)
Ê 2xyz œ 2x., x# z œ 2y., and x# y œ 2z. - Ê xyz œ x. Ê x œ 0 or yz œ . Ê . œ y since z œ 1.
CASE 1: x œ 0 and z œ 1 Ê y# 9 œ 0 (from g# ) Ê y œ „ 3 yielding the points a0ß „ 3ß 1b.
CASE 2: . œ y Ê x# z œ 2y# Ê x# œ 2y# (since z œ 1) Ê 2y# y# 1 10 œ 0 (from g# ) Ê 3y# 9 œ 0
#
Ê y œ „ È3 Ê x# œ 2 Š „ È3‹ Ê x œ „ È6 yielding the points Š „ È6ß „ È3ß "‹ .
Now f a!ß „ 3ß 1b œ 1 and f Š „ È6ß „ È3ß "‹ œ 6 Š „ È3‹ 1 œ 1 „ 6È3. Therefore the maximum of f is
1 6È3 at Š „ È6ß È3ß 1‹, and the minimum of f is 1 6È3 at Š „ È6ß È3ß "‹ .
38. (a) Let g" (xß yß z) œ x y z 40 œ 0 and g# (xß yß z) œ x y z œ 0 Ê ™ g" œ i j k , ™ g# œ i j k , and
™ w œ yzi xzj xyk so that ™ w œ - ™ g" . ™ g# Ê yzi xzj xyk œ -(i j k) .(i j k)
Ê yz œ - ., xz œ - ., and xy œ - . Ê yz œ xz Ê z œ 0 or y œ x.
CASE 1: z œ 0 Ê x y œ 40 and x y œ 0 Ê no solution.
CASE 2: x œ y Ê 2x z 40 œ 0 and 2x z œ 0 Ê z œ 20 Ê x œ 10 and y œ 10 Ê w œ (10)(10)(20)
œ 2000
â
â
âi j k â
â
â
" â œ 2i 2j is parallel to the line of intersection Ê the line is x œ 2t 10,
(b) n œ â " "
â
â
â " " " â
y œ 2t 10, z œ 20. Since z œ 20, we see that w œ xyz œ (2t 10)(2t 10)(20) œ a4t# 100b (20)
which has its maximum when t œ 0 Ê x œ 10, y œ 10, and z œ 20.
39. Let g" (Bß yß z) œ y x œ 0 and g# (xß yß z) œ x# y# z# 4 œ 0. Then ™ f œ yi xj 2zk , ™ g" œ i j , and
™ g# œ 2xi 2yj 2zk so that ™ f œ - ™ g" . ™ g# Ê yi xj 2zk œ -(i j) .(2xi 2yj 2zk)
Ê y œ - 2x., x œ - 2y., and 2z œ 2z. Ê z œ 0 or . œ 1.
CASE 1: z œ 0 Ê x# y# 4 œ 0 Ê 2x# 4 œ 0 (since x œ y) Ê x œ „ È2 and y œ „ È2 yielding the points
Š „ È2ß „ È2ß !‹ .
CASE 2: . œ 1 Ê y œ - 2x and x œ - 2y Ê x y œ 2(x y) Ê 2x œ 2(2x) since x œ y Ê x œ 0 Ê y œ 0
Ê z# 4 œ 0 Ê z œ „ 2 yielding the points a!ß !ß „ 2b .
Now, f a!ß !ß „ 2b œ 4 and f Š „ È2ß „ È2ß !‹ œ 2. Therefore the maximum value of f is 4 at a!ß !ß „ 2b and the
minimum value of f is 2 at Š „ È2ß „ È2ß !‹ .
40. Let f(xß yß z) œ x# y# z# be the square of the distance from the origin. We want to minimize f(xß yß z) subject
to the constraints g" (xß yß z) œ 2y 4z 5 œ 0 and g# (xß yß z) œ 4x# 4y# z# œ 0. Thus ™ f œ 2xi 2yj 2zk ,
™ g" œ 2j 4k , and ™ g# œ 8xi 8yj 2zk so that ™ f œ - ™ g" . ™ g# Ê 2xi 2yj 2zk
œ -(2j 4k) .(8xi 8yj 2zk) Ê 2x œ 8x., 2y œ 2- 8y., and 2z œ 4- 2z. Ê x œ 0 or . œ "4 .
CASE 1: x œ 0 Ê 4(0)# 4y# z# œ 0 Ê z œ „ 2y Ê 2y 4(2y) 5 œ 0 Ê y œ
Ê y œ 56 yielding the points ˆ!ß "# ß "‰ and ˆ!ß 56 ß 53 ‰ .
CASE 2: . œ
"
4
"
#
, or 2y 4(2y) 5 œ 0
Ê y œ - y Ê - œ 0 Ê 2z œ 4(0) 2z ˆ 4" ‰ Ê z œ 0 Ê 2y 4(0) œ 5 Ê y œ
#
4 ˆ #5 ‰
(0)# œ 4x#
Ê no solution.
"
Then f ˆ!ß "# ß 1‰ œ 54 and f ˆ!ß 56 ß 35 ‰ œ 25 ˆ 36
"9 ‰ œ
125
36
Ê the point ˆ!ß "# ß 1‰ is closest to the origin.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
5
#
and
856
Chapter 14 Partial Derivatives
41. ™ f œ i j and ™ g œ yi xj so that ™ f œ - ™ g Ê i j œ -(yi xj) Ê 1 œ y- and 1 œ x- Ê y œ x
Ê y# œ 16 Ê y œ „ 4 Ê (4ß 4) and (%ß 4) are candidates for the location of extreme values. But as x Ä _,
y Ä _ and f(xß y) Ä _; as x Ä _, y Ä 0 and f(xß y) Ä _. Therefore no maximum or minimum value
exists subject to the constraint.
4
42. Let f(Aß Bß C) œ ! (Axk Byk C zk )# œ C# (B C 1)# (A B C 1)# (A C 1)# . We want
k œ1
to minimize f. Then fA (Aß Bß C) œ 4A 2B 4C, fB (Aß Bß C) œ 2A 4B 4C 4, and
fC (Aß Bß C) œ 4A 4B 8C 2. Set each partial derivative equal to 0 and solve the system to get A œ "# ,
B œ 3# , and C œ "4 or the critical point of f is ˆ #" ß 3# ß "4 ‰ .
43. (a) Maximize f(aß bß c) œ a# b# c# subject to a# b# c# œ r# . Thus ™ f œ 2ab# c# i 2a# bc# j 2a# b# ck and
™ g œ 2ai 2bj 2ck so that ™ f œ - ™ g Ê 2ab# c# œ 2a-, 2a# bc# œ 2b-, and 2a# b# c œ 2cÊ 2a# b# c# œ 2a# - œ 2b# - œ 2c# - Ê - œ 0 or a# œ b# œ c# .
CASE 1: - œ 0 Ê a# b# c# œ 0.
#
$
CASE 2: a# œ b# œ c# Ê f(aß bß c) œ a# a# a# and 3a# œ r# Ê f(aß bß c) œ Š r3 ‹ is the maximum value.
(b) The point ŠÈaß Èbß Èc‹ is on the sphere if a b c œ r# . Moreover, by part (a), abc œ f ŠÈaß Èbß Èc‹
#
$
Ÿ Š r3 ‹ Ê (abc)"Î$ Ÿ
r#
3
œ
abc
3
, as claimed.
n
44. Let f(x" ß x# ß á ß xn ) œ ! ai xi œ a" x" a# x# á an xn and g(x" ß x# ß á ß xn ) œ x"# x## á xn# 1. Then we
i œ1
want ™ f œ - ™ g Ê a" œ -(2x" ), a# œ -(2x# ), á , an œ -(2xn ), - Á 0 Ê xi œ
n
n
iœ1
i œ1
"Î#
Ê 4-# œ ! a#i Ê 2- œ Œ! a#i
n
n
i œ1
i œ1
ai
2-
Ê f(x" ß x# ß á ß xn ) œ ! ai xi œ ! ai ˆ #a-i ‰ œ
Ê
"
#-
a#"
4- #
a##
4- #
an#
4- #
"Î#
á
n
n
i œ1
i œ1
! a#i œ Œ! a#i
the maximum value.
45-50. Example CAS commands:
Maple:
f := (x,y,z) -> x*y+y*z;
g1 := (x,y,z) -> x^2+y^2-2;
g2 := (x,y,z) -> x^2+z^2-2;
h := unapply( f(x,y,z)-lambda[1]*g1(x,y,z)-lambda[2]*g2(x,y,z), (x,y,z,lambda[1],lambda[2]) );
hx := diff( h(x,y,z,lambda[1],lambda[2]), x );
hy := diff( h(x,y,z,lambda[1],lambda[2]), y );
hz := diff( h(x,y,z,lambda[1],lambda[2]), z );
hl1 := diff( h(x,y,z,lambda[1],lambda[2]), lambda[1] );
hl2 := diff( h(x,y,z,lambda[1],lambda[2]), lambda[2] );
sys := { hx=0, hy=0, hz=0, hl1=0, hl2=0 };
q1 := solve( sys, {x,y,z,lambda[1],lambda[2]} );
q2 := map(allvalues,{q1});
for p in q2 do
eval( [x,y,z,f(x,y,z)], p );
``=evalf(eval( [x,y,z,f(x,y,z)], p ));
end do;
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
# (a)
#(b)
# (c)
# (d)
is
œ1
Section 14.9 Taylor's Formula for Two Variables
Mathematica: (assigned functions will vary)
Clear[x, y, z, lambda1, lambda2]
f[x_,y_,z_]:= x y y z
g1[x_,y_,z_]:= x2 y2 2
g2[x_,y_,z_]:= x2 z2 2
h = f[x, y, z] lambda1 g1[x, y, z] lambda2 g2[x, y, z];
hx= D[h, x]; hy= D[h, y]; hz= D[h,z]; hL1=D[h, lambda1]; hL2= D[h, lambda2];
critical=Solve[{hx==0, hy==0, hz==0, hL1==0, hL2==0, g1[x,y,z]==0, g2[x,y,z]==0},
{x, y, z, lambda1, lambda2}]//N
{{x, y, z}, f[x, y, z]}/.critical
14.9 TAYLOR'S FORMULA FOR TWO VARIABLES
1. f(xß y) œ xey Ê fx œ ey , fy œ xey , fxx œ 0, fxy œ ey , fyy œ xey
Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (0ß 0) "# cx# fxx (0ß 0) 2xyfxy (0ß 0) y# fyy (0ß 0)d
œ 0 x † 1 y † 0 "# ax# † 0 2xy † 1 y# † 0b œ x xy quadratic approximation;
fxxx œ 0, fxxy œ 0, fxyy œ ey , fyyy œ xey
Ê f(xß y) ¸ quadratic "6 cx$ fxxx (!ß !) 3x# yfxxy (0ß 0) 3xy# fxyy (!ß !) y$ fyyy (0ß 0)d
œ x xy "6 ax$ † 0 3x# y † 0 3xy# † 1 y$ † 0b œ x xy "# xy# , cubic approximation
2. f(xß y) œ ex cos y Ê fx œ ex cos y, fy œ ex sin y, fxx œ ex cos y, fxy œ ex sin y, fyy œ ex cos y
Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (!ß 0) "# cx# fxx (!ß !) 2xyfxy (!ß !) y# fyy (0ß 0)d
œ 1 x † 1 y † 0 "# cx# † 1 2xy † 0 y# † (1)d œ 1 x "# ax# y# b , quadratic approximation;
fxxx œ ex cos y, fxxy œ ex sin y, fxyy œ ex cos y, fyyy œ ex sin y
Ê f(xß y) ¸ quadratic "6 cx$ fxxx (0ß 0) 3x# yfxxy (!ß 0) 3xy# fxyy (0ß 0) y$ fyyy (0ß 0)d
œ 1 x "# ax# y# b 6" cx$ † 1 3x# y † 0 3xy# † (1) y$ † 0d
œ 1 x "# ax# y# b 6" ax$ 3xy# b , cubic approximation
3. f(xß y) œ y sin x Ê fx œ y cos x, fy œ sin x, fxx œ y sin x, fxy œ cos x, fyy œ 0
Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (!ß 0) "# cx# fxx (0ß 0) 2xyfxy (0ß 0) y# fyy (0ß 0)d
œ 0 x † 0 y † 0 "# ax# † 0 2xy † 1 y# † 0b œ xy, quadratic approximation;
fxxx œ y cos x, fxxy œ sin x, fxyy œ 0, fyyy œ 0
Ê f(xß y) ¸ quadratic "6 cx$ fxxx (0ß 0) 3x# yfxxy (!ß 0) 3xy# fxyy (0ß 0) y$ fyyy (0ß 0)d
œ xy "6 ax$ † 0 3x# y † 0 3xy# † 0 y$ † 0b œ xy, cubic approximation
4. f(xß y) œ sin x cos y Ê fx œ cos x cos y, fy œ sin x sin y, fxx œ sin x cos y, fxy œ cos x sin y,
fyy œ sin x cos y Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (0ß 0) "# cx# fxx (0ß 0) 2xyfxy (0ß 0) y# fyy (0ß 0)d
œ 0 x † 1 y † 0 "# ax# † 0 2xy † 0 y# † 0b œ x, quadratic approximation;
fxxx œ cos x cos y, fxxy œ sin x sin y, fxyy œ cos x cos y, fyyy œ sin x sin y
Ê f(xß y) ¸ quadratic "6 cx$ fxxx (0ß 0) 3x# yfxxy (!ß 0) 3xy# fxyy (0ß 0) y$ fyyy (0ß 0)d
œ x "6 cx$ † (1) 3x# y † 0 3xy# † (1) y$ † 0d œ x 6" ax$ 3xy# b, cubic approximation
5. f(xß y) œ ex ln (1 y) Ê fx œ ex ln (1 y), fy œ
ex
1y
, fxx œ ex ln (1 y), fxy œ
ex
1y
Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (0ß 0) "# cx# fxx (0ß 0) 2xyfxy (0ß 0) y# fyy (0ß 0)d
œ 0 x † 0 y † 1 "# cx# † 0 2xy † 1 y# † (1)d œ y "# a2xy y# b , quadratic approximation;
fxxx œ ex ln (1 y), fxxy œ
ex
1y
x
, fxyy œ (1 e y)# , fyyy œ
x
, fyy œ (1 e y)#
2ex
(1 y)$
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
857
858
Chapter 14 Partial Derivatives
Ê f(xß y) ¸ quadratic "6 cx$ fxxx (0ß 0) 3x# yfxxy (!ß 0) 3xy# fxyy (0ß 0) y$ fyyy (0ß 0)d
œ y "2 a2xy y# b 6" cx$ † 0 3x# y † 1 3xy# † (1) y$ † 2d
œ y "# a2xy y# b 6" a3x# y 3xy# 2y$ b , cubic approximation
4
2
(2x y 1)# , fxy œ (2x y 1)# ,
"
#
#
fyy œ (2x "
y 1)# Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (0ß 0) # cx fxx (0ß 0) 2xyfxy (0ß 0) y fyy (0ß 0)d
œ 0 x † 2 y † 1 "# cx# † (4) 2xy † (2) y# † (1)d œ 2x y "# a4x# 4xy y# b
œ (2x y) "# (2x y)# , quadratic approximation;
fxxx œ (2x 16y 1)$ , fxxy œ (2x 8y 1)$ , fxyy œ (2x 4y 1)$ , fyyy œ (2x 2y 1)$
Ê f(xß y) ¸ quadratic "6 cx$ fxxx (0ß 0) 3x# yfxxy (!ß 0) 3xy# fxyy (0ß 0) y$ fyyy (0ß 0)d
œ (2x y) "# (2x y)# 6" ax$ † 16 3x# y † 8 3xy# † 4 y$ † 2b
œ (2x y) "# (2x y)# 3" a8x$ 12x# y 6xy# y# b
œ (2x y) "# (2x y)# 3" (2x y)$ , cubic approximation
6. f(xß y) œ ln (2x y 1) Ê fx œ
2
2x y 1
, fy œ
"
#x y 1
, fxx œ
7. f(xß y) œ sin ax# y# b Ê fx œ 2x cos ax# y# b , fy œ 2y cos ax# y# b , fxx œ 2 cos ax# y# b 4x# sin ax# y# b ,
fxy œ 4xy sin ax# y# b , fyy œ 2 cos ax# y# b 4y# sin ax# y# b
Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (0ß 0) "# cx# fxx (0ß 0) 2xyfxy (0ß 0) y# fyy (0ß 0)d
œ 0 x † 0 y † 0 "# ax# † 2 2xy † 0 y# † 2b œ x# y# , quadratic approximation;
fxxx œ 12x sin ax# y# b 8x$ cos ax# y# b , fxxy œ 4y sin ax# y# b 8x# y cos ax# y# b ,
fxyy œ 4x sin ax# y# b 8xy# cos ax# y# b , fyyy œ 12y sin ax# y# b 8y$ cos ax# y# b
Ê f(xß y) ¸ quadratic "6 cx$ fxxx (0ß 0) 3x# yfxxy (!ß 0) 3xy# fxyy (0ß 0) y$ fyyy (0ß 0)d
œ x# y# "6 ax$ † 0 3x# y † 0 3xy# † 0 y$ † 0b œ x# y# , cubic approximation
8. f(xß y) œ cos ax# y# b Ê fx œ 2x sin ax# y# b , fy œ 2y sin ax# y# b ,
fxx œ 2 sin ax# y# b 4x# cos ax# y# b , fxy œ 4xy cos ax# y# b , fyy œ 2 sin ax# y# b 4y# cos ax# y# b
Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (0ß 0) "# cx# fxx (0ß 0) 2xyfxy (0ß 0) y# fyy (0ß 0)d
œ 1 x † 0 y † 0 "# cx# † 0 2xy † 0 y# † 0d œ 1, quadratic approximation;
fxxx œ 12x cos ax# y# b 8x$ sin ax# y# b , fxxy œ 4y cos ax# y# b 8x# y sin ax# y# b ,
fxyy œ 4x cos ax# y# b 8xy# sin ax# y# b , fyyy œ 12y cos ax# y# b 8y$ sin ax# y# b
Ê f(xß y) ¸ quadratic "6 cx$ fxxx (0ß 0) 3x# yfxxy (!ß 0) 3xy# fxyy (0ß 0) y$ fyyy (0ß 0)d
œ 1 "6 ax$ † 0 3x# y † 0 3xy# † 0 y$ † 0b œ 1, cubic approximation
9. f(xß y) œ
"
1xy
Ê fx œ
"
(1 x y)#
œ fy , fxx œ
Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (0ß 0)
2
(1 x y)$
"
#
# cx fxx (0ß 0)
œ fxy œ fyy
2xyfxy (0ß 0) y# fyy (0ß 0)d
œ 1 x † 1 y † 1 "# ax# † 2 2xy † 2 y# † 2b œ 1 (x y) ax# 2xy y# b
œ 1 (x y) (x y)# , quadratic approximation; fxxx œ
Ê f(xß y) ¸ quadratic "6 cx$ fxxx (0ß 0) 3x# yfxxy (!ß 0)
6
œ fxxy œ fxyy œ fyyy
(1 x y)%
#
3xy fxyy (0ß 0) y$ fyyy (0ß 0)d
$
œ 1 (x y) (x y)# "6 ax$ † 6 3x# y † 6 3xy# † 6 y † 6b
œ 1 (x y) (x y)# ax$ 3x# y 3xy# y$ b œ 1 (x y) (x y)# (x y)$ , cubic approximation
10. f(xß y) œ
fxy œ
"
1 x y xy
1
(" x y xy)#
Ê fx œ
, fyy œ
1y
(1 x y xy)#
, fy œ
1x
(" x y xy)#
, fxx œ
2(1 y)#
(1 x y xy)$
,
2(" x)#
(1 x y xy)$
Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (0ß 0) "# cx# fxx (0ß 0) 2xyfxy (0ß 0) y# fyy (0ß 0)d
œ 1 x † 1 y † 1 "# ax# † 2 2xy † 1 y# † 2b œ 1 x y x# xy y# , quadratic approximation;
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.10 Partial Derivatives with Constrained Variables
fxxx œ
6(1 y)$
(1 x y xy)%
, fxxy œ
[4(1 x y xy) 6(1 y)(1 x)](1 y)
(1 x y xy)%
,
$
[4(1 x y xy) 6(1 x)(1 y)](1 x)
x)
, fyyy œ (1 6(1
(1 x y xy)%
x y xy)%
Ê f(xß y) ¸ quadratic "6 cx$ fxxx (0ß 0) 3x# yfxxy (!ß 0) 3xy# fxyy (0ß 0)
œ 1 x y x# xy y# "6 ax$ † 6 3x# y † 2 3xy# † 2 y$ † 6b
#
#
$
#
#
$
fxyy œ
y$ fyyy (0ß 0)d
œ 1 x y x xy y x x y xy y , cubic approximation
11. f(xß y) œ cos x cos y Ê fx œ sin x cos y, fy œ cos x sin y, fxx œ cos x cos y, fxy œ sin x sin y,
fyy œ cos x cos y Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (0ß 0) "# cx# fxx (0ß 0) 2xyfxy (0ß 0) y# fyy (0ß 0)d
œ 1 x † 0 y † 0 "# cx# † (1) 2xy † 0 y# † (1)d œ 1
x#
#
y#
#
, quadratic approximation. Since all partial
derivatives of f are products of sines and cosines, the absolute value of these derivatives is less than or equal
to 1 Ê E(xß y) Ÿ "6 c(0.1)$ 3(0.1)$ 3(0.1)$ 0.1)$ d Ÿ 0.00134.
12. f(xß y) œ ex sin y Ê fx œ ex sin y, fy œ ex cos y, fxx œ ex sin y, fxy œ ex cos y, fyy œ ex sin y
Ê f(xß y) ¸ f(0ß 0) xfx (0ß 0) yfy (0ß 0) "# cx# fxx (0ß 0) 2xyfxy (0ß 0) y# fyy (0ß 0)d
œ 0 x † 0 y † 1 "# ax# † 0 2xy † 1 y# † 0b œ y xy , quadratic approximation. Now, fxxx œ ex sin y,
fxxy œ ex cos y, fxyy œ ex sin y, and fyyy œ ex cos y. Since kxk Ÿ 0.1, kex sin yk Ÿ ke0Þ1 sin 0.1k ¸ 0.11 and
kex cos yk Ÿ ke0Þ1 cos 0.1k ¸ 1.11. Therefore,
E(xß y) Ÿ "6 c(0.11)(0.1)$ 3(1.11)(0.1)$ 3(0.11)(0.1)$ (1.11)(0.1)$ d Ÿ 0.000814.
14.10 PARTIAL DERIVATIVES WITH CONSTRAINED VARIABLES
1. w œ x# y# z# and z œ x# y# :
Î x œ x(yß z) Ñ
y
yœy
Ä w Ê Š ``wy ‹ œ
(a) Œ Ä
z
z
Ï zœz Ò
œ 2x `` xy 2y Ê 0 œ 2x `` xy 2y Ê
`x
`y
œ
"
#y
`x
`z
œ
1
2x
`w `x
`x `z
`w `x
`x `z
œ 2x `` yx 2y `` yy
`w `y
`y `z
`w `z `x
`z `z ; `z
œ 0 and
`z
`z
œ 2x `` xz 2y `` yz
`w `y
`y `z
`w `z `y
`z `z ; `z
œ 0 and
`z
`z
œ 2x `` xz 2y `` yz
Ê ˆ ``wz ‰y œ (2x) ˆ #"x ‰ (2y)(0) (2z)(1) œ 1 2z
2. w œ x# y z sin t and x y œ t:
Î xœx Ñ
ÎxÑ
Ð yœy Ó
y
Ä Ð
Ä w Ê Š ``wy ‹ œ
(a)
Ó
zœz
xz
ÏzÒ
Ït œ x yÒ
ß
`t
`y
`z
`y
œ 0 and
"
Ê ˆ ``wz ‰x œ (2x)(0) (2y) Š 2y
‹ (2z)(1) œ 1 2z
Î x œ x(yß z) Ñ
y
yœy
Ä w Ê ˆ ``wz ‰y œ
(c) Œ Ä
z
Ï zœz Ò
Ê 1 œ 2x `` xz Ê
`w `z `z
`z `y ; `y
z
xœx Ñ
x
y œ y(xß z)
Ä w Ê ˆ ``wz ‰x œ
(b) Œ Ä
z
Ï zœz Ò
`y
`z
`w `y
`y `y
œ xy Ê Š ``wy ‹ œ (2x) ˆ xy ‰ (2y)(1) (2z)(0) œ 2y 2y œ 0
Î
Ê 1 œ 2y `` yz Ê
`w `x
`x `y
`w `x
`x `y
`w `y
`y `y
`w `z
`z `y
`w `t `x
`t `y ; `y
œ 0,
`z
`y
œ 0, and
œ 1 Ê Š ``wy ‹ œ (2x)(0) (1)(1) (1)(0) (cos t)(1) œ 1 cos t œ 1 cos (x y)
xßt
Îx œ t yÑ
ÎyÑ
Ð yœy Ó
z
Ä Ð
Ä w Ê Š ``wy ‹ œ
(b)
Ó
z
z
œ
zt
ÏtÒ
Ï tœt Ò
ß
Ê
`x
`y
œ
`t
`y
`y
`y
`w `x
`x `y
`w `y
`y `y
`w `z
`z `y
`w `t `z
`t `y ; `y
œ 0 and
`t
`y
œ0
œ 1 Ê Š ``wy ‹ œ (2x)(1) (1)(1) (1)(0) (cos t)(0) œ 1 2at yb œ 1 2y 2t
zßt
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
859
860
Chapter 14 Partial Derivatives
Î xœx Ñ
ÎxÑ
Ð yœy Ó
y
Ä Ð
Ä w Ê ˆ ``wz ‰x y œ
(c)
Ó
œ
z
z
ÏzÒ
Ït œ x yÒ
ß
`w `x
`x `z
`w `y
`y `z
`w `z
`z `z
`w `t `x
`t `z ; `z
œ 0 and
`y
`z
œ0
`w `z
`z `z
`w `t `y
`t `z ; `z
œ 0 and
`t
`z
œ0
`w `z
`z `t
`w `t `x
`t `t ; `t
œ 0 and
`z
`t
œ0
`w `z
`z `t
`w `t `y
`t `t ; `t
œ 0 and
`z
`t
œ0
Ê ˆ ``wz ‰x y œ (2x)(0) (1)(0) (1)(1) (cos t)(0) œ 1
ß
Îx œ t yÑ
ÎyÑ
Ð yœy Ó
z
Ä Ð
Ä w Ê ˆ ``wz ‰y t œ
(d)
Ó
zœz
ÏtÒ
Ï tœt Ò
ß
`w `x
`x `z
`w `y
`y `z
Ê ˆ ``wz ‰y t œ (2x)(0) (1)(0) (1)(1) (cos t)(0) œ 1
ß
Î xœx Ñ
ÎxÑ
Ð y œ t xÓ
z
Ä Ð
Ä w Ê ˆ ``wt ‰x z œ
(e)
Ó
zœz
ÏtÒ
Ï tœt Ò
ß
`w `x
`x `t
`w `y
`y `t
Ê ˆ ``wt ‰x z œ (2x)(0) (1)(1) (1)(0) (cos t)(1) œ 1 cos t
ß
Îx œ t yÑ
ÎyÑ
Ð yœy Ó
z
Ä Ð
Ä w Ê ˆ ``wt ‰y z œ
(f)
Ó
zœz
ÏtÒ
Ï tœt Ò
ß
`w `x
`x `t
`w `y
`y `t
Ê ˆ ``wt ‰y z œ (2x)(1) (1)(0) (1)(0) (cos t)(1) œ cos t 2x œ cos t 2(t y)
ß
3. U œ f(Pß Vß T) and PV œ nRT
Î PœP Ñ
P
VœV
Ä U Ê ˆ ``UP ‰V œ
(a) Œ Ä
V
PV
ÏT œ
Ò
`U `P
`P `P
`U `V
`V `P
`U `T
`T `P
œ
`U
`P
V ‰
‰
ˆ ` U ‰ ˆ nR
ˆ `` U
V (0) ` T
nR
œ
`U
`P
V ‰
ˆ ``UT ‰ ˆ nR
nRT
ÎP œ V Ñ
V
Ä U Ê ˆ ``UT ‰V œ
(b) Œ Ä
VœV
T
Ï TœT Ò
‰ `U
œ ˆ ``UP ‰ ˆ nR
V `T
`U `P
`P `T
4. w œ x# y# z# and y sin z z sin x œ 0
Î xœx Ñ
x
yœy
Ä w Ê ˆ ``wx ‰y œ
(a) Œ Ä
y
Ï z œ z(xß y) Ò
(y cos z)
Ê
`z
`x
(sin x)
ˆ ``wx ‰
yk (0ß1ß1)
`z
`x
z cos x œ 0 Ê
`z
`x
`x
`z
`w `x
`x `x
œ
`U `V
`V `T
`w `y
`y `x
z cos x
y cos z sin x .
#
`U `T
`T `T
‰ ˆ `U ‰
œ ˆ ``UP ‰ ˆ nR
V ` V (0)
`w `z `y
`z `x ; `x
œ 0 and
œ
1
1
œ1
œ (2x)
`x
`z
(2y)(0) (2z)(1)
At (0ß 1ß 1),
`z
`x
`U
`T
œ (2x)(1) (2y)(0) (2z)(1)k Ð0ß1ß1Ñ œ 21
Î x œ x(yß z) Ñ
y
yœy
Ä w Ê ˆ ``wz ‰y œ
(b) Œ Ä
z
Ï zœz Ò
œ (2x)
`y
`z y
x) `` xz œ
2z. Now (sin z)
Ê y cos z sin x (z cos
`w `x
`x `z
`w `y
`y `z
cos z sin x (z cos x)
0 Ê
`x
`z
œ
y cos z sin x
.
z cos x
`w `z
`z `z
`x
`z
`y
`z œ 0
(!ß "ß 1), `` xz œ (11)(1)0
œ 0 and
At
œ
"
1
Ê ˆ ``wz ‰Ck (!,"ß1Ñ œ 2(0) ˆ 1" ‰ 21 œ 21
5. w œ x# y# yz z$ and x# y# z# œ 6
Î xœx Ñ
x
yœy
(a) Œ Ä
Ä w Ê Š ``wy ‹ œ
y
x
Ï z œ z(xß y) Ò
`w `x
`x `y
`w `y
`y `y
`w `z
`z `y
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 14.10 Partial Derivatives with Constrained Variables
œ a2xy# b (0) a2x# y zb (1) ay 3z# b
`x
`y
`z
`y
œ 0 Ê 2y (2z)
`z
`y
œ0 Ê
`z
`y
`z
`y
`x
`y
`w `x
`x `y
`x
`y
a2x# y zb (1) ay 3z# b (0) œ a2x# yb
œ 0 Ê (2x)
`x
`y
`x
`y
2y œ 0 Ê
`w `y
`y `y
`x
`y
Now (2x)
`z
`y
œ yz . At (wß xß yß z) œ (4ß 2ß 1ß 1),
œ c(2)(2)# (1) (1)d c1 3(1)# d (1) œ 5
Î x œ x(yß z) Ñ
y
yœy
(b) Œ Ä
Ä w Ê Š ``wy ‹ œ
z
z
Ï zœz Ò
œ a2xy# b
`z
`y .
œ 2x# y z ay 3z# b
2y (2z)
`z
`y
œ 0 and
œ "1 œ 1 Ê Š ``wy ‹ ¹
x (4ß2ß1ßc1)
`w `z
`z `y
2x# y z. Now (2x)
œ yx . At (wß xß yß z) œ (4ß 2ß 1ß 1),
`x
`y
`x
`y
2y (2z)
œ "2 Ê Š ``wy ‹ ¹
z
œ (2)(2)(1) ˆ "# ‰ (2)(2)# (1) (1) œ 5
`z
`y
œ 0 and
(4ß2ß1ßc1)
#
6. y œ uv Ê 1 œ v
œv
`u
`y
u Š uv
`u
`y
u
`u
`y ‹
`v
`y ;
œ Šv
#
x œ u# v# and
u
v
#
‹
`u
`y
Ê
`u
`y
`x
`y
œ
œ 0 Ê 0 œ 2u
`u
`y
2v
`v
`y
At (uß v) œ ŠÈ2ß 1‹ ,
v
v# u# .
`v
`y
Ê
`u
`y
œ ˆ uv ‰
"
œ
#
1# ŠÈ2‹
`u
`y
Ê 1
œ 1
Ê Š `` uy ‹ œ 1
x
r
x œ r cos )
7. Œ Ä Œ
Ê ˆ ``xr ‰) œ cos ); x# y# œ r# Ê 2x 2y
)
y œ r sin )
Ê ``xr œ xr Ê ˆ ``xr ‰ œ È #x #
`y
`x
œ 2r
`r
`x
and
`y
`x
8. If x, y, and z are independent, then ˆ ``wx ‰y z œ
ß
`w `x
`x `x
`w `y
`y `x
`w `z
`z `x
`w `t
`t `x
œ (2x)(1) (2y)(0) (4)(0) (1) ˆ ``xt ‰ œ 2x ``xt . Thus x 2z t œ 25 Ê 1 0
Ê ˆ ``wx ‰ œ 2x 1. On the other hand, if x, y, and t are independent, then ˆ ``wx ‰
yßz
œ
Ê 1
`r
`x
x y
y
`w `x
`x `x
œ 0 Ê 2x œ 2r
`t
`x
œ0 Ê
`t
`x
œ 1
yßt
``wy `` yx ``wz `` xz ``wt ``xt œ (2x)(1) (2y)(0) 4 `` xz (1)(0) œ
2 `` xz 0 œ 0 Ê `` xz œ #" Ê ˆ ``wx ‰yßt œ 2x 4 ˆ #" ‰ œ 2x 2.
9. If x is a differentiable function of y and z, then f(xß yß z) œ 0 Ê
`f `x
`x `x
2x 4
`f `y
`y `x
`z
`x .
`f `z
`z `x
Thus, x 2z t œ 25
œ0 Ê
`f
`x
`f `y
`y `x
œ0
Ê Š `` xy ‹ œ `` f/f/`` yz . Similarly, if y is a differentiable function of x and z, Š `` yz ‹ œ `` f/f/`` xz and if z is a
z
x
differentiable function of x and y, ˆ `` xz ‰y œ `` f/f/`` xy . Then Š `` xy ‹ Š `` yz ‹ ˆ `` xz ‰y
z
œ Š
` f/` y ˆ
` f/` z ‰
` f/` x
` f/` z ‹ ` f/` x Š ` f/` y ‹
10. z œ z f(u) and u œ xy Ê
œ x ˆ1 y
df ‰
du
y ˆx
df ‰
du
`z
`x
x
œ 1.
œ1
df ` u
du ` x
œ1y
df
du ;
also
`z
`y
œ0
df ` u
du ` y
œx
df
du
so that x
`z
`x
y
œ 0 and
`x
`y
œ0 Ê
`g
`y
`z
`y
œx
11. If x and y are independent, then g(xß yß z) œ 0 Ê
`g `x
`x `y
`g `y
`y `y
`g `z
`z `y
`y
Ê Š `` yz ‹ œ `` g/
g/` z , as claimed.
x
12. Let x and y be independent. Then f(xß yß zß w) œ 0, g(xß yß zß w) œ 0 and
Ê `` xf `` xx `` yf `` yx `` zf `` xz ``wf ``wx œ `` xf `` zf `` xz ``wf ``wx
`g `x
`g `y
`g `z
`g `w
`g
`g `z
`g `w
`x `x `y `x `z `x `w `x œ `x `z `x `w `x œ 0
`y
`x
œ0
œ 0 and
imply
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
`g `z
`z `y
œ0
861
862
Chapter 14 Partial Derivatives
`f
`z
`g
`z
`z
`x
`z
`x
`f
`w
`g
`w
`w
`x
`w
`x
œ
œ
`f
`x
`g
`x
Ê ˆ `` xz ‰y œ
c ``xf
» c `g
»
`x
`f
`z
`g
`z
`f
`w
`g »
`w
`f
`w
`g »
`w
œ
`x
`y œ 0
`g
`g `z
`y `z `y
Likewise, f(xß yß zß w) œ 0, g(xß yß zß w) œ 0 and
œ
`f
`y
`f
`z
`g
`z
`z
`y
`z
`y
`f `z
`z `y
`f
`w
`g
`w
`w
`y
`w
`y
`f `w
`w `y
œ 0 and (similarly)
œ `` yf
œ
`g
`y
Ê Š ``wy ‹ œ
x
`g
`g `f
`w `x `w
`g `f
`f `g
`z `w `z `w
``xf
`f
`z
» `g
`z
`f
`z
» `g
`z
c ``yf
c `` gy »
`f
`w
`g »
`w
œ
œ
`f
`x
`f
`z
`g
`w
`g
`w
`g
`x
`f `g
`w `z
``wf
`f `x
`f `y
`f `z
`x `y `y `y `z `y
`g `w
` w ` y œ 0 imply
Ê
`g
`g `f
`y `z `y
`g `f
`f `g
`z `w `z `w
`` fz
œ
`f
`z
`f
`z
`g
`y
`g
`w
`g
`z
`f `g
`w `z
``yf
, as claimed.
`f `w
`w `y
, as claimed.
CHAPTER 14 PRACTICE EXERCISES
1. Domain: All points in the xy-plane
Range: z 0
Level curves are ellipses with major axis along the y-axis
and minor axis along the x-axis.
2. Domain: All points in the xy-plane
Range: 0 z _
Level curves are the straight lines x y œ ln z with
slope 1, and z 0.
3. Domain: All (xß y) such that x Á 0 and y Á 0
Range: z Á 0
Level curves are hyperbolas with the x- and y-axes
as asymptotes.
4. Domain: All (xß y) so that x# y
Range: z 0
0
Level curves are the parabolas y œ x# c, c
0.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 14 Practice Exercises
5. Domain: All points (xß yß z) in space
Range: All real numbers
Level surfaces are paraboloids of revolution with
the z-axis as axis.
6. Domain: All points (xß yß z) in space
Range: Nonnegative real numbers
Level surfaces are ellipsoids with center (0ß 0ß 0).
7. Domain: All (xß yß z) such that (xß yß z) Á (0ß !ß 0)
Range: Positive real numbers
Level surfaces are spheres with center (0ß 0ß 0) and
radius r 0.
8. Domain: All points (xß yß z) in space
Range: (0ß 1]
Level surfaces are spheres with center (0ß 0ß 0) and
radius r 0.
9.
lim
Ðxß yÑ Ä Ð1ß ln 2Ñ
ey cos x œ eln 2 cos 1 œ (2)(1) œ 2
2y
10.
lim
Ðxß yÑ Ä Ð0ß 0Ñ x cos y
11.
lim
#
#
Ðxß yÑ Ä Ð1ß 1Ñ x y
xÁ „y
12.
13.
14.
xy
x$ y$ 1
Ðxß yÑ Ä Ð1ß 1Ñ xy 1
lim
lim
P Ä Ð1ß 1ß eÑ
lim
œ
œ
20
0 cos 0
œ2
xy
lim
Ðxß yÑ Ä Ð1ß 1Ñ (x y)(x y)
xÁ „y
œ
œ
(xy 1) ax# y# xy 1b
xy 1
Ðx ß y Ñ Ä Ð 1 ß 1Ñ
lim
1
lim
Ðxß yÑ Ä Ð1ß 1Ñ x y
œ
lim
œ
Ðxß yÑ Ä Ð1ß 1Ñ
"
11
œ
"
#
ax# y# xy 1b œ 1# † 1# 1 † 1 1 œ 3
ln kx y zk œ ln k1 (1) ek œ ln e œ 1
P Ä Ð1 ß 1 ß 1 Ñ
tan" (x y z) œ tan" (1 (1) (1)) œ tan" (1) œ 14
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
863
864
Chapter 14 Partial Derivatives
15. Let y œ kx# , k Á 1. Then
kx#
œ
y
lim
#
Ðxß yÑ Ä Ð0ß 0Ñ x y
y Á x#
lim
#
#
axß kx# b Ä Ð0ß 0Ñ x kx
œ
k
1 k#
which gives different limits for
œ
1 k#
k
which gives different limits for
different values of k Ê the limit does not exist.
16. Let y œ kx, k Á 0. Then
x# y#
xy
Ðxß yÑ Ä Ð0ß 0Ñ
lim
x# (kx)#
x(kx)
(xß kxÑ Ä Ð0ß 0Ñ
œ
lim
xy Á 0
different values of k Ê the limit does not exist.
17. Let y œ kx. Then
x# y#
œ
lim
#
#
Ðxß yÑ Ä Ð0ß 0Ñ x y
x# k# x#
x # k# x#
1 k#
1 k#
œ
which gives different limits for different values
of k Ê the limit does not exist so f(0ß 0) cannot be defined in a way that makes f continuous at the origin.
18. Along the x-axis, y œ 0 and
sin (x y)
kxkkyk
lim
Ðxß yÑ Ä Ð0ß 0Ñ
œ lim
sin x
k xk
xÄ0
œœ
1, x 0
, so the limit fails to exist
", x 0
Ê f is not continuous at (0ß 0).
19.
`g
`r
œ cos ) sin ),
20.
`f
`x
œ
21.
`f
` R"
"
#
Š x# 2x
y# ‹
œ R"# ,
"
`f
` R#
`g
`)
œ r sin ) r cos )
y
‹
x#
y #
1 ˆx‰
Š
œ
œ R"# ,
x#
`f
` R$
#
x
y#
x#
y
y#
œ
xy
x# y#
`f
`y
,
œ
"
#
Š x# 2y
y# ‹
Š 1x ‹
y #
1 ˆx‰
œ
y
x# y#
x
x# y#
œ
xy
x# y#
œ R"#
$
22. hx (xß yß z) œ 21 cos (21x y 3z), hy (xß yß z) œ cos (21x y 3z), hz (xß yß z) œ 3 cos (21x y 3z)
23.
`P
`n
œ
RT
V
,
`P
`R
œ
nT
V
`P
`T
,
œ
nR
V
,
`P
`V
œ nRT
V#
24. fr (rß jß Tß w) œ 2r"# j É 1Tw , fj (rß jß Tß w) œ #r"j# É
25.
œ
"
4rj
É T1"w œ
`g
`x
œ
"
y
,
`g
`y
"
4rjT
œ1
, fT (rß jß Tß w) œ ˆ #"rj ‰ Š È"1w ‹ Š 2È" T ‹
T
1w
É 1Tw , fw (rß jß Tß w) œ ˆ #"rj ‰ É T1 ˆ "# w$Î# ‰ œ 4r"jw É 1Tw
Ê
x
y#
` #g
` x#
œ 0,
` #g
` y#
œ
2x
y$
,
` #g
` y` x
œ
` #g
` x` y
œ y"#
26. gx (xß y) œ ex y cos x, gy (xß y) œ sin x Ê gxx (xß y) œ ex y sin x, gyy (xß y) œ 0, gxy (xß y) œ gyx (xß y) œ cos x
27.
`f
`x
œ 1 y 15x#
2x
x# 1
,
`f
`y
œx Ê
` #f
` x#
œ 30x
22x#
ax# 1b#
,
` #f
` y#
œ 0,
` #f
` y` x
œ
` #f
` x` y
œ1
28. fx (xß y) œ 3y, fy (xß y) œ 2y 3x sin y 7ey Ê fxx (xß y) œ 0, fyy (xß y) œ 2 cos y 7ey , fxy (xß y) œ fyx (xß y) œ 3
29.
`w
`x
Ê
Ê
30.
`w
`x
Ê
Ê
œ y cos (xy 1),
`w
`y
œ x cos (xy 1),
dx
dt
œ et ,
dy
dt
dw
t
ˆ " ‰
dt œ [y cos (xy 1)]e [x cos (xy 1)] t1 ;
dw ¸
ˆ " ‰
dt tœ0 œ 0 † 1 [1 † (1)] 01 œ 1
œ ey ,
`w
`y
œ xey sin z,
dw
y "Î#
axey
dt œ e t
dw ¸
dt tœ1 œ 1 † 1 (2 † 1
`w
`z
œ y cos z sin z,
sin zb ˆ1
"‰
t
dx
dt
œ
"
t1
t œ 0 Ê x œ 1 and y œ 0
œ t"Î# ,
dy
dt
œ 1 "t ,
dz
dt
œ1
(y cos z sin z)1; t œ 1 Ê x œ 2, y œ 0, and z œ 1
0)(2) (0 0)1 œ 5
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 14 Practice Exercises
31.
`w
`x
Ê
Ê
33.
`w
`u
`w
`v
œ
`x
`u
`x
`v
œ
ˆ 1 x x#
`f
`x
œ y z,
`f
`y
œ x z,
œ
Ê
`w
`x
dw
dx
dw
dx
œ ˆ 1 x x#
`x
`r
`x
`s
œ 1,
œ cos s,
" ‰
u
x# 1 a2e cos vb ; u œ v œ 0 Ê
" ‰
`w ¸
u
x# 1 a2e sin vb Ê ` v Ð0ß0Ñ œ
`f
`z
œ y x,
dx
dt
df
dt œ (y z)(sin t) (x z)(cos
df ¸
dt tœ1 œ (sin 1 cos 2)(sin 1)
Ê
34.
œ cos (2x y),
`y
`r
`y
`s
œ s,
œr
`w
` r œ [2 cos (2x y)](1) [ cos (2x y)](s); r œ 1 and s œ 0 Ê x œ 1 and y œ 0
`w ¸
`w
` r Ð1ß0Ñ œ (2 cos 21) (cos 21 )(0) œ 2; ` s œ [2 cos (2x y)](cos s) [ cos (2x y)](r)
`w ¸
` s Ð1ß0Ñ œ (2 cos 21)(cos 0) (cos 21)(1) œ 2 1
Ê
32.
`w
`y
œ 2 cos (2x y),
œ
dw ` s
ds ` x
œ (5)
dw
ds
and
`w
`y
œ
dw ` s
ds ` y
œ sin t,
dy
dt
`w ¸
` u Ð0ß0Ñ
xœ2 Ê
ˆ 52
œ cos t,
dz
dt
"‰
5 (0)
œ ˆ 52 5" ‰ (2) œ
1 y cos xy
2y x cos xy
œ 2 sin 2t
(cos 1 cos 2)(cos 1) 2(sin 1 cos 1)(sin 2)
œ (1)
dw
ds
œ
`w
`x
Ê
dw
ds
5
`w
`y
œ5
œ
dy
dx ¹ Ð0ß1Ñ
1"
2
dw
ds
5
dw
ds
dy
dx ¹ Ð0ßln 2Ñ
1 y cos xy
œ FFxy œ 2y
x cos xy
dy
dx
xby
e
œ FFxy œ 2y
2x exby
ß
1‰
4
#
œ
i
j Ê f increases most rapidly in the direction u œ
È2
#
i
Ê f increases most rapidly in the direction u œ
u œ È12 i
1
È2
i
1
È2
"
È2
œ
È2
#
and decreases most
38. ™ f œ 2xec2y i 2x# ec2y j Ê ™ f k Ð1ß0Ñ œ #i #j Ê k ™ f k œ È2# (2)# œ 2È2; u œ
1
È2
#
œ "# i "# j Ê k ™ f k œ Ɉ "# ‰ ˆ "# ‰ œ
È2
# j
È
È
È
È
rapidly in the direction u œ #2 i #2 j ; (Du f)P! œ k ™ f k œ #2 and (Dcu f)P! œ #2 ;
7
u" œ kvvk œ È33i #4j4# œ 53 i 54 j Ê (Du" f)P! œ ™ f † u" œ ˆ "# ‰ ˆ 35 ‰ ˆ "# ‰ ˆ 45 ‰ œ 10
™f
k™f k
È2
#
dy
dx
2
œ 2 ln0 2
2 œ (ln 2 1)
37. ™ f œ ( sin x cos y)i (cos x sin y)j Ê ™ f k ˆ 14
È2
#
œ0
œ 1
36. F(xß y) œ 2xy exy 2 Ê Fx œ 2y exy and Fy œ 2x exy Ê
uœ
;
t) 2(y x)(sin 2t); t œ 1 Ê x œ cos 1, y œ sin 1, and z œ cos 2
Ê at (xß y) œ (!ß 1) we have
Ê at (xß y) œ (!ß ln 2) we have
2
5
œ0
35. F(xß y) œ 1 x y# sin xy Ê Fx œ 1 y cos xy and Fy œ 2y x cos xy Ê
œ
865
™f
k™f k
œ
1
È2
i
1
È2
j
j and decreases most rapidly in the direction
j ; (Du f)P! œ k ™ f k œ 2È2 and (Dcu f)P! œ 2È2 ; u" œ
v
kv k
œ
ij
È 1# 1#
œ
1
È2
i
1
È2
j
Ê (Du" f)P! œ ™ f † u" œ (2) Š È"2 ‹ (2) Š È"2 ‹ œ 0
2
3
6
39. ™ f œ Š 2x 3y
6z ‹ i Š 2x 3y 6z ‹ j Š 2x 3y 6z ‹ k Ê ™ f k Ð 1ß 1ß1Ñ œ 2i 3j 6k ;
uœ
™f
k™f k
œ
2i 3j 6k
È 2# 3# 6#
œ
2
7
i 73 j 76 k Ê f increases most rapidly in the direction u œ
2
7
i 37 j 67 k and
decreases most rapidly in the direction u œ 27 i 37 j 67 k ; (Du f)P! œ k ™ f k œ 7, (Du f)P! œ 7;
u" œ
v
kv k
œ
2
7
i 37 j 67 k Ê (Du" f)P! œ (Du f)P! œ 7
40. ™ f œ (2x 3y)i (3x 2)j (1 2z)k Ê ™ f k Ð0ß0ß0Ñ œ 2j k ; u œ
rapidly in the direction u œ
2
È5
j
"
È5
™f
k™f k
œ
2
È5
j
"
È5
k Ê f increases most
k and decreases most rapidly in the direction u œ È25 j
(Du f)P! œ k ™ f k œ È5 and (Du f)P! œ È5 ; u" œ
v
kvk
œ
ijk
È 1# 1# 1#
Ê (Du" f)P! œ ™ f † u" œ (0) Š È"3 ‹ (2) Š È"3 ‹ (1) Š È"3 ‹ œ
3
È3
œ
"
È3
i
"
È3
j
"
È3
k
œ È3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
È5
k;
;
866
Chapter 14 Partial Derivatives
41. r œ (cos 3t)i (sin 3t)j 3tk Ê v(t) œ (3 sin 3t)i (3 cos 3t)j 3k Ê v ˆ 13 ‰ œ 3j 3k
Ê u œ È"2 j
"
È2
k ; f(xß yß z) œ xyz Ê ™ f œ yzi xzj xyk ; t œ
Ê ™ f k Ð 1ß0ß1Ñ œ 1j Ê ™ f † u œ (1j) † Š È"2 j
"
È2
k‹ œ
1
3
yields the point on the helix (1ß 0ß 1)
1
È2
42. f(xß yß z) œ xyz Ê ™ f œ yzi xzj xyk ; at (1ß 1ß 1) we get ™ f œ i j k Ê the maximum value of
Du f k
œ k ™ f k œ È3
Ð1ß1ß1Ñ
43. (a) Let ™ f œ ai bj at (1ß 2). The direction toward (2ß 2) is determined by v" œ (2 1)i (2 2)j œ i œ u
so that ™ f † u œ 2 Ê a œ 2. The direction toward (1ß 1) is determined by v# œ (1 1)i (1 2)j œ j œ u
so that ™ f † u œ 2 Ê b œ 2 Ê b œ 2. Therefore ™ f œ 2i 2j ; fx a1, 2b œ fy a1, 2b œ 2.
(b) The direction toward (4ß 6) is determined by v$ œ (4 1)i (6 2)j œ 3i 4j Ê u œ 35 i 45 j
Ê ™f†uœ
14
5
.
44. (a) True
(b) False
(c) True
(d) True
45. ™ f œ 2xi j 2zk Ê
™ f k Ð0ß 1ß 1Ñ œ j 2k ,
™ f k Ð0ß0ß0Ñ œ j ,
™ f k Ð0ß 1ß1Ñ œ j 2k
46. ™ f œ 2yj 2zk Ê
™ f k Ð2ß2ß0Ñ œ 4j ,
™ f k Ð2ß 2ß0Ñ œ 4j ,
™ f k Ð2ß0ß2Ñ œ 4k ,
™ f k Ð2ß0ß 2Ñ œ 4k
47. ™ f œ 2xi j 5k Ê ™ f k Ð2ß 1ß1Ñ œ 4i j 5k Ê Tangent Plane: 4(x 2) (y 1) 5(z 1) œ 0
Ê 4x y 5z œ 4; Normal Line: x œ 2 4t, y œ 1 t, z œ 1 5t
48. ™ f œ 2xi 2yj k Ê ™ f k Ð1ß1ß2Ñ œ 2i 2j k Ê Tangent Plane: 2(x 1) 2(y 1) (z 2) œ 0
Ê 2x 2y z 6 œ 0; Normal Line: x œ 1 2t, y œ 1 2t, z œ 2 t
49.
`z
`x
œ
2x
x# y#
Ê
`z ¸
` x Ð0ß1ß0Ñ
œ 0 and
`z
`y
œ
2y
x# y#
Ê
`z
` y ¹ Ð0ß1ß0Ñ
œ 2; thus the tangent plane is
2(y 1) (z 0) œ 0 or 2y z 2 œ 0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 14 Practice Exercises
50.
`z
`x
œ 2x ax# y# b
#
`z ¸
` x ˆ1ß1ß 12 ‰
Ê
œ #" and
`z
`y
œ 2y ax# y# b
#
Ê
`z
` y ¹ ˆ1ß1ß 1 ‰
2
867
œ "# ; thus the tangent
plane is "# (x 1) "# (y 1) ˆz "# ‰ œ 0 or x y 2z 3 œ 0
51. ™ f œ ( cos x)i j Ê ™ f k Ð1ß1Ñ œ i j Ê the tangent
line is (x 1) (y 1) œ 0 Ê x y œ 1 1; the
normal line is y 1 œ 1(x 1) Ê y œ x 1 1
52. ™ f œ xi yj Ê ™ f k Ð1ß2Ñ œ i 2j Ê the tangent
line is (x 1) 2(y 2) œ 0 Ê y œ
"
#
x #3 ; the normal
line is y 2 œ 2(x 1) Ê y œ 2x 4
53. Let f(xß yß z) œ x# 2y 2z 4 and g(xß yß z) œ y 1. Then ™ f œ 2xi 2j 2kk a1 1 12 b œ 2i 2j 2k
â
â
â i j kâ
â
â
and ™ g œ j Ê ™ f ‚ ™ g œ â 2 2 2 â œ 2i 2k Ê the line is x œ 1 2t, y œ 1, z œ "# 2t
â
â
â0 " 0â
ß ß
54. Let f(xß yß z) œ x y# z 2 and g(xß yß z) œ y 1. Then ™ f œ i 2yj kk a 12 1 12 b œ i 2j k and
â
â
â i j kâ
â
â
™ g œ j Ê ™ f ‚ ™ g œ â 1 2 1 â œ i k Ê the line is x œ "# t, y œ 1, z œ "# t
â
â
â0 " 0â
ß ß
55. f ˆ 14 ß 14 ‰ œ
"
#
, fx ˆ 14 ß 14 ‰ œ cos x cos yk Ð1Î4ß1Î4Ñ œ
Ê L(xß y) œ
"
#
"# ˆx 14 ‰ "# ˆy 14 ‰ œ
"
#
"
#
"
#
, fy ˆ 14 ß 14 ‰ œ sin x sin yk Ð1Î4ß1Î4Ñ œ "#
x "# y; fxx (xß y) œ sin x cos y, fyy (xß y) œ sin x cos y, and
fxy (xß y) œ cos x sin y. Thus an upper bound for E depends on the bound M used for kfxx k , kfxy k , and kfyy k .
With M œ
È2
#
we have kE(xß y)k Ÿ
with M œ 1, kE(xß y)k Ÿ
"
#
"
#
Š
È2
ˆ¸
# ‹ x
#
14 ¸ ¸y 14 ¸‰ Ÿ
#
(1) ˆ¸x 14 ¸ ¸y 14 ¸‰ œ
"
#
È2
4
(0.2)# Ÿ 0.0142;
(0.2)# œ 0.02.
56. f(1ß 1) œ 0, fx (1ß 1) œ yk Ð1ß1Ñ œ 1, fy (1ß 1) œ x 6yk Ð1ß1Ñ œ 5 Ê L(xß y) œ (x 1) 5(y 1) œ x 5y 4;
fxx (xß y) œ 0, fyy (xß y) œ 6, and fxy (xß y) œ 1 Ê maximum of kfxx k , kfyy k , and kfxy k is 6 Ê M œ 6
Ê kE(xß y)k Ÿ
"
#
(6) akx 1k ky 1kb# œ
"
#
(6)(0.1 0.2)# œ 0.27
57. f(1ß 0ß 0) œ 0, fx (1ß 0ß 0) œ y 3zk Ð1ß0ß0Ñ œ 0, fy (1ß 0ß 0) œ x 2zk Ð1ß0ß0Ñ œ 1, fz (1ß 0ß 0) œ 2y 3xk Ð1ß0ß0Ñ œ 3
Ê L(xß yß z) œ 0(x 1) (y 0) 3(z 0) œ y 3z; f(1ß 1ß 0) œ 1, fx (1ß 1ß 0) œ 1, fy (1ß 1ß 0) œ 1, fz ("ß "ß !) œ 1
Ê L(xß yß z) œ 1 (x 1) (y 1) 1(z 0) œ x y z 1
58. f ˆ0ß !ß 14 ‰ œ 1, fx ˆ!ß 0ß 14 ‰ œ È2 sin x sin (y z)¹
ˆ0ß0ß 1 ‰
œ 0, fy ˆ!ß 0ß 14 ‰ œ È2 cos x cos (y z)¹
4
fz ˆ!ß 0ß 14 ‰ œ È2 cos x cos (y z)¹
ˆ0ß0ß 1 ‰
È2
#
œ 1 Ê L(xß yß z) œ 1 1(y 0) 1 ˆz 14 ‰ œ 1 y z
Ê L(xß yß z) œ
È2
È2
È2
ˆ1 1 ‰
ˆ1 1 ‰
# , fy 4 ß 4 ß 0 œ # , fz 4 ß 4 ß 0 œ #
È
È
È
È
È
#2 ˆy 14 ‰ #2 (z 0) œ #2 #2 x #2
, fx ˆ 14 ß 14 ß 0‰ œ
È2
#
È2
#
ˆx 14 ‰
œ 1,
4
4
f ˆ 14 ß 14 ß 0‰ œ
ˆ0ß0ß 1 ‰
y
È2
#
z
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1
4
;
868
Chapter 14 Partial Derivatives
59. V œ 1r# h Ê dV œ 21rh dr 1r# dh Ê dVk Ð1Þ5ß5280Ñ œ 21(1.5)(5280) dr 1(1.5)# dh œ 15,8401 dr 2.251 dh.
You should be more careful with the diameter since it has a greater effect on dV.
60. df œ (2x y) dx (x 2y) dy Ê df k Ð1ß2Ñ œ 3 dy Ê f is more sensitive to changes in y; in fact, near the point
(1ß 2) a change in x does not change f.
61. dI œ
"
R
dV
V
R#
"
100
dR Ê dI¸ Ð24ß100Ñ œ
dV
24
100#
dR Ê dI¸ dVœ1ßdRœ20 œ 0.01 (480)(.0001) œ 0.038,
" ‰
20 ‰
or increases by 0.038 amps; % change in V œ (100) ˆ 24
¸ 4.17%; % change in R œ ˆ 100
(100) œ 20%;
Iœ
24
100
œ 0.24 Ê estimated % change in I œ
dI
I
‚ 100 œ
0.038
0.24
‚ 100 ¸ 15.83% Ê more sensitive to voltage change.
62. A œ 1ab Ê dA œ 1b da 1a db Ê dAk Ð10ß16Ñ œ 161 da 101 db; da œ „ 0.1 and db œ „ 0.1
¸ ¸ 2.61
¸
Ê dA œ „ 261(0.1) œ „ 2.61 and A œ 1(10)(16) œ 1601 Ê ¸ dA
A ‚ 100 œ 1601 ‚ 100 ¸ 1.625%
63. (a) y œ uv Ê dy œ v du u dv; percentage change in u Ÿ 2% Ê kduk Ÿ 0.02, and percentage change in v Ÿ 3%
Ê kdvk Ÿ 0.03;
dy
y
Ÿ 2% 3% œ 5%
(b) z œ u v Ê dzz œ
œ
v du u dv
uv
du dv
uv
œ
œ
du
uv
Ê ¸ dzz ‚ 100¸ Ÿ ¸ du
u ‚ 100
64. C œ
Ê
dv
v
du
u
dv
uv
Þ
Þ
Þ
Þ
Þ
Þ
Ÿ
¸ du
Ê ¹ dy
y ‚ 100¹ œ u ‚ 100
du
u
dv
v
dv
v
¸ ¸ dv
¸
‚ 100¸ Ÿ ¸ du
u ‚ 100 v ‚ 100
(since u 0, v 0)
‚ 100¸ œ ¹ dy
y ‚ 100¹
(0.425)(7)
7
71.84w0 425 h0 725 Ê Cw œ 71.84w1 425 h0 725
2.975
5.075
dC œ 71.84w
1 425 h0 725 dw 71.84w0 425 h1 725
Þ
dv
v
Þ
and Ch œ
(0.725)(7)
71.84w0 425 h1 725
Þ
Þ
dh; thus when w œ 70 and h œ 180 we have
dCk Ð70ß180Ñ ¸ (0.00000225) dw (0.00000149) dh Ê 1 kg error in weight has more effect
65. fx (xß y) œ 2x y 2 œ 0 and fy (xß y) œ x 2y 2 œ 0 Ê x œ 2 and y œ 2 Ê (2ß 2) is the critical point;
#
œ 3 0 and fxx 0 Ê local minimum value
fxx (2ß 2) œ 2, fyy (#ß 2) œ 2, fxy (#ß 2) œ 1 Ê fxx fyy fxy
of f(#ß 2) œ 8
66. fx (xß y) œ 10x 4y 4 œ 0 and fy (xß y) œ 4x 4y 4 œ 0 Ê x œ 0 and y œ 1 Ê (0ß 1) is the critical point;
#
œ 56 0 Ê saddle point with f(0ß 1) œ 2
fxx (0ß 1) œ 10, fyy (0ß 1) œ 4, fxy (0ß 1) œ 4 Ê fxx fyy fxy
67. fx (xß y) œ 6x# 3y œ 0 and fy (xß y) œ 3x 6y# œ 0 Ê y œ 2x# and 3x 6 a4x% b œ 0 Ê x a1 8x$ b œ 0
Ê x œ 0 and y œ 0, or x œ "# and y œ "# Ê the critical points are (0ß 0) and ˆ "# ß "# ‰ . For (!ß !):
#
fxx (!ß !) œ 12xk Ð0ß0Ñ œ 0, fyy (!ß !) œ 12yk Ð0ß0Ñ œ 0, fxy (!ß 0) œ 3 Ê fxx fyy fxy
œ 9 0 Ê saddle point with
#
f(0ß 0) œ 0. For ˆ "# ß "# ‰: fxx œ 6, fyy œ 6, fxy œ 3 Ê fxx fyy fxy
œ 27 0 and fxx 0 Ê local maximum
"
"
"
value of f ˆ # ß # ‰ œ 4
68. fx (xß y) œ 3x# 3y œ 0 and fy (xß y) œ 3y# 3x œ 0 Ê y œ x# and x% x œ 0 Ê x ax$ 1b œ 0 Ê the critical
points are (0ß 0) and (1ß 1) . For (!ß !): fxx (!ß !) œ 6xk Ð0ß0Ñ œ 0, fyy (!ß !) œ 6yk Ð0ß0Ñ œ 0, fxy (!ß 0) œ 3
#
Ê fxx fyy fxy
œ 9 0 Ê saddle point with f(0ß 0) œ 15. For (1ß 1): fxx (1ß 1) œ 6, fyy (1ß 1) œ 6, fxy (1ß 1) œ 3
#
Ê fxx fyy fxy
œ 27 0 and fxx 0 Ê local minimum value of f(1ß 1) œ 14
69. fx (xß y) œ 3x# 6x œ 0 and fy (xß y) œ 3y# 6y œ 0 Ê x(x 2) œ 0 and y(y 2) œ 0 Ê x œ 0 or x œ 2 and
y œ 0 or y œ 2 Ê the critical points are (0ß 0), (0ß 2), (2ß 0), and (2ß 2) . For (!ß !): fxx (!ß !) œ 6x 6k Ð0ß0Ñ
#
œ 6, fyy (!ß !) œ 6y 6k Ð0ß0Ñ œ 6, fxy (!ß 0) œ 0 Ê fxx fyy fxy
œ 36 0 Ê saddle point with f(0ß 0) œ 0. For
#
(0ß 2): fxx (!ß 2) œ 6, fyy (0ß #) œ 6, fxy (!ß 2) œ 0 Ê fxx fyy fxy
œ 36 0 and fxx 0 Ê local minimum value of
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 14 Practice Exercises
869
#
f(!ß 2) œ 4. For (#ß 0): fxx (2ß 0) œ 6, fyy (#ß 0) œ 6, fxy (2ß 0) œ 0 Ê fxx fyy fxy
œ 36 0 and fxx 0
Ê local maximum value of f(2ß 0) œ 4. For (2ß 2): fxx (2ß 2) œ 6, fyy (2ß 2) œ 6, fxy (2ß 2) œ 0
#
Ê fxx fyy fxy
œ 36 0 Ê saddle point with f(2ß 2) œ 0.
70. fx (xß y) œ 4x$ 16x œ 0 Ê 4x ax# 4b œ 0 Ê x œ 0, 2, 2; fy (xß y) œ 6y 6 œ 0 Ê y œ 1. Therefore the critical
points are (0ß 1), (2ß 1), and (2ß 1). For (!ß 1): fxx (!ß 1) œ 12x# 16k Ð0ß1Ñ œ 16, fyy (!ß 1) œ 6, fxy (!ß 1) œ 0
#
Ê fxx fyy fxy
œ 96 0 Ê saddle point with f(0ß 1) œ 3. For (2ß 1): fxx (2ß 1) œ 32, fyy (2ß 1) œ 6,
#
fxy (2ß 1) œ 0 Ê fxx fyy fxy
œ 192 0 and fxx 0 Ê local minimum value of f(2ß 1) œ 19. For (#ß 1):
#
fxx (2ß 1) œ 32, fyy (#ß 1) œ 6, fxy (2ß 1) œ 0 Ê fxx fyy fxy
œ 192 0 and fxx 0 Ê local minimum value of
f(2ß 1) œ 19.
71. (i)
On OA, f(xß y) œ f(0ß y) œ y# 3y for 0 Ÿ y Ÿ 4
Ê f w (!ß y) œ 2y 3 œ 0 Ê y œ 3# . But ˆ!ß 3# ‰
is not in the region.
Endpoints: f(0ß 0) œ 0 and f(0ß 4) œ 28.
(ii) On AB, f(xß y) œ f(xß x 4) œ x# 10x 28
for 0 Ÿ x Ÿ 4 Ê f w (xß x 4) œ 2x 10 œ 0
Ê x œ 5, y œ 1. But (5ß 1) is not in the region.
Endpoints: f(4ß 0) œ 4 and f(!ß 4) œ 28.
(iii) On OB, f(xß y) œ f(xß 0) œ x# 3x for 0 Ÿ x Ÿ 4 Ê f w (xß 0) œ 2x 3 Ê x œ
critical point with f ˆ 3# ß !‰ œ 94 .
3
#
and y œ 0 Ê ˆ 3# ß 0‰ is a
Endpoints: f(0ß 0) œ 0 and f(%ß 0) œ 4.
(iv) For the interior of the triangular region, fx (xß y) œ 2x y 3 œ 0 and fy (xß y) œ x 2y 3 œ 0 Ê x œ 3
and y œ 3. But (3ß 3) is not in the region. Therefore the absolute maximum is 28 at (0ß 4) and the
absolute minimum is 94 at ˆ 3# ß !‰ .
On OA, f(xß y) œ f(0ß y) œ y# 4y 1 for
0 Ÿ y Ÿ 2 Ê f w (!ß y) œ 2y 4 œ 0 Ê y œ 2 and
x œ 0. But (0ß 2) is not in the interior of OA.
Endpoints: f(0ß 0) œ 1 and f(0ß 2) œ 5.
(ii) On AB, f(xß y) œ f(xß 2) œ x# 2x 5 for 0 Ÿ x Ÿ 4
Ê f w (xß 2) œ 2x 2 œ 0 Ê x œ 1 and y œ 2
Ê (1ß 2) is an interior critical point of AB with
f(1ß 2) œ 4. Endpoints: f(4ß 2) œ 13 and f(!ß 2) œ 5.
(iii) On BC, f(xß y) œ f(4ß y) œ y# 4y 9 for 0 Ÿ y Ÿ 2 Ê f w (4ß y) œ 2y 4 œ 0 Ê y œ # and x œ 4. But
(4ß 2) is not in the interior of BC. Endpoints: f(4ß 0) œ 9 and f(%ß 2) œ 13.
(iv) On OC, f(xß y) œ f(xß 0) œ x# 2x 1 for 0 Ÿ x Ÿ 4 Ê f w (xß 0) œ 2x 2 œ 0 Ê x œ 1 and y œ 0 Ê (1ß 0)
is an interior critical point of OC with f(1ß 0) œ 0. Endpoints: f(0ß 0) œ 1 and f(4ß 0) œ 9.
(v) For the interior of the rectangular region, fx (xß y) œ 2x 2 œ 0 and fy (xß y) œ 2y 4 œ 0 Ê x œ 1 and
y œ 2. But (1ß 2) is not in the interior of the region. Therefore the absolute maximum is 13 at (4ß 2)
and the absolute minimum is 0 at (1ß 0).
72. (i)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
870
73. (i)
Chapter 14 Partial Derivatives
On AB, f(xß y) œ f(2ß y) œ y# y 4 for
2 Ÿ y Ÿ 2 Ê f w (2ß y) œ 2y 1 Ê y œ "# and
x œ 2 Ê ˆ2ß "# ‰ is an interior critical point in AB
with f ˆ2ß "# ‰ œ 17
4 . Endpoints: f(2ß 2) œ 2 and
f(2ß 2) œ 2.
On BC, f(xß y) œ f(xß 2) œ 2 for 2 Ÿ x Ÿ 2
Ê f w (xß 2) œ 0 Ê no critical points in the interior of
BC. Endpoints: f(2ß 2) œ 2 and f(2ß 2) œ 2.
(iii) On CD, f(xß y) œ f(2ß y) œ y# 5y 4 for
2 Ÿ y Ÿ 2 Ê f w (2ß y) œ 2y 5 œ 0 Ê y œ 5# and x œ 2. But ˆ#ß 5# ‰ is not in the region.
(ii)
Endpoints: f(2ß 2) œ 18 and f(2ß 2) œ 2.
(iv) On AD, f(xß y) œ f(xß 2) œ 4x 10 for 2 Ÿ x Ÿ 2 Ê f w (xß 2) œ 4 Ê no critical points in the interior
of AD. Endpoints: f(2ß 2) œ 2 and f(2ß 2) œ 18.
(v) For the interior of the square, fx (xß y) œ y 2 œ 0 and fy (xß y) œ 2y x 3 œ 0 Ê y œ 2 and x œ 1
Ê (1ß 2) is an interior critical point of the square with f(1ß 2) œ 2. Therefore the absolute maximum
"‰
ˆ
is 18 at (2ß 2) and the absolute minimum is 17
4 at #ß # .
On OA, f(xß y) œ f(0ß y) œ 2y y# for 0 Ÿ y Ÿ 2
Ê f w (!ß y) œ 2 2y œ 0 Ê y œ 1 and x œ 0 Ê
(!ß 1) is an interior critical point of OA with
f(0ß 1) œ 1. Endpoints: f(0ß 0) œ 0 and f(0ß 2) œ 0.
(ii) On AB, f(xß y) œ f(xß 2) œ 2x x# for 0 Ÿ x Ÿ 2
Ê f w (xß 2) œ 2 2x œ 0 Ê x œ 1 and y œ 2
Ê (1ß 2) is an interior critical point of AB with
f(1ß 2) œ 1. Endpoints: f(0ß 2) œ 0 and f(2ß 2) œ 0.
(iii) On BC, f(xß y) œ f(2ß y) œ 2y y# for 0 Ÿ y Ÿ 2
Ê f w (2ß y) œ 2 2y œ 0 Ê y œ 1 and x œ 2
Ê (2ß 1) is an interior critical point of BC with f(2ß 1) œ 1. Endpoints: f(2ß 0) œ 0 and f(2ß 2) œ 0.
(iv) On OC, f(xß y) œ f(xß 0) œ 2x x# for 0 Ÿ x Ÿ 2 Ê f w (xß 0) œ 2 2x œ 0 Ê x œ 1 and y œ 0 Ê (1ß 0)
is an interior critical point of OC with f(1ß 0) œ 1. Endpoints: f(0ß 0) œ 0 and f(0ß 2) œ 0.
(v) For the interior of the rectangular region, fx (xß y) œ 2 2x œ 0 and fy (xß y) œ 2 2y œ 0 Ê x œ 1 and
y œ 1 Ê (1ß 1) is an interior critical point of the square with f(1ß 1) œ 2. Therefore the absolute maximum
is 2 at (1ß 1) and the absolute minimum is 0 at the four corners (0ß 0), (0ß 2), (2ß 2), and (2ß 0).
74. (i)
On AB, f(xß y) œ f(xß x 2) œ 2x 4 for
2 Ÿ x Ÿ 2 Ê f w (xß x 2) œ 2 œ 0 Ê no critical
points in the interior of AB. Endpoints: f(2ß 0) œ 8
and f(2ß 4) œ 0.
(ii) On BC, f(xß y) œ f(2ß y) œ y# 4y for 0 Ÿ y Ÿ 4
Ê f w (2ß y) œ 2y 4 œ 0 Ê y œ 2 and x œ 2
Ê (2ß 2) is an interior critical point of BC with
f(2ß 2) œ 4. Endpoints: f(2ß 0) œ 0 and f(2ß 4) œ 0.
(iii) On AC, f(xß y) œ f(xß 0) œ x# 2x for 2 Ÿ x Ÿ 2
Ê f w (xß 0) œ 2x 2 Ê x œ 1 and y œ 0 Ê (1ß 0) is an interior critical point of AC with f(1ß 0) œ 1.
Endpoints: f(2ß 0) œ 8 and f(2ß 0) œ 0.
(iv) For the interior of the triangular region, fx (xß y) œ 2x 2 œ 0 and fy (xß y) œ 2y 4 œ 0 Ê x œ 1 and
y œ 2 Ê (1ß 2) is an interior critical point of the region with f(1ß 2) œ 3. Therefore the absolute maximum
is 8 at (2ß 0) and the absolute minimum is 1 at (1ß 0).
75. (i)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 14 Practice Exercises
76. (i)
(ii)
871
On AB, faxß yb œ faxß xb œ 4x# 2x% 16 for
2 Ÿ x Ÿ 2 Ê f w axß xb œ 8x 8x$ œ 0 Ê x œ 0
and y œ 0, or x œ 1 and y œ 1, or x œ 1 and y œ 1
Ê a0ß 0b, a1ß 1b, a1ß 1b are all interior points of AB
with fa0ß 0b œ 16, fa1ß 1b œ 18, and fa1ß 1b œ 18.
Endpoints: fa2ß 2b œ 0 and fa2ß 2b œ 0.
On BC, faxß yb œ fa2ß yb œ 8y y% for 2 Ÿ y Ÿ 2
3
Ê f w a2ß yb œ 8 4y$ œ 0 Ê y œ È
2 and x œ 2
3
Ê Š2ß È
2‹ is an interior critical point of BC with
3
3
f Š2ß È
2‹ œ 6 È
2. Endpoints: fa2ß 2b œ 32 and fa2ß 2b œ 0.
3
(iii) On AC, faxß yb œ faxß 2b œ 8x x% for 2 Ÿ x Ÿ 2 Ê f w axß 2b œ 8 4x$ œ 0 Ê x œ È
2 and y œ 2
3
3
3
Ê ŠÈ
2ß 2‹ is an interior critical point of AC with f ŠÈ
2ß 2‹ œ 6 È
2. Endpoints:
fa2ß 2b œ 0 and fa2ß 2b œ 32.
(iv) For the interior of the triangular region, fx axß yb œ 4y 4x$ œ 0 and fy axß yb œ 4x 4y$ œ 0 Ê x œ 0 and
y œ 0, or x œ 1 and y œ 1 or x œ 1 and y œ 1. But neither of the points a0ß 0b and a1ß 1b, or a1ß 1b are
interior to the region. Therefore the absolute maximum is 18 at (1ß 1) and (1ß 1), and the absolute minimum is
32 at a2ß 2b.
On AB, f(xß y) œ f(1ß y) œ y$ 3y# 2 for
1 Ÿ y Ÿ 1 Ê f w (1ß y) œ 3y# 6y œ 0 Ê y œ 0
and x œ 1, or y œ 2 and x œ 1 Ê (1ß 0) is an
interior critical point of AB with f(1ß 0) œ 2; (1ß 2)
is outside the boundary. Endpoints: f(1ß 1) œ 2
and f(1ß 1) œ 0.
(ii) On BC, f(xß y) œ f(xß 1) œ x$ 3x# 2 for
1 Ÿ x Ÿ 1 Ê f w (xß 1) œ 3x# 6x œ 0 Ê x œ 0
and y œ 1, or x œ 2 and y œ 1 Ê (0ß 1) is an
interior critical point of BC with f(!ß 1) œ 2; (2ß 1) is outside the boundary. Endpoints: f("ß 1) œ 0 and
f("ß 1) œ 2.
(iii) On CD, f(xß y) œ f("ß y) œ y$ 3y# 4 for 1 Ÿ y Ÿ 1 Ê f w ("ß y) œ 3y# 6y œ 0 Ê y œ 0 and x œ 1, or
y œ 2 and x œ 1 Ê ("ß 0) is an interior critical point of CD with f("ß 0) œ 4; (1ß 2) is outside the boundary.
Endpoints: f(1ß 1) œ 2 and f("ß 1) œ 0.
(iv) On AD, f(xß y) œ f(xß 1) œ x$ 3x# 4 for 1 Ÿ x Ÿ 1 Ê f w (xß 1) œ 3x# 6x œ 0 Ê x œ 0 and y œ 1,
or x œ 2 and y œ 1 Ê (0ß 1) is an interior point of AD with f(0ß 1) œ 4; (#ß 1) is outside the
boundary. Endpoints: f(1ß 1) œ 2 and f("ß 1) œ 0.
(v) For the interior of the square, fx (xß y) œ 3x# 6x œ 0 and fy (xß y) œ 3y# 6y œ 0 Ê x œ 0 or x œ 2, and
y œ 0 or y œ 2 Ê (0ß 0) is an interior critical point of the square region with f(!ß 0) œ 0; the points (0ß 2),
(2ß 0), and (2ß 2) are outside the region. Therefore the absolute maximum is 4 at (1ß 0) and the
absolute minimum is 4 at (0ß 1).
77. (i)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
872
Chapter 14 Partial Derivatives
On AB, f(xß y) œ f(1ß y) œ y$ 3y for 1 Ÿ y Ÿ 1
Ê f w (1ß y) œ 3y# 3 œ 0 Ê y œ „ 1 and x œ 1
yielding the corner points (1ß 1) and (1ß 1) with
f(1ß 1) œ 2 and f(1ß 1) œ 2.
(ii) On BC, f(xß y) œ f(xß 1) œ x$ 3x 2 for
1 Ÿ x Ÿ 1 Ê f w (xß 1) œ 3x# 3 œ 0 Ê no
solution. Endpoints: f("ß 1) œ 2 and f("ß 1) œ 6.
(iii) On CD, f(xß y) œ f("ß y) œ y$ 3y 2 for
1 Ÿ y Ÿ 1 Ê f w ("ß y) œ 3y# 3 œ 0 Ê no
solution. Endpoints: f(1ß 1) œ 6 and f("ß 1) œ 2.
(iv) On AD, f(xß y) œ f(xß 1) œ x$ 3x for 1 Ÿ x Ÿ 1 Ê f w (xß 1) œ 3x# 3 œ 0 Ê x œ „ 1 and y œ 1
yielding the corner points (1ß 1) and (1ß 1) with f(1ß 1) œ 2 and f(1ß 1) œ 2
(v) For the interior of the square, fx (xß y) œ 3x# 3y œ 0 and fy (xß y) œ 3y# 3x œ 0 Ê y œ x# and
x% x œ 0 Ê x œ 0 or x œ 1 Ê y œ 0 or y œ 1 Ê (!ß 0) is an interior critical point of the square
region with f(0ß 0) œ 1; (1ß 1) is on the boundary. Therefore the absolute maximum is 6 at ("ß 1) and
the absolute minimum is 2 at (1ß 1) and (1ß 1).
78. (i)
79. ™ f œ 3x# i 2yj and ™ g œ 2xi 2yj so that ™ f œ - ™ g Ê 3x# i 2yj œ -(2xi 2yj) Ê 3x# œ 2x- and
2y œ 2y- Ê - œ 1 or y œ 0.
CASE 1: - œ 1 Ê 3x# œ 2x Ê x œ 0 or x œ 23 ; x œ 0 Ê y œ „ 1 yielding the points (0ß 1) and (!ß 1); x œ
Ê yœ „
È5
3
yielding the points Š 32 ß
È5
3 ‹
and Š 32 ß
2
3
È5
3 ‹.
CASE 2: y œ 0 Ê x# 1 œ 0 Ê x œ „ 1 yielding the points (1ß 0) and (1ß 0).
Evaluations give f a!ß „ 1b œ 1, f Š 23 ß „
È5
3 ‹
œ
23
27
, f("ß 0) œ 1, and f("ß 0) œ 1. Therefore the absolute
maximum is 1 at a!ß „ 1b and (1ß 0), and the absolute minimum is 1 at ("ß !).
80. ™ f œ yi xj and ™ g œ 2xi 2yj so that ™ f œ - ™ g Ê yi xj œ -(2xi 2yj) Ê y œ 2-x and
xy œ 2-y Ê x œ 2-(2-x) œ 4-# x Ê x œ 0 or 4-# œ 1.
CASE 1: x œ 0 Ê y œ 0 but (0ß 0) does not lie on the circle, so no solution.
CASE 2: 4-# œ 1 Ê - œ "# or - œ "# . For - œ "# , y œ x Ê 1 œ x# y# œ 2x# Ê x œ C œ „ È"2 yielding the
points Š È"2 ß È"2 ‹ and Š È"2 , È"2 ‹ . For - œ #" , y œ x Ê 1 œ x# y# œ 2x# Ê x œ „
"
È2
and
y œ x yielding the points Š È"2 ß È"2 ‹ and Š È"2 , È"2 ‹ .
Evaluations give the absolute maximum value f Š È"2 ß È"2 ‹ œ f Š È"2 ß È"2 ‹ œ
"
#
and the absolute minimum
value f Š È"2 ß È"2 ‹ œ f Š È"2 ß È"2 ‹ œ #" .
81. (i) f(xß y) œ x# 3y# 2y on x# y# œ 1 Ê ™ f œ 2xi (6y 2)j and ™ g œ 2xi 2yj so that ™ f œ - ™ g
Ê 2xi (6y 2)j œ -(2xi 2yj) Ê 2x œ 2x- and 6y 2 œ 2y- Ê - œ 1 or x œ 0.
CASE 1: - œ 1 Ê 6y 2 œ 2y Ê y œ "# and x œ „
È3
#
yielding the points Š „
È3
#
ß #" ‹ .
CASE 2: x œ 0 Ê y# œ 1 Ê y œ „ 1 yielding the points a!ß „ 1b .
Evaluations give f Š „
È3
#
ß "# ‹ œ
"
#
, f(0ß 1) œ 5, and f(!ß 1) œ 1. Therefore
"
#
and 5 are the extreme
values on the boundary of the disk.
(ii) For the interior of the disk, fx (xß y) œ 2x œ 0 and fy (xß y) œ 6y 2 œ 0 Ê x œ 0 and y œ "3
Ê ˆ!ß 13 ‰ is an interior critical point with f ˆ!ß 3" ‰ œ 3" . Therefore the absolute maximum of f on the
disk is 5 at (0ß 1) and the absolute minimum of f on the disk is "3 at ˆ!ß 3" ‰ .
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 14 Practice Exercises
873
82. (i) f(xß y) œ x# y# 3x xy on x# y# œ 9 Ê ™ f œ (2x 3 y)i (2y x)j and ™ g œ 2xi 2yj so that
™ f œ - ™ g Ê (2x 3 y)i (2y x)j œ -(2xi 2yj) Ê 2x 3 y œ 2x- and 2y x œ 2yÊ 2x(" -) y œ 3 and x 2y(1 -) œ 0 Ê 1 - œ
x
2y
x
and (2x) Š 2y
‹ y œ 3 Ê x# y# œ 3y
Ê x# œ y# 3y. Thus, 9 œ x# y# œ y# 3y y# Ê 2y# 3y 9 œ 0 Ê (2y 3)(y 3) œ 0
Ê y œ 3, 3# . For y œ 3, x# y# œ 9 Ê x œ 0 yielding the point (0ß 3). For y œ 3# , x# y# œ 9
Ê x#
9
4
œ 9 Ê x# œ
Ê xœ „
27
4
È
¸ 20.691, and f Š 3 # 3 , 3# ‹ œ 9
27È3
4
3È 3
#
È
. Evaluations give f(0ß 3) œ 9, f Š 3 # 3 ß 3# ‹ œ 9
27È3
4
¸ 2.691.
(ii) For the interior of the disk, fx (xß y) œ 2x 3 y œ 0 and fy (xß y) œ 2y x œ 0 Ê x œ 2 and y œ 1
Ê (2ß 1) is an interior critical point of the disk with f(2ß 1) œ 3. Therefore, the absolute maximum of f on
the disk is 9
27È3
4
È
at Š 3 # 3 ß 3# ‹ and the absolute minimum of f on the disk is 3 at (2ß 1).
83. ™ f œ i j k and ™ g œ 2xi 2yj 2zk so that ™ f œ - ™ g Ê i j k œ -(2xi 2yj 2zk) Ê 1 œ 2x-,
1 œ 2y-, 1 œ 2z- Ê x œ y œ z œ -" . Thus x# y# z# œ 1 Ê 3x# œ 1 Ê x œ „ È"3 yielding the points
Š È"3 ß È"3 ,
"
È3 ‹
and Š È"3 ,
f Š È"3 ß È"3 ß È"3 ‹ œ
3
È3
"
È3
, È"3 ‹ . Evaluations give the absolute maximum value of
œ È3 and the absolute minimum value of f Š È"3 ß È"3 ß È"3 ‹ œ È3.
84. Let f(xß yß z) œ x# y# z# be the square of the distance to the origin and g(xß yß z) œ x# zy 4. Then
™ f œ 2xi 2yj 2zk and ™ g œ 2xi zj yk so that ™ f œ - ™ g Ê 2x œ 2-x, 2y œ -z, and
2z œ -y Ê x œ 0 or - œ 1.
CASE 1: x œ 0 Ê zy œ 4 Ê z œ 4y and y œ 4z Ê 2 Š y4 ‹ œ -y and 2 ˆ 4z ‰ œ -z Ê
8
-
8
-
œ y# and
œ z# Ê y# œ z# Ê y œ „ z. But y œ x Ê z# œ 4 leads to no solution, so y œ z Ê z# œ 4
Ê z œ „ 2 yielding the points (0ß 2ß 2) and (0ß 2ß 2).
CASE 2: - œ 1 Ê 2z œ y and 2y œ z Ê 2y œ ˆ y# ‰ Ê 4y œ y Ê y œ 0 Ê z œ 0 Ê x# 4 œ 0 Ê
x œ „ 2 yielding the points (2ß 0ß 0) and (2ß !ß 0).
Evaluations give f(0ß 2ß 2) œ f(0ß 2ß 2) œ 8 and f(2ß 0ß 0) œ f(2ß !ß 0) œ 4. Thus the points (2ß 0ß 0) and
(2ß !ß 0) on the surface are closest to the origin.
85. The cost is f(xß yß z) œ 2axy 2bxz 2cyz subject to the constraint xyz œ V. Then ™ f œ - ™ g
Ê 2ay 2bz œ -yz, 2ax 2cz œ -xz, and 2bx 2cy œ -xy Ê 2axy 2bxz œ -xyz, 2axy 2cyz œ -xyz, and
2bxz 2cyz œ -xyz Ê 2axy 2bxz œ 2axy 2cyz Ê y œ ˆ bc ‰ x. Also 2axy 2bxz œ 2bxz 2cyz Ê z œ ˆ ca ‰ x.
Then x ˆ bc x‰ ˆ ca x‰ œ V Ê x$ œ
#
Height œ z œ ˆ ac ‰ Š cabV ‹
"Î$
#
c# V
ab
œ Š abcV ‹
#
Ê width œ x œ Š cabV ‹
"Î$
"Î$
#
, Depth œ y œ ˆ bc ‰ Š cabV ‹
"Î$
#
œ Š bacV ‹
"Î$
, and
.
86. The volume of the pyramid in the first octant formed by the plane is V(aß bß c) œ
"
3
ˆ "# ab‰ c œ
"
6
abc. The point
(2ß 1ß 2) on the plane Ê "b 2c œ 1. We want to minimize V subject to the constraint 2bc ac 2ab œ abc.
ac
ab
Thus, ™ V œ bc
6 i 6 j 6 k and ™ g œ (c 2b bc)i (2c 2a ac)j (2b a ab)k so that ™ V œ ac
ab
abc
Ê bc
6 œ -(c 2b bc), 6 œ -(2c 2a ac), and 6 œ -(2b a ab) Ê 6 œ -(ac 2ab abc),
abc
abc
6 œ -(2bc 2ab abc), and 6 œ -(2bc ac abc) Ê -ac œ 2-bc and 2-ab œ 2-bc. Now - Á 0 since
2
a
a Á 0, b Á 0, and c Á 0 Ê ac œ 2bc and ab œ bc Ê a œ 2b œ c. Substituting into the constraint equation gives
y
2
2
2
x
z
a a a œ 1 Ê a œ 6 Ê b œ 3 and c œ 6. Therefore the desired plane is 6 3 6 œ 1 or x 2y z œ 6.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
™g
874
Chapter 14 Partial Derivatives
87. ™ f œ (y z)i xj xk , ™ g œ 2xi 2yj , and ™ h œ zi xk so that ™ f œ - ™ g . ™ h
Ê (y z)i xj xk œ -(2xi 2yj) .(zi xk) Ê y z œ 2-x .z, x œ 2-y, x œ .x Ê x œ 0 or . œ 1.
CASE 1: x œ 0 which is impossible since xz œ 1.
CASE 2: . œ 1 Ê y z œ 2-x z Ê y œ 2-x and x œ 2-y Ê y œ (2-)(2-y) Ê y œ 0 or
4-# œ 1. If y œ 0, then x# œ 1 Ê x œ „ 1 so with xz œ 1 we obtain the points (1ß 0ß 1)
and (1ß 0ß 1). If 4-# œ 1, then - œ „ "# . For - œ "# , y œ x so x# y# œ 1 Ê x# œ "#
Ê xœ „
"
È2
with xz œ 1 Ê z œ „ È2, and we obtain the points Š È"2 ß È"2 ß È2‹ and
Š È"2 ß È"2 ß È2‹ . For - œ
"
#
, y œ x Ê x# œ
"
#
Ê xœ „
"
È2
with xz œ 1 Ê z œ „ È2,
and we obtain the points Š È"2 ß È"2 , È2‹ and Š È"2 ß È"2 ß È2‹ .
Evaluations give f(1ß 0ß 1) œ 1, f(1ß 0ß 1) œ 1, f Š È"2 ß È"2 ß È2‹ œ
f Š È"2 ß È"2 ß È2‹ œ
3
#
, and f Š È"2 ß È"2 ß È2‹ œ
3
#
"
#
, f Š È"2 ß È"2 , È2‹ œ
3
#
. Therefore the absolute maximum is
Š È"2 ß È"2 ß È2‹ and Š È"2 ß È"2 ß È2‹ , and the absolute minimum is
"
#
"
#
,
at
at Š È"2 ß È"2 ß È2‹ and
Š È"2 ß È"2 ß È2‹ .
88. Let f(xß yß z) œ x# y# z# be the square of the distance to the origin. Then ™ f œ 2xi 2yj 2zk ,
™ g œ i j k , and ™ h œ 4xi 4yj 2zk so that ™ f œ - ™ g . ™ h Ê 2x œ - 4x., 2y œ - 4y.,
and 2z œ - 2z. Ê - œ 2x(1 2.) œ 2y(1 2.) œ 2z(1 2.) Ê x œ y or . œ "# .
CASE 1: x œ y Ê z# œ 4x# Ê z œ „ 2x so that x y z œ 1 Ê x x 2x œ 1 or x x 2x œ 1
(impossible) Ê x œ "4 Ê y œ "4 and z œ "# yielding the point ˆ "4 ß "4 ß "# ‰ .
CASE 2: . œ
"
#
Ê - œ 0 Ê 0 œ 2z(1 1) Ê z œ 0 so that 2x# 2y# œ 0 Ê x œ y œ 0. But the origin
(!ß 0ß 0) fails to satisfy the first constraint x y z œ 1.
Therefore, the point ˆ "4 ß 4" ß "# ‰ on the curve of intersection is closest to the origin.
89. (a) y, z are independent with w œ x# eyz and z œ x# y# Ê
œ a2xeyz b
`x
`y
`w
`y
`w `x
`w `y
`w `z
`x `y `y `y `z `y
œ 2x `` xy 2y Ê `` xy œ yx
œ
azx# eyz b (1) ayx# eyz b (0); z œ x# y# Ê 0
; therefore,
Š ``wy ‹ œ a2xeyz b ˆ xy ‰ zx# eyz œ a2y zx# b eyz
z
(b) z, x are independent with w œ x# eyz and z œ x# y# Ê
œ a2xeyz b (0) azx# eyz b
`y
`z
`w
`z
œ
`w `x
`x `z
ayx# eyz b (1); z œ x# y# Ê 1 œ 0 2y
1
ˆ ``wz ‰ œ azx# eyz b Š 2y
‹ yx# eyz œ x# eyz Šy
x
`w `y
`w `z
`y `z `z `z
`y
`y
"
` z Ê ` z œ #y
; therefore,
z
2y ‹
(c) z, y are independent with w œ x# eyz and z œ x# y# Ê
`w
`z
œ
œ a2xeyz b `` xz azx# eyz b (0) ayx# eyz b (1); z œ x# y# Ê 1
1 ‰
ˆ ``wz ‰ œ a2xeyz b ˆ 2x
yx# eyz œ a1 x# yb eyz
`w `x
`w `y
`w `z
`x `z `y `z `z `z
œ 2x `` xz 0 Ê `` xz œ #"x
; therefore,
y
90. (a) T, P are independent with U œ f(Pß Vß T) and PV œ nRT Ê ``UT œ ``UP `` TP
‰ ˆ ``VT ‰ ˆ ``UT ‰ (1); PV œ nRT Ê P ``VT œ nR Ê ``VT œ
œ ˆ ``UP ‰ (0) ˆ `` U
V
ˆ ``UT ‰ œ ˆ `` U
‰ ˆ nR
‰
V
P
P
`U
`T
`U `V
`U `T
`V `T `T `T
nR
P ; therefore,
`U `P
`U `V
(b) V, T are independent with U œ f(Pß Vß T) and PV œ nRT Ê `` U
V œ `P `V `V `V
U‰
œ ˆ ``UP ‰ ˆ ``VP ‰ ˆ `` V
(1) ˆ ``UT ‰ (0); PV œ nRT Ê V ``VP P œ (nR) ˆ ``VT ‰ œ 0 Ê
ˆ `` U
‰
V T
œ
ˆ ``UP ‰ ˆ VP ‰
`U
`V
`U `T
`T `V
`P
P
`V œ V
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
; therefore,
Chapter 14 Practice Exercises
875
91. Note that x œ r cos ) and y œ r sin ) Ê r œ Èx# y# and ) œ tan" ˆ yx ‰ . Thus,
`w
`x
œ
`w `r
`r `x
`w `)
`) `x
œ ˆ ``wr ‰ Š Èx#x y# ‹ ˆ ``w) ‰ Š x#yy# ‹ œ (cos ))
`w
`r
ˆ sinr ) ‰
`w
`)
`w
`y
œ
`w `r
`r `y
`w `)
` ) )y
œ ˆ ``wr ‰ Š Èx#y y# ‹ ˆ ``w) ‰ Š x# x y# ‹ œ (sin ))
`w
`r
ˆ cosr ) ‰
`w
`)
92. zx œ fu
93.
`u
`y
`v
`x
fv
œ afu afv , and zy œ fu
`u
`x œ a
" `w
" `w
a `x œ b `y
œ b and
Ê
94.
`u
`x
`w
`x
œ
and
`w
`z
œ
2
rs
œ
Ê
œ
2
x# y# 2z
and
`w
`s
œ
`w
dw
` x œ du
b ``wx œ a
"
(r s)#
`w `x
`x `s
`w `y
`y `s
Solving this system yields
Ê ae cos vb
`u
`y
`u
`x
ae sin vb
u
`v
`y
dw ` u
du ` y
œ
2(r s)
2 ar# 2rs s# b
œ
"
rs
œ
`w `x
`x `r
`g
`)
œ
Ê
`f `x
`x `)
` #g
` )#
`f `y
`y `)
œ (r sin )) Š `` x)
`y
`) ‹
"
rs
rs
(r s)#
`w
`y
`w `z
`z `r
dw
du
œ
œ
Ê
`f
`x
`v
`y
"
rs
œ
dw
du
2(r s)
#(r s)#
and
" `w
b `y
œ
rs
(r s)#
œ
dw
du
,
’ (r " s)# “ (2s) œ
2r 2s
(r s)#
2
rs
y œ 0 Ê aeu sin vb
`u
`x
aeu cos vb
`v
`x
œ 0.
Similarly, e cos v x œ 0
u
`u
`y
œ eu cos v. Therefore Š `` ux i
(r cos ))
œ
rs
(r s)#
’ (r " s)# “ (2r) œ
`v
u
` x œ 1; e sin v
`v
u
sin v.
` x œ e
aeu cos vb
`u
`y
`v
`y
j‹ † Š `` vx i
œ 1. Solving this
`v
`y
j‹
u
cos vb jd œ 0 Ê the vectors are orthogonal Ê the angle
`f
`x
(r cos )) Š ``x`fy
sin vb i ae
` #f ` y
` y` x ` ) ‹
" `w
a `x
2y
x# y# 2z
œ 0 and e sin v y œ 0 Ê aeu sin vb
u
`x
`)
`w `y
`y `r
,
œb
u
u
#
œ
cos v and
œ eu sin v and
œ (r sin )) Š `` xf#
and
aeu sin vb
`u
`y
œ (r sin ))
dw
du
`w `z
`z `s
œ cae cos vb i ae sin vb jd † cae
between the vectors is the constant 1# .
96.
œ bfu bfv
œ
`u
`x
u
œe
u
second system yields
`v
`y
fv
`w
`y
œa
`w
`r
Ê
95. eu cos v x œ 0 Ê aeu cos vb
u
`u
`x
`w
`y
2(r s)
(r s)# (r s)# 4rs -
œ
2x
x# y# 2z
Ê
`u
`y
;
`f
`y
(r cos ))
(r cos )) (r cos )) Š `` x)
`y
`) ‹
#
`x
`)
` #f ` y
` y# ` ) ‹
(r sin ))
`f
`y
(r sin ))
œ (r sin ) r cos ))(r sin ) r cos )) (r cos ) r sin )) œ (2)(2) (0 2) œ 4 2 œ 2 at
(rß )) œ ˆ2ß 1# ‰ .
97. (y z)# (z x)# œ 16 Ê ™ f œ 2(z x)i 2(y z)j 2(y 2z x)k ; if the normal line is parallel to the
yz-plane, then x is constant Ê `` xf œ 0 Ê 2(z x) œ 0 Ê z œ x Ê (y z)# (z z)# œ 16 Ê y z œ „ 4.
Let x œ t Ê z œ t Ê y œ t „ 4. Therefore the points are (tß t „ 4ß t), t a real number.
98. Let f(xß yß z) œ xy yz zx x z# œ 0. If the tangent plane is to be parallel to the xy-plane, then ™ f is
perpendicular to the xy-plane Ê ™ f † i œ 0 and ™ f † j œ 0. Now ™ f œ (y z 1)i (x z)j (y x 2z)k
so that ™ f † i œ y z 1 œ 0 Ê y z œ 1 Ê y œ 1 z, and ™ f † j œ x z œ 0 Ê x œ z. Then
z(1 z) (" z)z z(z) (z) z# œ 0 Ê z 2z# œ 0 Ê z œ "# or z œ 0. Now z œ "# Ê x œ "# and y œ
Ê ˆ "# ß "# ß "# ‰ is one desired point; z œ 0 Ê x œ 0 and y œ 1 Ê (0ß 1ß 0) is a second desired point.
99. ™ f œ -(xi yj zk) Ê
`f
`x
œ -x Ê f(xß yß z) œ
"
#
-x# g(yß z) for some function g Ê -y œ
`f
`y
œ
`g
`y
-y# h(z) for some function h Ê -z œ `` zf œ `` gz œ hw (z) Ê h(z) œ #" -z# C for some arbitrary
constant C Ê g(yß z) œ "# -y# ˆ "# -z# C‰ Ê f(xß yß z) œ "# -x# "# -y# "# -z# C Ê f(0ß 0ß a) œ "# -a# C
Ê g(yß z) œ
"
#
and f(0ß 0ß a) œ
"
#
-(a)# C Ê f(0ß 0ß a) œ f(0ß 0ß a) for any constant a, as claimed.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
#
876
Chapter 14 Partial Derivatives
ß
f(0 su" ß 0 su# ß 0 su$ )f(0ß 0ß 0)
œ lim
sÄ0
‰
100. ˆ df
ds u (0 0 0)
ß ß
,s0
s
És# u#" s# u## s# u#$ 0
œ lim
sÄ0
s
,s0
sÉu#" u## u#$
œ lim
œ lim kuk œ 1;
s
sÄ0
sÄ0
however, ™ f œ Èx# xy# z# i Èx# yy# z# j Èx# zy# z# k fails to exist at the origin (0ß 0ß 0)
101. Let f(xß yß z) œ xy z 2 Ê ™ f œ yi xj k . At (1ß 1ß 1), we have ™ f œ i j k Ê the normal line is
x œ 1 t, y œ 1 t, z œ 1 t, so at t œ 1 Ê x œ 0, y œ 0, z œ 0 and the normal line passes through the origin.
102. (b) f(xß yß z) œ x# y# z# œ 4
Ê ™ f œ 2xi 2yj 2zk Ê at (2ß 3ß 3)
the gradient is ™ f œ 4i 6j 6k which is
normal to the surface
(c) Tangent plane: 4x 6y 6z œ 8 or
2x 3y 3z œ 4
Normal line: x œ 2 4t, y œ 3 6t, z œ 3 6t
CHAPTER 14 ADDITIONAL AND ADVANCED EXERCISES
fx (0ß h) fx (0ß 0)
h
1. By definition, fxy (!ß 0) œ lim
hÄ0
so we need to calculate the first partial derivatives in the
numerator. For (xß y) Á (0ß 0) we calculate fx (xß y) by applying the differentiation rules to the formula for
fy (xß y) œ
00
h
œ lim
hÄ0
2.
`w
`x
x$ xy#
x# y#
x# y y$
x# y#
ax# y# b (2x) ax# y# b (2x)
ax # y # b #
4x# y$
Ê fx (0ß h)
ax # y # b#
f(0ß0)
For (xß y) œ (0ß 0) we apply the definition: fx (!ß 0) œ lim f(hß 0)
œ lim 0 h 0 œ
h
hÄ0
hÄ0
f (hß 0) fy (!ß 0)
fxy (0ß 0) œ lim hh 0 œ 1. Similarly, fyx (0ß 0) œ lim y
,
so
for (xß y) Á
h
hÄ0
hÄ0
f(xß y): fx (xß y) œ
(xy)
4x$ y#
ax# y# b#
Ê fy (hß 0) œ
h$
h#
x# y y $
x# y#
œ
$
œ hh# œ h.
0. Then by definition
(0ß 0) we have
œ h; for (xß y) œ (0ß 0) we obtain fy (0ß 0) œ lim
h0
h
œ 0. Then by definition fyx (0ß 0) œ lim
hÄ0
œ 1 ex cos y Ê w œ x ex cos y g(y);
`w
`y
hÄ0
f(0ß h) f(!ß 0)
h
œ 1. Note that fxy (0ß 0) Á fyx (0ß 0) in this case.
œ ex sin y gw (y) œ 2y ex sin y Ê gw (y) œ 2y
Ê g(y) œ y# C; w œ ln 2 when x œ ln 2 and y œ 0 Ê ln 2 œ ln 2 eln 2 cos 0 0# C Ê 0 œ 2 C
Ê C œ 2. Thus, w œ x ex cos y g(y) œ x ex cos y y# 2.
3. Substitution of u u(x) and v œ v(x) in g(uß v) gives g(u(x)ß v(x)) which is a function of the independent
variable x. Then, g(uß v) œ 'u f(t) dt Ê
v
œ Š ``u
#
#
fzz œ Š ddr#f ‹ ˆ ``zr ‰
` #r
` y#
œ
` g du
` u dx
` g dv
` v dx
œ Š ``u
#
df ` # r
dr ` x#
'uv f(t) dt‹ dxdu Š ``v 'uv f(t) dt‹ dxdv
'vu f(t) dt‹ dudx Š ``v 'uv f(t) dt‹ dvdx œ f(u(x)) dudx f(v(x)) dvdx œ f(v(x)) dvdx f(u(x)) dudx
4. Applying the chain rules, fx œ
Ê
dg
dx
œ
#
df ` r
dr ` z#
x# z#
3
ˆÈx# y# z# ‰
df ` r
dr ` x
#
Ê fxx œ Š ddr#f ‹ ˆ ``xr ‰
. Moreover,
; and
`r
`z
œ
`r
`x
œ
x
È x # y # z#
z
È x # y # z#
Ê
` #r
` z#
Ê
œ
#
` r
` x#
œ
#
#
. Similarly, fyy œ Š ddr#f ‹ Š ``yr ‹
#
#
y z
3
ˆÈx# y# z# ‰
x# y#
3
ˆÈ x # y # z # ‰
;
`r
`y
œ
y
È x # y # z#
. Next, fxx fyy fzz œ 0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
df ` # r
dr ` y#
and
Chapter 14 Additional and Advanced Exercises
#
#
df
dr
d
dr
x#
(f w ) œ ˆ 2r ‰ f w , where f w œ
œ Cr# Ê f(r) œ Cr b œ
y # z# ‰
Ê
df
dr
y#
d# f
x # y # z# ‰
x# y#
#
‰
Š ddr#f ‹ Š x# yz # z# ‹ ˆ df
dr Œ ˆÈ
Ê
y # z#
#
‰
Ê Š ddr#f ‹ Š x# xy# z# ‹ ˆ df
dr Œ ˆÈ
3
df
f
3
877
x # z#
Š dr# ‹ Š x# y# z# ‹ ˆ dr ‰ Œ ˆÈx# y# z# ‰3
d# f
dr#
œ0 Ê
df
Š Èx# 2y# z# ‹
d# f
dr#
œ0 Ê
df
dr
2 df
r dr
œ0
œ 2 rdr Ê ln f w œ 2 ln r ln C Ê f w œ Cr# , or
w
w
b for some constants a and b (setting a œ C)
a
r
5. (a) Let u œ tx, v œ ty, and w œ f(uß v) œ f(u(tß x)ß v(tß y)) œ f(txß ty) œ tn f(xß y), where t, x, and y are
independent variables. Then ntnc1 f(xß y) œ ``wt œ ``wu ``ut ``wv ``vt œ x ``wu y ``wv . Now,
`w
`w `u
`w `v
`w
`w
ˆ `w ‰
ˆ `w ‰
ˆ " ‰ ˆ ``wx ‰ . Likewise,
` x œ ` u ` x ` v ` x œ ` u (t) ` v (0) œ t ` u Ê ` u œ t
`w
`y
œ
`w `u
`u `y
ntnc1 f(xß y) œ x
`w
`x
Ê
œ
`f
`x
`w `v
`v `y
`w
`u
`w
`v
y
`w
`y
and
œ ˆ ``wu ‰ (0) ˆ ``wv ‰ (t) Ê
œ
`f
`x
Ê nf(xß y) œ x
Also from part (a),
œ
`
`y
œ t#
ˆt
`w ‰
`v
` #w
` v` u
œt
` #w
` x#
`
`x
œ
x
y
ˆ ``wx ‰
`
`x
`w ‰
`u
t
œ
`f
`x
`w
`v .
` w `v
` v` u ` t
` w `u
` u# ` t
` #w ` u
` u` v ` y
Ê ˆ t"# ‰
`w
`u
#
` #w
` x#
œ ˆ "t ‰ Š ``wy ‹ . Therefore,
œ ˆ xt ‰ ˆ ``wx ‰ ˆ yt ‰ Š ``wy ‹. When t œ 1, u œ x, v œ y, and w œ f(xß y)
(b) From part (a), ntnc1 f(xß y) œ x
n(n 1)tnc2 f(xß y) œ x
`w
`v
y
` #w ` v
` v# ` y
` #w
` u#
ˆt
` #w
` v#
œ t#
, ˆ t"# ‰
` #w
` y#
`f
`y
, as claimed.
Differentiating with respect to t again we obtain
#
œ
y
œ
` #w ` u
` u` v ` t
œt
, and
` #w
` v#
` #w ` v
` v# ` t
y
` #w ` u
` u# ` x
t
` #w
` y` x
`
`y
œ
, and ˆ t"# ‰
œ x#
` #w
` u#
2xy
` #w ` v
` v` u ` x
œ
#
t# `` uw#
ˆ ``wx ‰ œ
`
`y
ˆt
` #w
` y` x
œ
`w ‰
`u
,
` #w
` u` v
` #w
` y#
œt
œ
y#
`
`y
` #w
` v#
.
Š ``wy ‹
` #w ` u
` u# ` y
t
` #w ` v
` v` u ` y
` #w
` v` u
‰ Š ``y`wx ‹ Š yt# ‹ Š `` yw# ‹ for t Á 0. When t œ 1, w œ f(xß y) and
Ê n(n 1)tnc2 f(xß y) œ Š xt# ‹ Š `` xw# ‹ ˆ 2xy
t#
#
#
#
#
#
#
#
#
we have n(n 1)f(xß y) œ x# Š `` xf# ‹ 2xy Š ``x`fy ‹ y# Š `` yf# ‹ as claimed.
6. (a) lim
rÄ0
sin 6r
6r
œ lim
tÄ0
sin t
t
œ 1, where t œ 6r
f(0 hß 0) f(0ß 0)
h
hÄ0
36 sin 6h
lim
œ0
12
hÄ0
(b) fr (0ß 0) œ lim
œ
f(rß ) h) f(rß ))
h
hÄ0
(c) f) (rß )) œ lim
ˆ sin6h6h ‰ 1
h
hÄ0
œ lim
œ lim
hÄ0
œ lim
hÄ0
6 cos 6h 6
12h
(applying l'Hopital's
rule twice)
s
ˆ sin6r6r ‰ ˆ sin6r6r ‰
h
hÄ0
œ lim
œ lim
0
hÄ0 h
7. (a) r œ xi yj zk Ê r œ krk œ Èx# y# z# and ™ r œ
(b) rn œ ˆÈx# y# z# ‰
sin 6h 6h
6h#
œ0
x
È x # y # z#
i
y
È x # y # z#
j
z
È x # y # z#
kœ
r
r
n
ÐnÎ2Ñ
1
(d) dr œ dxi dyj dzk Ê r † dr œ x dx y dy z dz, and dr œ rx dx ry dy rz dz œ
x
r
Ê ™ arn b œ nx ax# y# z# b
(c) Let n œ 2 in part (b). Then
"
#
ÐnÎ2Ñ 1
ÐnÎ2Ñ
i ny ax# y# z# b
j nz ax# y# z# b
k œ nrn 2 r
#
™ ar# b œ r Ê ™ ˆ "# r# ‰ œ r Ê r# œ #" ax# y# z# b is the function.
1
dx
y
r
dy
z
r
dz
Ê r dr œ x dx y dy z dz œ r † dr
(e) A œ ai bj ck Ê A † r œ ax by cz Ê ™ (A † r) œ ai bj ck œ A
8. f(g(t)ß h(t)) œ c Ê 0 œ
df
dt
œ
` f dx
` x dt
` f dy
` y dt
œ Š `` xf i
`f
`y
j‹ † Š dx
dt i
dy
dt
j‹ , where
dx
dt
i
dy
dt
j is the tangent vector
Ê ™ f is orthogonal to the tangent vector
9. f(xß yß z) œ xz# yz cos xy 1 Ê ™ f œ az# y sin xyb i (z x sin xy)j (2xz y)k Ê ™ f(0ß 0ß 1) œ i j
Ê the tangent plane is x y œ 0; r œ (ln t)i (t ln t)j tk Ê rw œ ˆ "t ‰ i (ln t 1)j k ; x œ y œ 0, z œ 1
Ê t œ 1 Ê rw (1) œ i j k . Since (i j k) † (i j) œ rw (1) † ™ f œ 0, r is parallel to the plane, and
r(1) œ 0i 0j k Ê r is contained in the plane.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
878
Chapter 14 Partial Derivatives
10. Let f(xß yß z) œ x$ y$ z$ xyz Ê ™ f œ a3x# yzb i a3y# xzb j a3z# xyb k Ê ™ f(0ß 1ß 1) œ i 3j 3k
$
Ê the tangent plane is x 3y 3z œ 0; r œ Š t4 2‹ i ˆ 4t 3‰ j (cos (t 2)) k
#
Ê rw œ Š 3t4 ‹ i ˆ t4# ‰ j (sin (t 2)) k ; x œ 0, y œ 1, z œ 1 Ê t œ 2 Ê rw (2) œ 3i j . Since
rw (2) † ™ f œ 0 Ê r is parallel to the plane, and r(2) œ i k Ê r is contained in the plane.
11.
`z
`x
œ 3x# 9y œ 0 and
`z
`y
œ 3y# 9x œ 0 Ê y œ
"
3
#
x# and 3 ˆ "3 x# ‰ 9x œ 0 Ê
"
3
x% 9x œ 0
Ê x ax$ 27b œ 0 Ê x œ 0 or x œ 3. Now x œ 0 Ê y œ 0 or (!ß 0) and x œ 3 Ê y œ 3 or (3ß 3). Next
` #z
` x#
œ 6x,
` #z
` y#
œ 6y, and
and for (3ß 3),
` #z ` #z
` x# ` y#
` #z
` x` y
#
` #z ` #z
` x# ` y#
œ 9. For (!ß 0),
#
` #z
` x#
Š ``x`zy ‹ œ 243 0 and
#
#
Š ``x`zy ‹ œ 81 Ê no extremum (a saddle point),
œ 18 0 Ê a local minimum.
12. f(xß y) œ 6xyeÐ2x3yÑ Ê fx (xß y) œ 6y(1 2x)eÐ2x3yÑ œ 0 and fy (xß y) œ 6x(1 3y)eÐ2x3yÑ œ 0 Ê x œ 0 and
y œ 0, or x œ "# and y œ 3" . The value f(0ß 0) œ 0 is on the boundary, and f ˆ "# ß "3 ‰ œ e"2 . On the positive y-axis,
f(0ß y) œ 0, and on the positive x-axis, f(xß 0) œ 0. As x Ä _ or y Ä _ we see that f(xß y) Ä 0. Thus the absolute
maximum of f in the closed first quadrant is e"2 at the point ˆ #" ß 3" ‰ .
13. Let f(xß yß z) œ
P! (x! ß y! ß y! ) is
y#
x#
a# b#
!‰
ˆ 2x
a# x
z#
c# 1
!‰
ˆ 2y
b# y
Ê ™fœ
ˆ 2zc#! ‰ z
2y
2x
a# i b# j
#
2y#!
!
œ 2x
a# b#
#
2z
c# k Ê an equation of the plane tangent
2z#!
ˆ x! ‰
ˆ y! ‰
ˆ z! ‰
c# œ 2 or a# x b# y c# z œ 1.
#
at the point
#
The intercepts of the plane are Š xa! ß 0ß 0‹ , Š0ß by! ß 0‹ and Š!ß !ß zc! ‹ . The volume of the tetrahedron formed by the
#
#
#
plane and the coordinate planes is V œ ˆ "3 ‰ ˆ #" ‰ Š xa! ‹ Š by! ‹ Š cz! ‹ Ê we need to maximize V(xß yß z) œ
subject to the constraint f(xß yß z) œ
#
"
and ’ (abc)
6 “ Š xyz# ‹ œ
2z
c#
x#
a#
y#
b#
#
z#
c#
"
œ 1. Thus, ’ (abc)
6 “ Š x# yz ‹ œ
2x
a#
(abc)#
6
#
"
-, ’ (abc)
6 “ Š xy# z ‹ œ
(xyz)"
2y
b#
-,
-. Multiply the first equation by a# yz, the second by b# xz, and the third by c# xy. Then equate
the first and second Ê a# y# œ b# x# Ê y œ
substitute into f(xß yß z) œ 0 Ê x œ
a
È3
b
a
x, x 0; equate the first and third Ê a# z# œ c# x# Ê z œ ca x, x 0;
Ê yœ
Ê zœ
b
È3
c
È3
Ê Vœ
È3
#
abc.
14. 2(x u) œ -, 2(y v) œ -, 2(x u) œ ., and 2(y v) œ 2.v Ê x u œ v y, x u œ .# , and
y v œ .v Ê x u œ .v œ .# Ê v œ
"
#
or . œ 0.
CASE 1: . œ 0 Ê x œ u, y œ v, and - œ 0; then y œ x 1 Ê v œ u 1 and v# œ u Ê v œ v# 1
1 „ È1 4
Ê
#
"
"
"
"
#
v œ # and u œ v Ê u œ 4 ; x 4 œ #
Ê y œ 78 . Then f ˆ 8" ß 87 ß "4 ß "# ‰ œ ˆ 8"
Ê v# v 1 œ 0 Ê v œ
CASE 2:
no real solution.
"
4 œ
#
2 ˆ 38 ‰
y and y œ x 1 Ê x
#
#
"4 ‰ ˆ 78 #" ‰ œ
Ê 2x œ 4" Ê x œ 8"
Ê the minimum distance is 38 È2.
x
"
#
(Notice that f has no maximum value.)
15. Let (x! ß y! ) be any point in R. We must show
lim
Ðhß kÑ Ä Ð0ß 0Ñ
lim
Ðxß yÑ Ä Ðx! ß y! Ñ
f(xß y) œ f(x! ß y! ) or, equivalently that
kf(x! hß y! k) f(x! ß y! )k œ 0. Consider f(x! hß y! k) f(x! ß y! )
œ [f(x! hß y! k) f(x! ß y! k)] [f(x! ß y! k) f(x! ß y! )]. Let F(x) œ f(xß y! k) and apply the Mean Value
Theorem: there exists 0 with x! 0 x! h such that Fw (0 )h œ F(x! h) F(x! ) Ê hfx (0ß y! k)
œ f(x! hß y! k) f(x! ß y! k). Similarly, k fy (x! ß () œ f(x! ß y! k) f(x! ß y! ) for some ( with
y! ( y! k. Then kf(x! hß y! k) f(x! ß y! )k Ÿ khfx (0ß y! k)k kkfy (x! ß ()k . If M, N are positive real
numbers such that kfx k Ÿ M and kfy k Ÿ N for all (xß y) in the xy-plane, then kf(x! hß y! k) f(x! ß y! )k
Ÿ M khk N kkk . As (hß k) Ä 0, kf(x! hß y! k) f(x! ß y! )k Ä 0 Ê
lim
kf(x! hß y! k) f(x! ß y! )k
Ðhß kÑ Ä Ð0ß 0Ñ
œ 0 Ê f is continuous at (x! ß y! ).
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 14 Additional and Advanced Exercises
16. At extreme values, ™ f and v œ
dr
dt
df
dt
are orthogonal because
œ ™f†
879
œ 0 by the First Derivative Theorem for
dr
dt
Local Extreme Values.
17.
`f
`x
œ 0 Ê f(xß y) œ h(y) is a function of y only. Also,
Moreover,
`f
`y
œ
`g
`x
`g
`y
œ
`f
`x
œ 0 Ê g(xß y) œ k(x) is a function of x only.
Ê hw (y) œ kw (x) for all x and y. This can happen only if hw (y) œ kw (x) œ c is a constant.
Integration gives h(y) œ cy c" and k(x) œ cx c# , where c" and c# are constants. Therefore f(xß y) œ cy c"
and g(xß y) œ cx c# . Then f(1ß 2) œ g(1ß 2) œ 5 Ê 5 œ 2c c" œ c c# , and f(0ß 0) œ 4 Ê c" œ 4 Ê c œ
Ê c# œ
9
#
. Thus, f(xß y) œ
"
#
y 4 and g(xß y) œ
"
#
"
#
x #9 .
18. Let g(xß y) œ Du f(xß y) œ fx (xß y)a fy (xß y)b. Then Du g(xß y) œ gx (xß y)a gy (xß y)b
œ fxx (xß y)a# fyx (xß y)ab fxy (xß y)ba fyy (xß y)b# œ fxx (xß y)a# 2fxy (xß y)ab fyy (xß y)b# .
19. Since the particle is heat-seeking, at each point (xß y) it moves in the direction of maximal temperature
increase, that is in the direction of ™ T(xß y) œ aec2y sin xb i a2ec2y cos xb j . Since ™ T(xß y) is parallel to
2ec2y cos x
ec2y sin x œ
È
œ 2 ln #2
the particle's velocity vector, it is tangent to the path y œ f(x) of the particle Ê f w (x) œ
2 cot x.
Integration gives f(x) œ 2 ln ksin xk C and f ˆ 14 ‰ œ 0 Ê 0 œ 2 ln ¸sin 14 ¸ C Ê C
œ ln Š È22 ‹
#
œ ln 2. Therefore, the path of the particle is the graph of y œ 2 ln ksin xk ln 2.
20. The line of travel is x œ t, y œ t, z œ 30 5t, and the bullet hits the surface z œ 2x# 3y# when
30 5t œ 2t# 3t# Ê t# t 6 œ 0 Ê (t 3)(t 2) œ 0 Ê t œ 2 (since t 0). Thus the bullet hits the
surface at the point (2ß 2ß 20). Now, the vector 4xi 6yj k is normal to the surface at any (xß yß z), so that
n œ 8i 12j k is normal to the surface at (2ß 2ß 20). If v œ i j 5k , then the velocity of the particle
†25 ‰
after the ricochet is w œ v 2 projn v œ v Š 2knvk†#n ‹ n œ v ˆ 2209
n œ (i j 5k) ˆ 400
209 i
œ 191
209 i
391
209
j
995
209
600
209
j
50
209
k‰
k.
21. (a) k is a vector normal to z œ 10 x# y# at the point (!ß 0ß 10). So directions tangential to S at (!ß 0ß 10) will
be unit vectors u œ ai bj . Also, ™ T(xß yß z) œ (2xy 4) i ax# 2yz 14b j ay# 1b k
Ê ™ T(!ß 0ß 10) œ 4i 14j k . We seek the unit vector u œ ai bj such that Du T(0ß 0ß 10)
œ (4i 14j k) † (ai bj) œ (4i 14j) † (ai bj) is a maximum. The maximum will occur when ai bj
has the same direction as 4i 14j , or u œ È"53 (2i 7j).
(b) A vector normal to S at (1ß 1ß 8) is n œ 2i 2j k . Now, ™ T(1ß 1ß 8) œ 6i 31j 2k and we seek the unit
vector u such that Du T(1ß 1ß 8) œ ™ T † u has its largest value. Now write ™ T œ v w , where v is parallel
to ™ T and w is orthogonal to ™ T. Then Du T œ ™ T † u œ (v w) † u œ v † u w † u œ w † u. Thus
Du T(1ß 1ß 8) is a maximum when u has the same direction as w . Now, w œ ™ T Š ™knTk#†n ‹ n
62 2 ‰
œ (6i 31j 2k) ˆ 124
(2i 2j k) œ ˆ6
41
œ 98
9 i
127
9
j
58
9
k Ê uœ
w
kwk
152 ‰
i
9
ˆ31
152 ‰
j
9
ˆ2
76 ‰
9 k
"
œ È29,097
(98i 127j 58k).
22. Suppose the surface (boundary) of the mineral deposit is the graph of z œ f(xß y) (where the z-axis points up into the air).
Then `` xf i `` yf j k is an outer normal to the mineral deposit at (xß y) and `` xf i `` yf j points in the direction of steepest
ascent of the mineral deposit. This is in the direction of the vector
`f
`x
i
`f
`y
j at (0ß 0) (the location of the 1st borehole)
that the geologists should drill their fourth borehole. To approximate this vector we use the fact that (0ß 0ß 1000),
(0ß 100ß 950), and (100ß !ß 1025) lie on the graph of z œ f(xß y). The plane containing these three points is a good
â
â
j
k â
â i
â
â
"00 50 â
approximation to the tangent plane to z œ f(xß y) at the point (0ß 0ß 0). A normal to this plane is â 0
â
â
25 â
â "00 0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
880
Chapter 14 Partial Derivatives
œ 2500i 5000j 10,000k, or i 2j 4k. So at (0ß 0) the vector
geologists should drill their fourth borehole in the direction of
"
È5
`f
`x
`f
`y
i
j is approximately i 2j . Thus the
(i 2j) from the first borehole.
23. w œ ert sin 1x Ê wt œ rert sin 1x and wx œ 1ert cos 1x Ê wxx œ 1# ert sin 1x; wxx œ
positive constant determined by the material of the rod Ê 1# ert sin 1x œ
"
c#
"
c#
wt , where c# is the
arert sin 1xb
# #
Ê ar c# 1# b ert sin 1x œ 0 Ê r œ c# 1# Ê w œ ec 1 t sin 1x
24. w œ ert sin kx Ê wt œ rert sin kx and wx œ kert cos kx Ê wxx œ k# ert sin kx; wxx œ
Ê k# ert sin kx œ
"
c#
"
c#
wt
# #
arert sin kxb Ê ar c# k# b ert sin kx œ 0 Ê r œ c# k# Ê w œ ec k t sin kx.
# #
Now, w(Lß t) œ 0 Ê ec k t sin kL œ 0 Ê kL œ n1 for n an integer Ê k œ
n1
L
# # #
#
Ê w œ ec n 1 tÎL sin ˆ nL1 x‰ .
# # #
#
As t Ä _, w Ä 0 since ¸sin ˆ nL1 x‰¸ Ÿ 1 and ec n 1 tÎL Ä 0.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
CHAPTER 15 MULTIPLE INTEGRALS
15.1 DOUBLE AND ITERATED INTEGRALS OVER RECTANGLES
1.
'12 '04 2xy dy dx œ '12 cx y# d 40 dx œ '12 16x dx œ c8 x# d 21
2.
'02 'c11 ax yb dy dx œ '02 xy 12 y# ‘ ""
3.
'c01 'c11 (x y 1) dx dy œ 'c01 ’ x2
4.
'01 '01 Š1 x 2 y ‹ dx dy œ '01 ’x x6
5.
'03 '02 a4 y# b dy dx œ '03 ’4y y3 “ # dx œ '03 163 dx œ 163 x‘30
6.
'03 'c02 ax# y 2xyb dy dx œ '03 ’ x 2y
7.
'01 '01 1 yx y dx dy œ '01 clnl1 x yld"0 dy œ '01 lnl1 yldy œ cy lnl1 yl y lnl1 yld 10 œ 2 ln 2 1
8.
'14 '04 ˆ 2x Èy‰ dx dy œ '14 41 x2 xÈy‘ !4 dy œ '14 ˆ4 4 y1/2 ‰dy œ 4y 38 y3/2 ‘41
9.
'0ln 2 '1ln 5 e2x y dy dx œ '0ln 2 ce2x y dln" 5 dx œ '0ln 2 a5e2x e2x 1 b dx œ 52 e2x "# e2x 1 ‘0ln 2
10.
'01 '12 x y ex dy dx œ '01 "# x y2 ex ‘2" dx œ '01 32 x ex dx œ 32 x ex 32 ex ‘10
11.
'c21 '01Î2 y sin x dx dy œ 'c21 cy cos xd10 Î2 dy œ 'c21 y dy œ "# y2 ‘2 1 œ 32
12.
'121 '01 asin x cos yb dx dy œ '121 ccos x x cos yd01 dy œ '121 a2 1 cos yb dy œ c2y 1 sin yd121
13.
' ' a6 y# 2 xbdA œ ' ' a6 y# 2 xb dy dx œ ' c2 y3 2 x yd20 dx œ ' a16 4 xb dx œ c16 x 2 x2 d10 œ 14
0
0
0
0
#
#
dx œ '0 2x dx œ c x# d 0 œ 4
2
"
yx x“
#
3
2
dy œ 'c1 (2y 2) dy œ cy# 2yd " œ 1
0
"
"
x y#
2 “0 dy
œ '0 Š 65
1
$
!
# #
1
œ 24
!
xy# “
#
!
y#
2 ‹dy
œ ’ 56 y
œ
2
3
œ 16
dx œ '0 a4x 2x# b dx œ ’2x#
3
2
1
y3
6 “0
œ
1
3
2x$
3 “!
œ0
œ
92
3
œ 32 a5 eb
3
2
œ 21
1
R
14.
''
R
Èx
y2 dA
œ '0
4
'12 Èy x dy dx œ '04 ’ Èyx “2 dx œ '04 "# x1Î2 dx œ 31 x3Î2 ‘40
2
1
8
3
' ' x y cos y dA œ ' ' x y cos y dy dx œ ' cx y sin y x cos yd10 dx œ ' a2xb dx œ cx2 dc1 1 œ 0
c1 0
c1
c1
1
15.
œ
1
1
1
R
' ' y sinax yb dA œ ' ' y sinax yb dy dx œ ' cy cosax yb sinax ybd10 dx
c1 0
c1
0
16.
1
0
R
œ 'c1 asinax 1b 1 cosax 1b sin xbdx œ ccosax 1b 1 sinax 1b cos xdc0 1 œ 4
0
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
882
Chapter 15 Multiple Integrals
' ' ex y dA œ ' ' ex y dy dx œ ' cex y dln0 2 dx œ ' aex ln 2 ex b dx œ cex ln 2 ex dln0 2 œ
0
0
0
0
ln 2
17.
ln 2
ln 2
ln 2
R
' ' x y ex y2 dA œ ' ' x y ex y2 dy dx œ ' ’ "# ex y2 “ dx œ ' ˆ "# ex "# ‰ dx œ "# ex "# x‘20 œ "# ae2 3b
0
0
0
0
2
18.
1
1
2
''
R
20.
''
R
2
0
R
19.
"
#
x y3
x2 1 dA
œ '0
y
x2 y2 1 dA
1
'02 xx y 1 dy dx œ '01 ’ 4axx y 1b “2 dx œ '01 x 4x 1 dx œ c2 lnlx2 1ld10
3
4
2
2
2
0
œ '0
1
œ 2 ln 2
'01 ax yby 1 dx dy œ '01 ctan1 ax ybd10 dy œ '01 tan1 y dy œ y tan1 y "# lnl1 y2 l‘10
2
21.
'12 '12
22.
'01 '01 y cos xy dx dy œ '01 csin xyd 1! dy œ '01 sin 1y dy œ 1" cos 1y‘ "! œ 1" (1 1) œ 12
1
xy
dy dx œ '1
2
"
x
(ln 2 ln 1) dx œ (ln 2) '1
2
"
x
œ
1
4
"# ln 2
dx œ (ln 2)#
"
"
23. V œ ' ' fax, yb dA œ 'c1 'c1 ax2 y2 b dy dx œ 'c1 x2 y 31 y3 ‘ 1 dx œ 'c1 ˆ2 x2 32 ‰ dx œ 32 x3 32 x‘ 1 œ
1
1
1
1
R
24. V œ ' ' fax, yb dA œ '0
2
R
œ
8
3
'02 a16 x2 y2 b dy dx œ '02 16 y x2 y 13 y3 ‘20 dx œ '02 ˆ 883 2 x2 ‰ dx œ 883 x 23 x3 ‘20
160
3
25Þ V œ ' ' fax, yb dA œ '0
'01 a2 x yb dy dx œ '01 2 y x y "# y2 ‘ "! dx œ '01 ˆ 32 x‰ dx œ 32 x "# x2 ‘ "! œ 1
26Þ V œ ' ' fax, yb dA œ '0
'02 y2 dy dx œ '04 ’ y4 “2 dx œ '04 1 dx œ cxd40 œ 4
1
R
4
R
27Þ V œ ' ' fax, yb dA œ '0
2
0
1Î2
R
'01Î4 2 sin x cos y dy dx œ '01Î2 c2 sin x sin yd01Î4 dx œ '01Î2 ŠÈ2 sin x‹ dx œ ’È2 cos x“1Î2
0
œ È2
28. V œ ' ' fax, yb dA œ '0
1
R
'02 a4 y2 b dy dx œ '01 4 y 13 y3 ‘20 dx œ '01 ˆ 163 ‰ dx œ 163 x‘ "! œ 163
15.2 DOUBLE INTEGRALS OVER GENERAL REGIONS
1.
2.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 15.2 Double Integrals Over General Regions
3.
4.
5.
6.
7.
8.
9. (a)
'!# 'x8 dy dx
3
(b)
'!8 '0y
10. (a)
'!3 '02x dy dx
(b)
'!6 'y3Î2 dx dy
11. (a)
'!3 'x3x dy dx
(b)
'!9 'yÈÎ3y dx dy
12. (a)
'!# '1e dy dx
(b)
'1e 'ln2 y dx dy
13. (a)
'!9 '0
2
x
1Î3
dx dy
2
Èx
(b)
dy dx
'0 'y dx dy
3
9
2
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
883
884
Chapter 15 Multiple Integrals
14. (a)
'!1Î4 'tan1 x dy dx
(b)
15. (a)
'01 '0tan
16. (a)
dx dy
'!ln 3 'e1c dy dx
x
'1Î3 'ln y dx dy
1
(b)
c1 y
ln 3
'!1 '01 dy dx '1e 'ln1 x dy dx
'01 '0e dx dy
y
(b)
17. (a)
(b)
18. (a)
'!1 'x3 2x dy dx
'01 '0y dx dy '13 '0a3 ybÎ2 dx dy
'21 'xx 2 dy dx
2
'0 'Èy dx dy '13 'yÈy2 dx dy
1
(b)
19.
Èy
'01 '0x (x sin y) dy dx œ '01 c x cos yd x! dx
1
1
œ '0 (x x cos x) dx œ ’ x2 (cos x x sin x)“
#
œ
1#
#
!
2
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 15.2 Double Integrals Over General Regions
20.
'01 '0sin x y dy dx œ '01 ’ y2 “ sin x dx œ '01 "# sin# x dx
#
!
œ
21.
"
4
'01 (1 cos 2x) dx œ "4 x "2 sin 2x‘ !1 œ 14
'1ln 8 '0ln yexby dx dy œ '1ln 8 cexbyd !ln y dy œ '1ln 8 ayey eyb dy
œ c(y 1)ey ey d 1ln 8 œ 8(ln 8 1) 8 e
œ 8 ln 8 16 e
'12 'yy
#
22.
dx dy œ '1 ay# yb dy œ ’ y3
2
$
œ ˆ 83 2‰ ˆ "3 "# ‰ œ
7
3
œ
3
#
5
6
'01 '0y 3y$ exy dx dy œ '01 c3y# exy d 0y
#
23.
#
y#
# “"
#
dy
œ '0 Š3y# ey 3y# ‹ dy œ ’ey y$ “ œ e 2
1
$
"
$
!
24.
Èx
'14 '0
œ
3
#
3
#
eyÎÈx dy dx œ
'14 32 Èx eyÎÈx ‘ 0Èx dx
%
(e 1) '1 Èx dx œ 23 (e 1) ˆ 32 ‰ x$Î# ‘ " œ 7(e 1)
4
25.
'12 'x2x
26.
'01 '01cx ax# y# b dy dx œ '01 ’x# y y3 “ "
x
y
dy dx œ '1 cx ln yd x2x dx œ (ln 2) '1 x dx œ
2
2
$
x
0
$
œ ’ x3
27.
x%
4
"
(1x)%
1# “ !
œ ˆ "3
"
4
#
œ '0 Š "# u
1
u#
#
vÈ u “
ln 2
dx œ '0 ’x# (1 x)
1
0‰ ˆ0 0
'01 '01cu ˆv Èu‰ dv du œ '01 ’ v2
3
#
"
u
0
u"Î# u$Î# ‹ du œ ’ u2
" ‰
1#
œ
(1x)$
3 “
dx œ '0 ’x# x$
1
(1x)$
3 “
dx
"
6
du œ '0 ’ 12u# u Èu(1 u)“ du
1
u#
#
u$
6
#
"
32 u$Î# 25 u&Î# “ œ
!
"
#
"
#
"
6
2
3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
2
5
œ #"
2
5
"
œ 10
885
886
28.
Chapter 15 Multiple Integrals
'12 '0ln t es ln t ds dt œ '12 ces ln td 0ln t dt œ '12 (t ln t ln t) dt œ ’ t2
#
œ (2 ln 2 1 2 ln 2 2) ˆ "4 1‰ œ
29.
"
4
ln t
t#
4
t ln t t“
#
"
'c02 'vcv 2 dp dv œ 2'c02 cpd vv dv œ 2'c02 2v dv
œ 2 cv# d c2 œ 8
0
30.
È1cs
'01 '0
#
È1cs
8t dt ds œ '0 c4t# d 0
1
œ '0 4 a1 s# b ds œ 4 ’s
1
31.
#
ds
"
s$
3 “!
œ
8
3
'c11ÎÎ33 '0sec t 3 cos t du dt œ ' 11ÎÎ33 c(3 cos t)ud 0sec t
1Î3
œ 'c1Î3 3 dt œ 21
32.
'03Î2 '14 2u 4 v 2u dv du œ '03Î2 2u v 4 ‘ 14 2u du
3Î2
$Î2
œ '0 a3 2ub du œ c3u u# d ! œ 92
#
33.
'24 '0Ð4
y)Î2
34.
' 02 '0x2 dy dx
dx dy
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 15.2 Double Integrals Over General Regions
35.
'01 'xx dy dx
36.
'01 '1cy1cydx dy
37.
'1e 'ln1ydx dy
38.
'12 '0ln x dy dx
39.
'09 '0
40.
'04 '0
#
È
1
2
È9cy
È4cx
16x dx dy
y dy dx
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
887
888
Chapter 15 Multiple Integrals
È1cx
41.
'c11 '0
42.
'c22 '0
43.
'01 'ee x y dx dy
44.
'01Î2 '0sin
45.
'1e 'ln3 x ax ybdy dx
46.
'01Î3 'tan 3x Èx y dy dx
È4cy
#
3y dy dx
#
6x dx dy
y
c1 y
x y2 dx dy
3
È
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 15.2 Double Integrals Over General Regions
47.
48.
'01 'x1 siny y dy dx œ '01 '0y siny y dx dy œ '01 sin y dy œ 2
'02 'x2 2y# sin xy dy dx œ '02 '0y2y# sin xy dx dy
2
2
œ '0 c2y cos xyd 0y dy œ '0 a2y cos y# 2yb dy
#
œ c sin y# y# d ! œ 4 sin 4
49.
'01 'y1 x# exy dx dy œ '01 '0x x# exy dy dx œ '01 cxexyd 0x dx
œ '0 axex xb dx œ ’ "2 ex
1
È4cy
'02 '04cx 4xey dy dx œ '04 '0
#
50.
2y
œ '0 ’ #x(4ey) “
4
51.
'02
# 2y
Èln 3 Èln 3
'y/2
Èln 3
œ '0
52.
"
x#
# “!
#
#
È4cy
0
dy œ '0
4 2y
e
Èln 3
ex dx dy œ '0
#
#
#
Èln 3
$
dy dx œ '0
1
'03y
#
e2
#
dx dy
2y
%
dy œ ’ e4 “ œ
#
2xex dx œ cex d 0
'03 'È1xÎ3 ey
xe2y
4 y
œ
!
'02x ex
#
e) "
4
dy dx
œ eln 3 1 œ 2
$
ey dx dy
œ '0 3y# ey dy œ cey d ! œ e 1
1
53.
$
$
"
'01Î16 'y1Î2 cos a161x& b dx dy œ '01Î2 '0x
"Î%
%
cos a161x& b dy dx
161x b
œ '0 x% cos a161x& b dx œ ’ sin a80
“
1
1Î2
&
"Î#
!
œ
"
801
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
889
890
54.
Chapter 15 Multiple Integrals
'08 'È2x
$
œ '0
2
55.
dy dx œ '0
2
"
y % 1
y$
y % 1
dy œ
"
4
'0y y "1 dx dy
$
%
#
cln ay% 1bd ! œ
ln 17
4
' ' ay 2x# b dA
R
xb1
œ 'c1 'cxc1 ay 2x# b dy dx '0
0
1
'x1cc1x ay 2x# b dy dx
x "
1x
œ 'c1 "2 y# 2x# y‘ x1 dx '0 2" y# 2x# y‘ x1 dx
0
1
œ 'c1 "# (x 1)# 2x# (x 1) "# (x 1)# 2x# (x 1)‘dx
0
'0 "# (1 x)# 2x# (1 x) "# (x 1)# 2x# (x 1)‘ dx
1
œ 4 'c1 ax$ x# b dx 4 '0 ax$ x# b dx
0
1
%
œ 4 ’ x4
56.
0
x$
3 “ c1
"
x$
3 “!
%
4 ’ x4
%
œ 4 ’ (41)
(1)$
3 “
3
4 ˆ 4" 3" ‰ œ 8 ˆ 12
4 ‰
12
8
œ 12
œ 32
' ' xy dA œ ' ' xy dy dx ' ' xy dy dx
0
x
2Î3 x
2Î3
R
2x
2Î3
1
2x
2
œ '0 "2 xy# ‘ x dx '2Î3 2" xy# ‘ x
1
x
2 x
dx
œ '0 ˆ2x$ "# x$ ‰ dx '2Î3 "# x(2 x)# "# x$ ‘ dx
2Î3
1
œ '0
2Î3
3
#
x$ dx '2Î3 a2x x# b dx
1
2Î3
"
2‰
8 ‰‘
‰ ˆ
4 ˆ 2 ‰ ˆ 27
œ 38 x% ‘ 0 x# 23 x$ ‘ #Î$ œ ˆ 38 ‰ ˆ 16
œ
81 1 3 9 3
57. V œ '0
1
'x2cx ax# y# b dy dx œ '01 ’x# y y3 “ 2cx dx œ '01 ’2x# 7x3
$
$
x
œ ˆ 23
7
12
2cx#
58. V œ 'c2 'x
1
œ ˆ 23
"
5
" ‰
12
ˆ0 0
4cx#
1
œ
x# dy dx œ 'c2 cx# yd x
32
5
16 ‰
4
(2x)$
3 “
È 4 cx
40
œ ˆ 60
2
7x%
12
13
81
"
(2x)%
12 “ !
12
60
15 ‰
60
ˆ 320
60
384
60
240 ‰
60
œ
189
60
œ
63
20
4cx
(x 4) dy dx œ 'c4 cxy 4yd 3x
dx œ 'c4 cx a4 x# b 4 a4 x# b 3x# 12xd dx
1
1
#
#
(3 y) dy dx œ '0 ’3y
2
œ ’ 32 xÈ4 x# 6 sin" ˆ x# ‰ 2x
61. V œ '0
$
œ
"
"
'0
16 ‰
81
dx œ ’ 2x3
1
1
2
ˆ 36
81
dx œ 'c2 a2x# x% x$ b dx œ 23 x$ 15 x& 14 x% ‘ #
œ 'c4 ax$ 7x# 8x 16b dx œ 41 x% 37 x$ 4x# 16x‘ % œ ˆ 4"
60. V œ '0
27
81
4
3
2cx#
1
4" ‰ ˆ 16
3
59. V œ 'c4 '3x
16 ‰
12
6
81
È 4c x
y#
2 “0
#
x$
6 “!
#
7
3
‰
12‰ ˆ 64
3 64 œ
dx œ '0 ’3È4 x# Š 4#x ‹“ dx
2
œ 6 ˆ 1# ‰ 4
#
8
6
œ 31
16
6
œ
918
3
'03 a4 y# b dx dy œ '02 c4x y# xd $! dy œ '02 a12 3y# b dy œ c12y y$ d !# œ 24 8 œ 16
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
157
3
"
4
œ
625
12
Section 15.2 Double Integrals Over General Regions
62. V œ '0
2
'04cx
#
2
œ 8x 43 x$
63. V œ '0
2
4cx#
a4 x# yb dy dx œ '0 ’a4 x# b y
"
10
#
x& ‘ ! œ 16
32
3
32
10
œ
y#
2 “!
48032096
30
œ
"
#
a4 x# b dx œ '0 Š8 4x#
2
#
!
xb1
1
1Îx
2
66. V œ 4 '0
1Î3
'x1cc1x (3 3x) dy dx œ 6 'c01 a1 x# b dx 6 '01 (1 x)# dx œ 4 2 œ 6
2
2
"
x
ˆ1 x" ‰‘dx œ 2 '1 ˆ1 x" ‰ dx œ 2 cx ln xd #"
2
'0sec x a1 y# b dy dx œ 4 '01Î3 ’y y3 “ sec x dx œ 4 '01Î3 Šsec x sec3 x ‹ dx
$
$
0
1Î$
c7 ln ksec x tan xk sec x tan xd !
œ
’7 ln Š2 È3‹ 2È3“
2
3
67.
68.
'1_ 'ec1 x"y dy dx œ '1_ ’ lnx y “ "
x
$
$
ec x
_
dx œ '1 ˆ x$x ‰ dx œ lim
bÄ_
1/ ˆ1cx ‰
È1cx
1
70. 'c1 'c1/È1cx (2y 1) dy dx œ 'c1 cy# ydº
1
1/
# 1Î#
#
#
c1/ a1c
x# b1Î#
œ 4 lim c csin" b 0d œ 21
bÄ1
71.
dx
128
15
65. V œ '1 'c1Îx (x 1) dy dx œ '1 cxy yd 1Î1xÎx dx œ '1 1
œ 2(1 ln 2)
69.
x%
#‹
%
0
2
3
2
#
2
x
'02cx a12 3y# b dy dx œ '02 c12y y$ d #
dx œ '0 c24 12x (2 x)$ d dx œ ’24x 6x# (24x) “ œ 20
!
64. V œ 'c1 'cxc1 (3 3x) dy dx '0
œ
dx œ '0
_ _
'c_
' _ ax 1b"ay 1b -dx dy œ 2 '0_ Š y 21 ‹ Š
#
#
#
lim
bÄ_
dx œ 'c1 È 2
x" ‘ b œ lim
1
1
1 x #
bÄ_
ˆ "b 1‰ œ 1
dx œ 4 lim c csin" xd !
bÄ1
b
tan" b tan" 0‹ dy œ 21 lim
bÄ_
'0b y "1 dy
#
œ 21 Š lim tan" b tan" 0‹ œ (21) ˆ 1# ‰ œ 1#
bÄ_
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
891
892
72.
Chapter 15 Multiple Integrals
'0_ '0_ xecÐx
2yÑ
_
_
cxex ex d b0 dy œ '0 e2y lim
bÄ_
œ '0 ec2y dy œ
73.
_
dx dy œ '0 e2y lim
"
# b lim
Ä_
abeb eb 1b dy
bÄ_
aec2b 1b œ
"
#
' ' f(xß y) dA ¸ "4 f ˆ "# ß 0‰ 8" f(0ß 0) 8" f ˆ "4 ß 0‰ œ "4 ˆ "# ‰ 8" ˆ0 "4 ‰ œ 323
R
74.
' ' f(xß y) dA ¸ "4 ’f ˆ 47 ß 114 ‰ f ˆ 94 ß 114 ‰ f ˆ 74 ß 134 ‰ f ˆ 94 ß 134 ‰“ œ
R
75. The ray ) œ
1
6
"
16
(29 31 33 35) œ
128
16
œ8
meets the circle x# y# œ 4 at the point ŠÈ3ß 1‹ Ê the ray is represented by the line y œ
È
È
È
$Î#
3
4cx
3
x# b
' ' f(xß y) dA œ ' ' È È4x# dy dx œ ' ’a4 x# b Èx3 È4 x# “ dx œ ”4x x3$ a4È
0
xÎ 3
0
3 3 •
#
R
76.
'2_ '02 ax xb "(y1)
#
bÄ_
bÄ_
77. V œ '0
1
0
cln (x 1) ln xd 2b œ 6 lim
lim
bÄ_
_
dx œ 6 '2
bÄ_
dx
x(x1)
[ln (b 1) ln b ln 1 ln 2]
$
x
7x
3
œ ˆ 23
" ‰
1#
7
12
$
(2x)$
3 “
$
dx œ ’ 2x3
ˆ0 0
16 ‰
12
œ
œ '0
'2
œ
2 tan
ˆ1 1" ‰ y
1y# dy
1
1
ˆ 21 ‰ ln 5
"
21
ˆ2 1
1 y #
"
"
21 2 tan
y‰
21
2
%
7x
12
"
(2x)%
12 “ !
4
3
'02 atan" 1x tan" xb dx œ '02 'x1x 1"y
œ 2 tan
3 ‰
x # x
'x2cx ax# y# b dy dx œ '01 ’x# y y3 “ 2cx dx
œ '0 ’2x#
2
2
"Î$
0
ln ˆ1 "b ‰ ln 2“ œ 6 ln 2
1
78.
_
1)
' ˆ x#3x
dy dx œ '2 ’ 3(y
ax# xb “ dx œ 2
'2b ˆ x" 1 "x ‰ dx œ 6
œ 6 lim
œ 6 ’ lim
_
#Î$
È3
dy dx œ '0
2
#
'yyÎ1
"
1y #
dx dy '2
21
# #
"
‰
dy œ ˆ 12"
y
1 cln a1 y bd ! 2 tan
"
21
"
21
ln a1 41# b 2 tan" 2
#
ln a1 41 b
"
#1
"
#1
'y2Î1 1"y
#
dx dy
21
ln a1 y# b‘ 2
ln 5
ln 5
#
79. To maximize the integral, we want the domain to include all points where the integrand is positive and to
exclude all points where the integrand is negative. These criteria are met by the points (xß y) such that
4 x# 2y# 0 or x# 2y# Ÿ 4, which is the ellipse x# 2y# œ 4 together with its interior.
80. To minimize the integral, we want the domain to include all points where the integrand is negative and to
exclude all points where the integrand is positive. These criteria are met by the points (xß y) such that
x# y# 9 Ÿ 0 or x# y# Ÿ 9, which is the closed disk of radius 3 centered at the origin.
81. No, it is not possible. By Fubini's theorem, the two orders of integration must give the same result.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
x
È3
œ
. Thus,
20È3
9
Section 15.2 Double Integrals Over General Regions
82. One way would be to partition R into two triangles with the
line y œ 1. The integral of f over R could then be written
as a sum of integrals that could be evaluated by integrating
first with respect to x and then with respect to y:
' ' f(xß y) dA
R
œ '0
1
'22ccÐ2yyÎ2Ñ f(xß y) dx dy '12 'y2c1ÐyÎ2Ñ f(xß y) dx dy.
Partitioning R with the line x œ 1 would let us write the
integral of f over R as a sum of iterated integrals with
order dy dx.
83.
' bb ' bb e
x# y#
dx dy œ '
b
'
e
b
b
b
#
y#
e
x#
dx dy œ ' b e
b
#
y#
Œ' b e
b
x#
dx dy œ Œ' b e
b
x#
dx Œ' b e
b
y#
dy
#
#
#
#
œ Œ'cb ecx dx œ Œ2 '0 ecx dx œ 4 Œ'0 ecx dx ; taking limits as b Ä _ gives the stated result.
b
84.
'01 '03 (yx1)
dy dx œ '0
3
#
œ
b
#Î$
"
3 b lim
Ä 1c
'0
b
dy
(y1)#Î$
'01 (yx1)
dx dy œ '0
3
#
#Î$
"
3
b
'b
3
lim
b Ä 1b
dy
(y1)#Î$
œ
"
(y1)#Î$
lim
b Ä 1c
$
"
’ x3 “ dy œ
!
"
3
'03 (ydy1)
#Î$
(y 1)"Î$ ‘ b lim (y 1)"Î$ ‘ 3
0
b
b Ä 1b
3
3
œ ’ lim c (b 1)"Î$ (1)"Î$ “ ’ lim b (b 1)"Î$ (2)"Î$ “ œ (0 1) Š0 È
2‹ œ 1 È
2
bÄ1
bÄ1
85-88. Example CAS commands:
Maple:
f := (x,y) -> 1/x/y;
q1 := Int( Int( f(x,y), y=1..x ), x=1..3 );
evalf( q1 );
value( q1 );
evalf( value(q1) );
89-94. Example CAS commands:
Maple:
f := (x,y) -> exp(x^2);
c,d := 0,1;
g1 := y ->2*y;
g2 := y -> 4;
q5 := Int( Int( f(x,y), x=g1(y)..g2(y) ), y=c..d );
value( q5 );
plot3d( 0, x=g1(y)..g2(y), y=c..d, color=pink, style=patchnogrid, axes=boxed, orientation=[-90,0],
scaling=constrained, title="#89 (Section 15.2)" );
r5 := Int( Int( f(x,y), y=0..x/2 ), x=0..2 ) + Int( Int( f(x,y), y=0..1 ), x=2..4 );
value( r5);
value( q5-r5 );
85-94. Example CAS commands:
Mathematica: (functions and bounds will vary)
You can integrate using the built-in integral signs or with the command Integrate. In the Integrate command, the
integration begins with the variable on the right. (In this case, y going from 1 to x).
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
893
894
Chapter 15 Multiple Integrals
Clear[x, y, f]
f[x_, y_]:= 1 / (x y)
Integrate[f[x, y], {x, 1, 3}, {y, 1, x}]
To reverse the order of integration, it is best to first plot the region over which the integration extends. This can be done
with ImplicitPlot and all bounds involving both x and y can be plotted. A graphics package must be loaded. Remember to
use the double equal sign for the equations of the bounding curves.
Clear[x, y, f]
< y/(x^2+y^2);
a,b := 0,1;
f1 := x -> x;
f2 := x -> 1;
plot3d( f(x,y), y=f1(x)..f2(x), x=a..b, axes=boxed, style=patchnogrid, shading=zhue, orientation=[0,180], title="#47(a)
(Section 15.4)" );
# (a)
q1 := eval( x=a, [x=r*cos(theta),y=r*sin(theta)] );
# (b)
q2 := eval( x=b, [x=r*cos(theta),y=r*sin(theta)] );
q3 := eval( y=f1(x), [x=r*cos(theta),y=r*sin(theta)] );
q4 := eval( y=f2(x), [x=r*cos(theta),y=r*sin(theta)] );
theta1 := solve( q3, theta );
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
904
Chapter 15 Multiple Integrals
theta2 := solve( q1, theta );
r1 := 0;
r2 := solve( q4, r );
plot3d(0,r=r1..r2, theta=theta1..theta2, axes=boxed, style=patchnogrid, shading=zhue, orientation=[-90,0],
title="#47(c) (Section 15.4)" );
fP := simplify(eval( f(x,y), [x=r*cos(theta),y=r*sin(theta)] ));
# (d)
q5 := Int( Int( fP*r, r=r1..r2 ), theta=theta1..theta2 );
value( q5 );
Mathematica: (functions and bounds will vary)
For 47 and 48, begin by drawing the region of integration with the FilledPlot command.
Clear[x, y, r, t]
< x^2*y^2*z;
q1 := Int( Int( Int( F(x,y,z), y=-sqrt(1-x^2)..sqrt(1-x^2) ), x=-1..1 ), z=0..1 );
value( q1 );
Mathematica: (functions and bounds will vary)
Clear[f, x, y, z];
f:= x2 y2 z
Integrate[f, {x,1,1}, {y,Sqrt[1 x2 ], Sqrt[1 x2 ]}, {z, 0, 1}]
N[%]
topolar={x Ä r Cos[t], y Ä r Sin[t]};
fp= f/.topolar //Simplify
Integrate[r fp, {t, 0, 21}, {r, 0, 1},{z, 0, 1}]
N[%]
15.6 MOMENTS AND CENTERS OF MASS
1. M œ '0
1
'x2cx 3 dy dx œ 3'01 a2 x# xb dx œ 7# ; My œ '01 'x2cx
#
œ 3'0 a2x x$ x# b dx œ
1
Ê xœ
2. M œ $ '0
3
Iy œ $ '0
3
3. M œ '0
2
œ
"
#
and y œ
5
14
'03
'0
5
4
; Mx œ '0
'x2cx
#
3y dy dx œ
3x dy dx œ 3 '0 cxyd x2cx dx
1
'01 cy# d x2cx
3
#
#
dx œ
3
#
#
'01 a4 5x# x% b dx œ 195
38
35
dy dx œ $ '0 3 dx œ 9$ ; Ix œ $ '0
3
3
x dy dx œ $ '0 cx
3
1
#
3
#
#
$
yd !
'03 y# dy dx œ $ '03 ’ y3 “ 3 dx œ 27$ ;
$
0
dx œ $ '0 3x dx œ 27$
3
#
'y4Î2ydx dy œ '02 Š4 y y# ‹ dy œ 143 ; My œ '02 'y4Î2y
#
#
#
4cy
x dx dy œ
"
#
'02 cx# d y4 Îy2 dy
#
'0 Š16 8y y# y4 ‹ dy œ 128
' 'y Î2 y dx dy œ '0 Š4y y# y# ‹ dy œ 103
15 ; Mx œ 0
2
2
%
2
$
#
Ê xœ
4. M œ '0
3
64
35
and y œ
5
7
'03cx dy dx œ '03 (3 x) dx œ 9# ; My œ '03 '03cx x dy dx œ '03 cxyd 03cx dx œ '03 a3x x# b dx œ 9#
Ê x œ 1 and y œ 1, by symmetry
Èa cx
5. M œ '0 '0
a
#
Ê xœyœ
#
4a
31
dy dx œ
1a#
4
; My œ
Èa cx
'0a '0
#
#
a
a
È# #
x dy dx œ '0 cxyd 0 a cx dx œ '0 xÈa# x# dx œ
, by symmetry
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
a$
3
909
910
Chapter 15 Multiple Integrals
6. M œ '0
'0sin x dy dx œ '01 sin x dx œ 2; Mx œ '01 '0sin x y dy dx œ "# '01 cy# d 0sin x dx œ "# '01 sin# x dx
1
œ
"
4
'01 (1 cos 2x) dx œ 14
È4cx
1
#
Ê xœ
and y œ
È4cx
7. Ix œ 'c2 'cÈ4cx# y# dy dx œ 'c2 ’ y3 “
dx œ
cÈ4cx#
I o œ I x I y œ 81
2
8. Iy œ '1
21
#
2
$
#
#
ˆ
œ
'c22 a4 x# b$Î# dx œ 41; Iy œ 41, by symmetry;
#
‰
ex
0
b Ä _
'c_ e
0
2x
_
10. My œ '0
dx œ
'0e
'b0 ex dx œ 1
0
'b0 xex dx œ
lim
"
#
2
3
'0 sin x Îx x# dy dx œ '121 asin# x 0b dx œ "# '121 (1 cos 2x) dx œ 1#
9. M œ 'c_ '0 dy dx œ ' _ ex dx œ lim
b Ä _
œ
1
8
x# Î2
cxex ex d b0 œ 1
lim
b Ä _
"
lim
# bÄ
_
'b0 e2x dx œ "4
x dy dx œ lim
bÄ_
ycy#
'0b xe
x# Î2
eb œ 1; My œ ' _ '0 x dy dx œ ' _ xex dx
b Ä _
ex
0
abeb eb b œ 1; Mx œ 'c_ '0 y dy dx
b Ä _
2
Ê x œ 1 and y œ
dx œ lim
bÄ_
ycy#
#
"
ex# Î2
"
4
b
1‘ 0 œ 1
ycy#
2
%
ycy#
&
Ix œ '0 'cy y# (x y) dx dy œ '0 ’ x 2y xy$ “
dy œ '0 Š y2 2y& 2y% ‹ dy œ
cy
2
È3Î2
2
È12
12. M œ 'cÈ3Î2 '4y#
4y#
# #
È3Î2
È12
5x dx dy œ 5 ' È3Î2 ’ x2 “
4y#
#
4y#
ex
0
lim
y
11. M œ '0 'cy (x y) dx dy œ '0 ’ x2 xy“
dy œ '0 Š y2 2y$ 2y# ‹ dy œ ’ 10
cy
2
0
lim
2
dy œ
5
#
'
y%
#
64
105
#
2y$
3 “!
œ
8
15
;
;
È
' È33ÎÎ22 a12 4y# 16y% b dy œ 23È3
'x2cx (6x 3y 3) dy dx œ '01 6xy 3# y# 3y‘ x2cx dx œ '01 a12 12x# b dx œ 8;
1
2cx
1
1
2cx
My œ '0 'x x(6x 3y 3) dy dx œ '0 a12x 12x$ b dx œ 3; Mx œ '0 'x y(6x 3y 3) dy dx
1
3
17
œ '0 a14 6x 6x# 2x$ b dx œ 17
# Ê x œ 8 and y œ 16
13. M œ '0
1
14. M œ '0
1
'y2ycy (y 1) dx dy œ '01 a2y 2y$ b dy œ "# ; Mx œ '01 'y2ycy
#
#
#
My œ '0
1
#
2ycy#
'y
#
x(y 1) dx dy œ '0 a2y# 2y% b dy œ
1
œ 2 '0 ay$ y& b dy œ
1
4
15
Ê xœ
8
15
y(y 1) dx dy œ '0 a2y# 2y% b dy œ
1
and y œ
8
15
; Ix œ '0
1
2ycy#
'y
#
4
15
;
y# (y 1) dx dy
"
6
15. M œ '0
'06 (x y 1) dx dy œ '01 (6y 24) dy œ 27; Mx œ '01 '06 y(x y 1) dx dy œ '01 y(6y 24) dy œ 14;
1
6
1
14
' 1' 6 #
My œ '0 '0 x(x y 1) dx dy œ '0 (18y 90) dy œ 99 Ê x œ 11
3 and y œ 27 ; Iy œ 0 0 x (x y 1) dx dy
1
‰
œ 216 '0 ˆ y3 11
6 dy œ 432
1
16. M œ 'c1 'x# (y 1) dy dx œ 'c1 Š x# x# 3# ‹ dx œ
1
œ
48
35
1
1
%
; My œ 'c1 'x# x(y 1) dy dx œ 'c1 Š 3x
#
1
œ 'c1 Š 3x2
1
#
x'
2
1
1
x% ‹ dx œ
x&
#
32
15
; Mx œ 'c1 'x# y(y 1) dy dx œ 'c1 Š 56
1
1
x$ ‹ dx œ 0 Ê x œ 0 and y œ
1
9
14
x'
3
x%
#‹
dx
; Iy œ 'c1 'x# x# (y 1) dy dx
16
35
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
1
1
Section 15.6 Moments and Centers of Mass
17. M œ 'c1 '0 (7y 1) dy dx œ 'c1 Š 7x# x# ‹ dx œ
x#
1
1
%
31
15
; Mx œ 'c1 '0 y(7y 1) dy dx œ 'c1 Š 7x3
x#
1
1
My œ 'c1 '0 x(7y 1) dy dx œ 'c1 Š 7x# x$ ‹ dx œ 0 Ê x œ 0 and y œ
1
x
#
1
œ 'c1 Š 7x# x% ‹ dx œ
1
'
18. M œ '0
My œ '0
'c1
1
x ˆ1
x ‰
20
dy dx œ '0 Š2x
20
y
x#
10 ‹
dx œ
Ê xœ
2000
3
1
1
y
; Iy œ '0 'cy x# (y 1) dx dy œ
1
1
y
y
"
3
1
3
#
x
and y œ 0; Ix œ '0
20
100
9
21. Ix œ '0 '0
a
b
œ 'c3 'c2 ’ 8y3
4
#
œ 'c3 'c2 ’ (4 812y)
3
4
$
Ð4
2yÑÎ3
Iz œ 'c3 'c2 'c4Î3
3
4
23. M œ 4 '0
1
œ 2 '0
8(2 y)$
81
x# (4 2y)
3
64
81 “
4x#
3
11
30
aa# c# b and Iz œ
M
3
y
4
1
$
c$ b
3 ‹
Ð4
2yÑÎ3
6
5
dx œ
abc ab# c# b
3
ay# z# b dz dy dx
3
3
ax# y# b dz dy dx œ '
'
3
4
16
15
aa# b# b , by symmetry
dy dx œ 'c3 ˆ12x#
3
;
1
Ê Io œ Ix Iy œ
3
64
81 “
7
6
6
5
Ð4
2yÑÎ3
' ' '
dy dx œ 'c3 104
3 dx œ 208; Iy œ c3 c2 c4Î3
dy dx
ax# y# b ˆ 83
2
4
ax# z# b dz dy dx
32 ‰
3
dx œ 280;
2y ‰
3
dy dx œ 12 ' 3 ax# 2b dx œ 360
3
'01 '4y4 dz dy dx œ 4 '01 '01 a4 4y# b dy dx œ 16 '01 23 dx œ 323 ; Mxy œ 4 '01 '01 '4y4
#
#
z dz dy dx
12
'0 a16 16y% b dy dx œ 128
'0 dx œ 128
5
5 Ê z œ 5 , and x œ y œ 0, by symmetry;
1
1
4
1
1
64y
7904
%
' 1 1976
‰
Ix œ 4 '0 '0 '4y ay# z# b dz dy dx œ 4 '0 '0 ’ˆ4y# 64
3 Š4y 3 ‹“ dy dx œ 4 0 105 dx œ 105 ;
1
1
1
'
#
Iy œ 4 '0
1
œ
4832
63
'01 '4y4 ax# z# b dz dy dx œ 4 '01 '01 ’ˆ4x# 643 ‰ Š4x# y# 64y3 ‹“ dy dx œ 4 '01 ˆ 83 x# 128
‰ dx
7
'
#
; Iz œ 4 '0
1
œ 16 '0 Š 2x3
1
#
'01 '4y4 ax# y# b dz dy dx œ 16 '01 '01 ax# x# y# y# y% b dy dx
2
15 ‹
#
dx œ
ŠÈ4 x# ‹Î2
256
45
24. (a) M œ 'c2 'ŠcÈ4cx#‹Î2 '0
2
ŠÈ4 x# ‹Î2
2 x
Myz œ 'c2 'ŠcÈ4cx#‹Î2 '0
2
dz dy dx œ '
2 x
2
2
'ŠŠ È44 xx ‹‹ÎÎ22 (2 x) dy dx œ ' 22 (2 x) ŠÈ4 x# ‹ dx œ 41;
x dz dy dx œ '
È
#
#
2
2
ŠÈ4 x# ‹Î2
'Š
È4 x# ‹Î2
;
; Ix œ '0 'cy y# a3x# 1b dx dy œ '0 a2y& 2y$ b dy œ 56 ;
32
45
3
2y$
3
x ‰
20
1
y
is the top of the wedge Ê Ix œ 'c3 'c2 'c4Î3
4 2y
3
y
Ê Io œ Ix Iy œ
$
22. The plane z œ
3
1
'0c ay# z# b dz dy dx œ '0a '0b Šcy# c3 ‹ dy dx œ '0a Š cb3
M
3
y# ˆ1
dx œ 0;
; Ix œ '0 'cy y# (y 1) dx dy œ '0 a2y% 2y$ b dy
1
1
ab# c# b where M œ abc; Iy œ
"
1
1
Iy œ '0 'cy x# a3x# 1b dx dy œ 2 '0 ˆ 35 y& 3" y$ ‰ dy œ
y
'c1
1
; Mx œ '0 'cy y a3x# 1b dx dy œ '0 a2y% 2y# b dy œ
y
1
7
10
'01 a2y% 2y$ b dy œ 103
My œ '0 'cy x a3x# 1b dx dy œ 0 Ê x œ 0 and y œ
1
M
3
1
y
1
20. M œ '0 'cy a3x# 1b dx dy œ '0 a2y$ 2yb dy œ
œ
; Iy œ 'c1 '0 x# (7y 1) dy dx
13
31
; Mx œ '0 'cy y(y 1) dx dy œ 2 ' ay$ y# b dy œ
0
5
3
My œ '0 'cy x(y 1) dx dy œ '0 0 dy œ 0 Ê x œ 0 and y œ
1
;
#
'020 ˆ1 20x ‰ dx œ 20
1
9
10
13
15
7
5
19. M œ '0 'cy (y 1) dx dy œ '0 a2y# 2yb dy œ
œ
dx œ
#
20
2
3
x%
2‹
'c11 ˆ1 20x ‰ dy dx œ '020 ˆ2 10x ‰ dx œ 60; Mx œ '020 'c11 y ˆ1 20x ‰ dy dx œ '020 ’ˆ1 #x0 ‰ Š y# ‹“ "
20
œ
&
'
911
x(2 x) dy dx œ '
2
2
x(2 x) ŠÈ4 x# ‹ dx œ 21;
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
912
Chapter 15 Multiple Integrals
ŠÈ4 x# ‹Î2
Mxz œ 'c2 'ŠcÈ4cx#‹Î2 '0
2
œ
2 x
'c2 (2 x) ’ 44x
2
"
#
#
4 x #
4 “
ŠÈ4 x# ‹Î2
(b) Mxy œ 'c2 'ŠcÈ4cx#‹Î2 '0
2
œ 51 Ê z œ
25. (a) M œ 4 '0
2
Mxy œ '0
2 x
È4cx
#
È
#
y(2 x) dy dx
#
dx œ 0 Ê x œ "# and y œ 0
"
#
z dz dy dx œ
' 22 'ŠŠ È44 xx ‹‹ÎÎ22
È
#
#
' 22 (2 x)# ŠÈ4 x# ‹ dx
"
#
(2 x)# dy dx œ
'x4 y dz dy dx œ 4 '01Î2 '02 'r 4 r dz dr d) œ 4 '01Î2 '02 a4r r$ b dr d) œ 4 '01Î2 4 d) œ 81;
#
#
#
'0 'r zr dz dr d) œ '0 '0
2
Š 4 x ‹Î2
'
2 Š È4 x ‹Î2
2
5
4
'0
21
y dz dy dx œ '
21
4
#
(b) M œ 81 Ê 41 œ '0
21
Èc
'0 'r
c
#
2
r
#
a16 r% b dr d) œ
r dz dr d) œ '0
21
Èc
'0
32
3
'021 d) œ 6431
acr r$ b dr d) œ '0
21
Ê zœ
8
3
c#
4
c# 1
#
d) œ
, and x œ y œ 0, by symmetry
Ê c# œ 8 Ê c œ 2È2,
since c 0
26. M œ 8; Mxy œ 'c1 '3 'c1 z dz dy dx œ 'c1 '3 ’ z2 “ dy dx œ 0; Myz œ 'c1 '3 'c1 x dz dy dx
"
1
5
1
1
5
"
#
1
5
1
œ 2 'c1 '3 x dy dx œ 4 'c1 x dx œ 0; Mxz œ 'c1 '3 'c1 y dz dy dx œ 2 'c1 '3 y dy dx œ 16 'c1 dx œ 32
1
5
1
1
5
1
Ê x œ 0, y œ 4, z œ 0; Ix œ ' ' ' ay# z# b dz dy dx œ ' ' ˆ2y# 23 ‰ dy dx œ 32 ' 100 dx œ 400
3 ;
1
5
1
1
c1
3
5
1
1
c1
c1
5
1
3
1
5
1
1
5
1
Iy œ 'c1 '3 'c1 ax# z# b dz dy dx œ 'c1 '3 ˆ2x# 23 ‰ dy dx œ 43 'c1 a3x# 1b dx œ 16
3 ;
1
5
1
1
5
1
400
‰
Iz œ 'c1 '3 'c1 ax# y# b dz dy dx œ 2 'c1 '3 ax# y# b dy dx œ 2 'c1 ˆ2x# 98
3 dx œ 3
Ð2 yÑÎ2
27. The plane y 2z œ 2 is the top of the wedge Ê IL œ 'c2 'c2 'c1
2
œ 'c2 'c2 ’ (y 6)#(4 y)
2
Mœ
"
#
4
#
(2 y)$
24
4
4
$
49
3 ‹
dt œ 1386;
(3)(6)(4) œ 36
2
"
#
c(y 6)# z# d dz dy dx
#
"3 “ dy dx; let t œ 2 y Ê IL œ 4 'c2 Š 13t
24 5t 16t
Ð2 yÑÎ2
28. The plane y 2z œ 2 is the top of the wedge Ê IL œ 'c2 'c2 'c1
œ
c1
4
c(x 4)# y# d dz dy dx
'c22 'c42 ax# 8x 16 y# b (4 y) dy dx œ 'c22 a9x# 72x 162b dx œ 696; M œ "# (3)(6)(4) œ 36
'02cx '02cxcy 2x dz dy dx œ '02 '02cx a4x 2x# 2xyb dy dx œ '02 ax$ 4x# 4xb dx œ 43
2
2cx
2cxcy
2
2cx
2
8
8
Mxy œ '0 '0 '0
2xz dz dy dx œ '0 '0 x(2 x y)# dy dx œ '0 x(23 x) dx œ 15
; Mxz œ 15
by
2
2cx
2cxcy
2
2cx
2
#
symmetry; Myz œ '0 '0 '0
2x# dz dy dx œ '0 '0 2x# (2 x y) dy dx œ '0 a2x x# b dx œ 16
15
29. (a) M œ '0
2
(b)
$
Ê xœ
30. (a) M œ '0
2
4
5
, and y œ z œ
È
'0 x '04cx
(b) Myz œ '0
2
Ê xœ
œ
256È2k
231
œ
k
4
#
È
kxy dz dy dx œ k'0
'0 x '04cx
5
4
2
#
2
2
È
'0 x '04cx
40È2
77
#
1
'01 '01
xy a4 x# b dy dx œ
Èx
'0
2
; Mxy œ '0
2
È
'0 x '04cx
#
Ê zœ
(x y z 1) dz dy dx œ '0
1
'02 a4x# x% b dx œ 32k
15
k
#
x# y a4 x# b dy dx œ
kxy# dz dy dx œ k'0
'02 a16x# 8x% x' b dx œ 256k
105
31. (a) M œ '0
Èx
'0
kx# y dz dy dx œ k '0
; Mxz œ '0
Ê yœ
2
5
Èx
'0
k
#
'02 a4x$ x& b dx œ 8k3
xy# a4 x# b dy dx œ
kxyz dz dy dx œ '0
2
Èx
'0
k
3
'02 ˆ4x&Î# x*Î# ‰ dx
#
xy a4 x# b dy dx
8
7
'01
ˆx y 3# ‰ dy dx œ ' (x 2) dx œ
0
1
5
#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 15.6 Moments and Centers of Mass
(b) Mxy œ '0
1
'01 '01 z(x y z 1) dz dy dx œ "# '01 '01 ˆx y 53 ‰ dy dx œ "# '01 ˆx 136 ‰ dx œ 43
Ê Mxy œ Myz œ Mxz œ
(c) Iz œ '0
1
4
3
, by symmetry Ê x œ y œ z œ
8
15
'01 '01 ax# y# b (x y z 1) dz dy dx œ '01 '01 ax# y# b ˆx y 3# ‰ dy dx
œ '0 ˆx$ 2x# "3 x 43 ‰ dx œ
1
Ê I x œ Iy œ Iz œ
11
6
11
6
, by symmetry
32. The plane y 2z œ 2 is the top of the wedge.
Ð2 yÑÎ2
(a) M œ 'c1 'c2 'c1
1
4
4
1
Ð2 yÑÎ2
(b) Myz œ 'c1 'c2 'c1
1
(x 1) dz dy dx œ 'c1 'c2 (x 1) ˆ2 y# ‰ dy dx œ 18
Ð2 yÑÎ2
Mxz œ 'c1 'c2 'c1
1
4
Ð2 yÑÎ2
Mxy œ 'c1 'c2 'c1
1
4
Ð2 yÑÎ2
(c) Ix œ 'c1 'c2 'c1
1
4
Ð2 yÑÎ2
Iy œ 'c1 'c2 'c1
1
4
Ð2 yÑÎ2
Iz œ 'c1 'c2 'c1
1
33. M œ '0
1
4
Èz
'zc1c1z '0
4
x(x 1) dz dy dx œ 'c1 'c2 x(x 1) ˆ2 y# ‰ dy dx œ 6;
1
4
y(x 1) dz dy dx œ 'c1 'c2 y(x 1) ˆ2 y# ‰ dy dx œ 0;
1
z(x 1) dz dy dx œ
"
#
4
'c11 'c42 (x 1) Š y4
#
y‹ dy dx œ 0 Ê x œ
(x 1) ay# z# b dz dy dx œ 'c1 'c2 (x 1) ’2y#
1
4
(x 1) ax# z# b dz dy dx œ 'c1 'c2 (x 1) ’2x#
1
4
1
3
, and y œ z œ 0
$
3" ˆ" 2y ‰ “ dy dx œ 45;
"
3
y$
#
"
3
x# y
#
$
3" ˆ" y2 ‰ “ dy dx œ 15;
(x 1) ax# y# b dz dy dx œ 'c1 'c2 (x 1) ˆ2 y# ‰ ax# y# b dy dx œ 42
1
(2y 5) dy dx dz œ '0
1
4
'zc1c1z ˆz 5Èz‰ dx dz œ '01 2 ˆz 5Èz‰ (1 z) dz
"
$Î#
œ 2 '0 ˆ5z"Î# z 5z$Î# z# ‰ dz œ 2 10
"# z# 2z&Î# 3" z$ ‘ ! œ 2 ˆ 93 3# ‰ œ 3
3 z
1
È4cx
16c2 ˆx# by# ‰
34. M œ 'c2 'cÈ4cx# '2 ax#by# b
2
œ 4 '0
21
35. (a) x œ
#
2
'02 r a4 r# b r dr d) œ 4 '021 ’ 4r3
$
Myz
M
œ0 Ê
È4cx
Èx# y# dz dy dx œ ' ' È # Èx# y# c16 4 ax# y# bd dy dx
c2 c 4cx
#
r5 “ d) œ 4 '0
&
#
21
!
64
15
d) œ
5121
15
' ' ' x$ (xß yß z) dx dy dz œ 0 Ê Myz œ 0
R
(b) IL œ ' ' ' kv hik# dm œ ' ' ' k(x h) i yjk# dm œ ' ' ' ax# 2xh h# y# b dm
D
D
D
œ ' ' ' ax# y# b dm 2h ' ' ' x dm h# ' ' ' dm œ Ix 0 h# m œ Ic m h# m
Þ
D
D
36. IL œ Ic m mh# œ
Þ
Þ
2
5
ma# ma# œ
7
5
ma#
#
37. (a) (xß yß z) œ ˆ #a ß #b ß #c ‰ Ê Iz œ Ic m abc ŠÉ a4
Þ
œ
abc aa# b# b
3
abc aa# b# b
4
#
(b) IL œ IcÞmÞ abc ŒÉ a4
œ
abc aa# 7b# b
3
Ð4
3
4
Þ
#
b#
4‹
Ê IcÞmÞ œ Iz
# b #
abc aa# b# b
; RcÞmÞ œ É IcMÞmÞ œ É a 12
1#
#
#
#
#
#
ˆ b# 2b‰# œ abc aa12 b b abc aa 4 9b b
abc aa# b# b
4
œ
IL
; RL œ É M
œ Éa
2yÑÎ3
38. M œ 'c3 'c2 'c4Î3
Þ
D
dz dy dx œ '
#
œ
abc a4a# 28b# b
1#
7b#
3
%
4
3
y
2
2
'
'
(4
y)
dy
dx
œ
4y
’
“
3
3
2
3
2
3
3
#
#
#
dx œ 12 ' 3 dx œ 72;
3
x œ y œ z œ 0 from Exercise 22 Ê Ix œ IcÞmÞ 72 ŠÈ0# 0# ‹ œ IcÞmÞ Ê IL œ IcÞmÞ 72 ŠÉ16
‰ œ 1488
œ 208 72 ˆ 160
9
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
16
9 ‹
#
913
914
Chapter 15 Multiple Integrals
15.7 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
1.
'021 '01 'r
È2cr
#
dz r dr d) œ '0
21
œ '0 Š 2 3 23 ‹ d) œ
21
2.
È
'021 '03 'r Î318cr
4.
21
24r#
1
'03 ’r a18 r# b"Î# r3 “ dr d) œ '021 ’ "3 a18 r# b$Î# 12r “ $ d)
$
%
!
$
’ 12)1#
21
21
)&
201% “ !
È
'01 '0 Î ' 3È44 rr
) 1
dz r dr d) œ '0
%
ˆ
#‰
'0 Î2 a3r 24r$ b dr d) œ '02 32 r# 6r% ‘ !Î2
)
"Î#
) Î1
!
1
1
)
1
d) œ
'02 Š 4)1
1
3
#
#
#
'0 Î
) 1
œ 4 '0 Š 21)#
1
#
3 dz r dr d) œ 3 '0
21
"
#
c9 a4 r# b a4 r# bd r dr d) œ 4 '0
)%
41 % ‹
1
d)
'0 Î a4r r$ b dr d)
) 1
371
15
d) œ
'01 ’r a2 r# b"Î# r# “ dr d) œ 3 '021 ’ a2 r# b"Î# r3 “ " d)
$
!
21
œ 3 '0 ŠÈ2 43 ‹ d) œ 1 Š6È2 8‹
6.
'021 '01 'c11ÎÎ22 ar# sin# ) z# b dz r dr d) œ '021 '01 ˆr$ sin# ) 12r ‰ dr d) œ '021 Š sin4 ) 24" ‹ d) œ 13
7.
'021 '03 '0zÎ3
8.
'c11 '021 '01bcos
9.
'01 '0 z '021 ar# cos# ) z# b r d) dr dz œ '01 '0
#
r$ dr dz d) œ '0
21
)
21
z
3
'03 324
dz d) œ '0 20
d) œ 3101
%
4r dr d) dz œ 'c1 '0 2(1 cos ))# d) dz œ 'c1 61 d) œ 121
1
21
È
1
È
1
Èz
œ '0 ’ 14r 1r# z# “
10.
4) %
161% ‹
171
5
1
1
'021 '01 'r 2cr
œ
z dz r dr d) œ '0
#
#
œ 4 '0 ’2r# r4 “
5.
3
2
)
3
#
!
91 Š8È2 7‹
'021 '0 Î2 '03
œ
$
41 ŠÈ2 "‹
dz r dr d) œ '0
#
#
œ
3.
$Î#
'01 ’r a2 r# b"Î# r# “ dr d) œ '021 ’ "3 a2 r# b$Î# r3 “ " d)
%
Èz
!
1
#
$
È
#
$Î#
r# sin 2)
4
dz œ '0 Š 14z 1z$ ‹ dz œ ’ 112z
'02 'rc24cr '021 (r sin ) 1) r d) dz dr œ '02 'rc24cr
œ 21 ’ "3 a4 r# b
#
’ r2)
r$
3
#
z# )“
"
1 z%
4 “!
#1
!
r dr dz œ '0
1
Èz
'0
a1r$ 21rz# b dr dz
1
3
œ
21r dz dr œ 21'0 ’r a4 r# b
2
"Î#
r# 2r“ dr
#
r# “ œ 21 38 4 3" (4)$Î# ‘ œ 81
!
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 15.7 Triple Integrals in Cylindrical and Spherical Coordinates
È4cr
'021 '01 '0
11. (a)
È3
#
dz r dr d)
È4cz
'021 '0 '01 r dr dz d) '021 'È23 '0
(b)
È4cr
'01 '0
(c)
#
#
r dr dz d)
'021 r d) dz dr
'021 '01 'r 2cr dz r dr d)
#
12. (a)
È2cz
(b)
'021 '01 '0z r dr dz d) '021 '12 '0
(c)
'0 'r '0
2cr#
1
21
r dr dz d)
r d) dz dr
13.
'c11ÎÎ22 '0cos '03r
14.
'c11ÎÎ22 '01 '0r cos
15.
'01 '02 sin '04cr sin
17.
'c1ÎÎ22 '11cos '04
19.
'0 Î4 '0sec '02
21.
'01 '01 '02 sin 9 3# sin 9 d3 d9 d) œ 83 '01 '01 sin% 9 d9 d) œ 83 '01 Š’ sin 94cos 9 “ 1 34 '01 sin# 9 d9‹ d)
)
)
#
)
1
)
Î
r$ dz dr d) œ ' 1Î2 '0 r% cos ) dr d) œ
)
1
f(rß )ß z) dz r dr d)
1 2
1
"
5
'
Î
1 2
1Î2
cos ) d) œ
2
5
f(rß )ß z) dz r dr d)
16.
' ÎÎ22 '03 cos '05
f(rß )ß z) dz r dr d)
18.
' 1ÎÎ22 'cos2 cos '03
20.
' ÎÎ42 '0csc '02
)
r sin )
f(rß )ß z) dz r dr d)
1
)
r cos )
1
1
)
r sin )
)
1
)
r sin )
1
23.
24.
f(rß )ß z) dz r dr d)
f(rß )ß z) dz r dr d)
$
!
1
1
1
1
1
œ 2 '0 '0 sin# 9 d9 d) œ '0 ) sin#2) ‘ ! d) œ '0 1 d) œ 1#
22.
f(rß )ß z) dz r dr d)
'021 '01Î4 '02 (3 cos 9) 3# sin 9 d3 d9 d) œ '021 '01Î4 4 cos 9 sin 9 d9 d) œ '021 c2 sin# 9d 1! Î% d) œ '021 d) œ 21
'021 '01 '0Ð1 cos 9ÑÎ2 3# sin 9 d3 d9 d) œ 24" '021 '01 (1 cos 9)$ sin 9 d9 d) œ 96" '021 c(1 cos 9)% d 1! d)
21
" '
"
1
' 21
œ 96
a2% 0b d) œ 16
96 0 d) œ 6 (21) œ 3
0
'031Î2 '01 '01 53$ sin$ 9 d3 d9
œ
5
6
d) œ
5
4
'031Î2 '01 sin$ 9 d9 d) œ 54 '031Î2 Š’ sin 93cos 9 “ 1 23 '01 sin 9 d9‹ d)
'031Î2 c cos 9d 1! d) œ 53 '031Î2 d) œ 5#1
#
!
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
915
916
25.
Chapter 15 Multiple Integrals
'021 '01Î3 'sec2 9 33# sin 9 d3 d9
d) œ '0
21
œ '0 (4 2) ˆ8 "# ‰‘ d) œ
21
5
#
'01Î3 a8 sec$ 9b sin 9 d9 d) œ '021 8 cos 9 "2 sec# 9‘ !1Î$ d)
'021 d) œ 51
26.
'021 '01Î4 '0sec 9 3$ sin 9 cos 9 d3 d9 d) œ "4 '021 '01Î4 tan 9 sec# 9 d9 d) œ "4 '021 "2 tan# 9‘ !1Î% d) œ "8 '021 d) œ 14
27.
2
0
'02 'c01 '11ÎÎ42 3$ sin 29 d9 d) d3 œ '02 ' 01 3$ cos229 ‘ 11Î#
d) d3 œ '0 ' 1 3#
Î%
28.
'11ÎÎ63 'csc2 csc9 9 '021 3# sin 9 d) d3 d9 œ 21 '11ÎÎ63 'csc2 csc9 9 3# sin 9 d3 d9 œ 231 '11ÎÎ63 c3$ sin 9d csc2 csc9 9 d9 œ 1431 '11ÎÎ63 csc# 9 d9 œ
29.
'01 '01 '01Î4 123 sin$ 9 d9 d) d3 œ '01 '01 Œ123 ’ sin 39 cos 9 “ 1Î% 83 '01Î4 sin 9 d9 d) d3
$
2 $
3 1
#
%
#
d3 œ ’ 138 “ œ 21
!
#
!
œ '0
1
œ
30.
d) d3 œ '0
'0 Š È23
83 ccos 9d ! ‹ d) d3 œ '0
1
1
1Î%
2
'0 Š83 10È3 ‹ d) d3 œ 1'01 Š83 10È3 ‹ d3 œ 1 ’43# È53
1
#
2
“
2
2
"
!
Š4È2 5‹ 1
È2
'11ÎÎ62 ' 11ÎÎ22 'csc2 9 53% sin$ 9 d3 d) d9 œ '11ÎÎ62 ' 11ÎÎ22 a32 csc& 9b sin$ 9 d) d9 œ '11ÎÎ62 ' 11ÎÎ22 a32 sin$ 9 csc# 9b d) d9
œ 1 '1Î6 a32 sin$ 9 csc# 9b d9 œ 1 ’ 32 sin 39 cos 9 “
1Î2
œ
#
È
1 Š 3224 3 ‹
641
3
ccos
1Î#
9d 1Î'
1 ŠÈ 3‹ œ
È3
3
1Î#
1Î'
1 ˆ 6431 ‰ Š
641
3
'11ÎÎ62 sin 9 d9 1 ccot 9d 11Î#
Î'
È3
# ‹
œ
331È3
3
œ 111È3
31. (a) x# y# œ 1 Ê 3# sin# 9 œ 1, and 3 sin 9 œ 1 Ê 3 œ csc 9; thus
'021 '01Î6 '02 3# sin 9 d3 d9 d) '021 '11ÎÎ62 '0csc 9 3# sin 9 d3 d9 d)
'021 '12 '1sinÎ6
(b)
3# sin 9 d9 d3 d) '0
21
'02 '01Î6 3# sin 9 d9 d3 d)
'021 '01Î4 '0sec 9 3# sin 9 d3 d9 d)
'021 '01 '01Î4 3# sin 9 d9 d3 d)
32. (a)
(b)
'0
21
33. V œ '0
21
œ
" Ð1Î3Ñ
"
3
È
'1 2 'cos1Î4" Ð"Î3Ñ 3# sin 9 d9 d3 d)
'01Î2 'cos2 9 3# sin 9 d3 d9 d) œ "3 '021 '01Î2 a8 cos$ 9b sin 9 d9 d)
'021 ’8 cos 9 cos4 9 “ 1Î# d) œ 3" '021 ˆ8 4" ‰ d) œ ˆ 3112 ‰ (21) œ 3161
%
!
'021 '01Î2 a3 cos 9 3 cos# 9 cos$ 9b sin 9 d9 d)
21
21
1Î#
111
' 21
ˆ 11 ‰
œ 3" '0 3# cos# 9 cos$ 9 14 cos% 9‘ ! d) œ 3" '0 ˆ 32 1 "4 ‰ d) œ 11
12 0 d) œ 12 (21) œ 6
34. V œ '0
'01Î2 '11
35. V œ '0
1
9)
'01 '01ccos 9 3# sin 9 d3 d9 d) œ "3 '021 '01 (1 cos 9)$ sin 9 d9 d) œ 3" '021 ’ (" cos
“ d)
4
21
21
œ
"
12
(2)
cos 9
3# sin 9 d3 d9 d) œ
"
3
%
%
'021 d) œ 34 (21) œ 831
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
!
281
3È 3
Section 15.7 Triple Integrals in Cylindrical and Spherical Coordinates
36. V œ '0
21
œ
"
12
21
'11ÎÎ42 '02 cos 9
38. V œ '0
21
1Î2
(c) 8 '0
2
%
!
3# sin 9 d3 d9 d) œ
8
3
'021 '11ÎÎ42 cos$ 9 sin 9 d9 d) œ 83 '021 ’ cos4 9 “ 1Î# d)
%
1Î%
'01Î2 '02 3# sin 9 d3 d9 d)
È4cx
'0
#
È4cx cy
'0
È
'01Î2 '03Î 2 'r
#
È9
r#
(b) 8'0
1Î2
È4cr
'02 '0
41. (a) V œ '0
21
È3cx
dz r dr d)
È4cx cy
(c) V œ 'cÈ3 'cÈ3cx# '1
(d) V œ '0
21
œ
È3
'0
(b)
'01Î2 '01Î4 '03 3# sin 9 d3 d9 d)
#
#
’r a4 r# b
(b) V œ '0
21
È
È
1‹ d) œ
2
È4cr
'0 3 '1
#
91 Š2 È2‹
4
dz r dr d)
#
"Î#
dz dy dx
r“ dr d) œ '0 ” a4 3r b
21
# $Î#
#
r# •
'021 d) œ 531
5
6
dz r dr d)
dz dy dx
'01Î3 'sec2 9 3# sin 9 d3 d9 d)
È3
#
#
'01Î2 '01Î4 '03 3# sin 9 d3 d9 d) œ 9 '01Î2 '01Î4 sin 9 d9 d) œ 9 '01Î2 Š È"
(c)
È$
!
d) œ '0 Š 3"
21
3
#
4$Î#
3 ‹
d)
'01 '0 1cr r# dz r dr d)
1Î2
21
1
Iz œ '0 '0 '0 a3# sin# 9b a3# sin 9b d3 d9 d), since r# œ x# y# œ 3# sin# 9 cos# ) 3# sin# 9 sin# ) œ 3# sin# 9
42. (a) Iz œ '0
21
(c) Iz œ '0
œ
2
15
#
!
(21) œ
41
15
43. V œ 4 '0
'01 'r 4 14r
44. V œ 4'0
'01 ' 1Èr1
1Î2
œ 4 '0
ˆ "#
45. V œ '31Î2 '0
21
9
4
#
%
1Î2
1Î2
#
'01Î2 "5 sin$ 9 d9 d) œ "5 '021 Œ’ sin 93cos 9 “ 1Î# 32 '01Î2 sin 9 d9 d) œ 152 '021 c cos 9d !1Î# d)
21
œ
1Î#
9)
'021 '01Î2 (1 cos 9)$ sin 9 d9 d) œ 3" '021 ’ (" cos
d)
“
4
21
4 '
81
'11ÎÎ32 '02 3# sin 9 d3 d9 d) œ 83 '021 '11ÎÎ32 sin 9 d9 d) œ 83 '021 c cos 9d 11Î#
Î$ d) œ 3 0 d) œ 3
39. (a) 8'0
(b)
"
3
3# sin 9 d3 d9 d) œ
'021 d) œ "6 (21) œ 13
" ‰
ˆ 83 ‰ ˆ 16
40. (a)
cos 9
'021 d) œ 12" (21) œ 16
37. V œ '0
œ
'01Î2 '01
917
"
3
3 cos )
0œ
'01 a5r 4r$ r& b dr d) œ 4 '01Î2 ˆ 5# 1 "6 ‰ d) œ 4 '01Î2 d) œ 831
1 Î2
r#
dz r dr d) œ 4 '0
"‰
3
'0cr sin
)
d) œ 2'0 d) œ
1Î2
'01 Šr r# rÈ1r# ‹ dr d) œ 4 '01Î2 ’ r2
#
2 ˆ 1# ‰
dz r dr d) œ '31Î2 '0
21
3 cos )
r$
3
"3 a1 r# b
c3 cos )
œ 18 Œ’ cos
#
'0r dz r dr d) œ 2 '1Î2 '0c3 cos
1
1
) sin )
“
3
1Î#
2
3
$Î# "
“ d)
!
œ1
r# sin ) dr d) œ '31Î2 a9 cos$ )b (sin )) d) œ 94 cos% )‘ $1Î#
21
9
4
46. V œ 2 '1Î2 '0
1
1Î2
dz r dr d) œ 4 '0
)
r# dr d) œ
2
3
'1Î2 27 cos$ ) d)
1
'11Î2 cos ) d) œ 12 csin )d 11Î# œ 12
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
#1
918
Chapter 15 Multiple Integrals
47. V œ '0
1Î2
È1
'0sin '0
)
r#
1 2
Î
rÈ1r# dr d) œ '0 ’ "3 a1 r# b
)
1 2
$Î# sin )
“
!
d)
#
!
œ csin
2
9
48. V œ '0
1Î2
1Î#
)d !
1
6
'0cos '03
)
È1
œ
1Î#
Î
1 2
1Î2
32 ccos )d !
4 3 1
18
dz r dr d) œ '0
r#
œ '0 ’ a1 cos# )b
1
#
'0sin
'01Î2 ’a1 sin# )b$Î# 1“ d) œ "3 '01Î2 acos$ ) 1b d) œ "3 Œ’ cos )3 sin ) “ 1Î# 32 '01Î2 cos ) d) 3) ‘ 1! Î#
œ "3
œ
Î
dz r dr d) œ '0
$Î#
1
#
œ
'0cos
)
Î
3rÈ1r# dr d) œ '0 ’ a1 r# b
1 2
1“ d) œ '0 a1 sin$ )b d) œ ’)
1Î2
2
3
œ
'12Î13Î3 '0a 3# sin 9 d3 d9 d) œ '021 '12Î13Î3
50. V œ '0
'01Î2 '0a 3# sin 9 d3 d9 d) œ a3 '01Î6 '01Î2
51. V œ '0
'01Î3 'sec2 9
1Î6
21
“
!
d)
'01Î2 sin ) d)
2
3
31 4
6
49. V œ '0
21
1Î#
sin# ) cos )
“
3
!
$Î# cos )
$
a$
3
sin 9 d9 d) œ
'021 c cos 9d #11Î$Î$ d) œ a3 '021 ˆ "# "# ‰ d) œ 213a
a$
3
sin 9 d9 d) œ
a$
3
$
'01Î6 d) œ a181
$
3# sin 9 d3 d9 d)
'021 '01Î3 a8 sin 9 tan 9 sec# 9b d9 d)
21
1Î$
œ "3 '0 8 cos 9 "2 tan# 9‘ ! d)
21
21
œ "3 '0 4 #" (3) 8‘ d) œ 3" '0 #5 d) œ 65 (21) œ 531
œ
"
3
52. V œ 4 '0
1Î2
œ
28
3
'01Î4 'sec2 sec9 9 3# sin 9 d3 d9 d)
œ
4
3
'01Î2 '01Î4 a8 sec$ 9 sec$ 9b sin 9 d9 d)
'01Î2 '01Î4 sec$ 9 sin 9 d9 d) œ 283 '01Î2 '01Î4 tan 9 sec# 9 d9 d) œ 283 '01Î2 2" tan# 9‘ !1Î% d) œ 143 '01Î2 d) œ 731
53. V œ 4 '0
'01 '0r
#
54. V œ 4 '0
'01 'r r
#
55. V œ 8 '0
'1 2 '0r
56. V œ 8 '0
'1 2 '0
1Î2
1Î2
1Î2
1Î2
dz r dr d) œ 4 '0
1Î2
1
#
È
1Î2
dz r dr d) œ 4 '0
dz r dr d) œ 8 '0
1Î2
È2
È
r#
58. V œ '0
'02 '04cr cos cr sin
21
È2
'1
1Î2
'02 '04cr sin
)
'01 r dr d) œ 2 '01Î2 d) œ 1
dz r dr d) œ 8 '0
57. V œ '0
21
'01 r$ dr d) œ '01Î2 d) œ 1#
dz r dr d) œ '0
21
)
)
r# dr d) œ 8 Š 2
È2
'1
È2"
‹
3
È
'01Î2 d) œ 41 Š2 3 2"‹
1Î2
rÈ2 r# dr d) œ 8 '0 ’ "3 a2 r# b
$Î#
1Î2
'01 '4r5
#
r#
È#
1
d) œ
8
3
'01Î2 d) œ 431
'02 a4r r# sin )b dr d) œ 8 '02 ˆ1 sin3 ) ‰ d) œ 161
1
dz r dr d) œ '0
21
'02 c4r r# (cos ) sin ))d dr d) œ 83 '02
1
(3 cos ) sin )) d) œ 161
59. The paraboloids intersect when 4x# 4y# œ 5 x# y# Ê x# y# œ 1 and z œ 4
Ê V œ 4 '0
“
1Î2
dz r dr d) œ 4 '0
'01 a5r 5r$ b dr d) œ 20 '01Î2 ’ r2
#
1Î2
r4 “ d) œ 5'0
%
"
!
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
d) œ
51
#
$
Section 15.7 Triple Integrals in Cylindrical and Spherical Coordinates
60. The paraboloid intersects the xy-plane when 9 x# y# œ 0 Ê x# y# œ 9 Ê V œ 4 '0
1Î2
œ 4 '0
1Î2
'1
3
61. V œ 8 '0
21
'0
#
$
%
1Î2
17 ‰
4
"
È4cr
'01 '0
21
œ 83
a9r r$ b dr d) œ 4 '0 ’ 9r2 r4 “ d) œ 4 '0 ˆ 81
4
1Î2
#
dz r dr d) œ 8 '0
21
ˆ3$Î# 8‰ d) œ
'13 '09cr
#
919
dz r dr d)
d) œ 64 '0 d) œ 321
1Î2
'01 r a4 r# b"Î# dr d) œ 8 '021 ’ "3 a4 r# b$Î# “ " d)
!
41 Š8 3È3‹
3
62. The sphere and paraboloid intersect when x# y# z# œ 2 and z œ x# y# Ê z# z 2 œ 0
Ê (z 2)(z 1) œ 0 Ê z œ 1 or z œ 2 Ê z œ 1 since z 0. Thus, x# y# œ 1 and the volume is
given by the triple integral V œ 4 '0
1Î2
œ 4 '0 ’ "3 a2 r# b
1Î2
63. average œ
"
#1
œ
È2
r#
#
1Î2
dz r dr d) œ 4 '0
r4 “ d) œ 4 '0 Š 2 3 2
$Î#
%
"
1Î2
È
!
'021 '01 'c11
"
#1
r# dz dr d) œ
È
'021 '01 'cÈ11ccrr
"
ˆ 431 ‰
64. average œ
'01 'r
#
#
7
12 ‹
'01 ’r a2 r# b"Î# r$ “ dr d)
1 Š8 È 2 7 ‹
d) œ
6
'021 '01 2r# dr d) œ 3"1 '021 d) œ 23
r# dz dr d) œ
3
41
'021 '01 2r# È1 r# dr d)
'021 ’ "8 sin" r "8 rÈ1 r# a1 2r# b“ " d) œ 1631 '021 ˆ 1# 0‰ d) œ 323 '021 d) œ ˆ 323 ‰ (21) œ 3161
3
21
!
65. average œ
"
ˆ 431 ‰
'021 '01 '01 3$ sin 9 d3 d9 d) œ 1631 '021 '01 sin 9 d9 d) œ 831 '021 d) œ 43
66. average œ
"
ˆ 231 ‰
'021 '01Î2 '01 3$ cos 9 sin 9 d3 d9 d) œ 831 '021 '01Î2
œ
3
161
cos 9 sin 9 d9 d) œ
3
81
'021 ’ sin2 9 “ 1Î# d)
#
!
'021 d) œ ˆ 1631 ‰ (21) œ 38
67. M œ 4 '0
'01 '0r dz r dr d) œ 4 '01Î2 '01 r# dr d) œ 43 '01Î2 d) œ 231 ; Mxy œ '021 '01 '0r z dz r dr d)
21
1
21
œ "# '0 '0 r$ dr d) œ 18 '0 d) œ 14 Ê z œ MM œ ˆ 14 ‰ ˆ 231 ‰ œ 38 , and x œ y œ 0, by symmetry
1Î2
xy
68. M œ '0
'02 '0r dz r dr d) œ '01Î2 '02 r# dr d) œ 83 '01Î2 d) œ 431 ; Myz œ '01Î2 '02 '0r x dz r dr d)
2
2
r
2
1Î2
1Î2
1Î2
1Î2
œ '0 '0 r$ cos ) dr d) œ 4 '0 cos ) d) œ 4; Mxz œ '0 '0 '0 y dz r dr d) œ '0 '0 r$ sin ) dr d)
2
r
2
1Î2
1Î2
1Î2
1Î2
M
œ 4 '0 sin ) d) œ 4; Mxy œ '0 '0 '0 z dz r dr d) œ "# '0 '0 r$ dr d) œ 2 '0 d) œ 1 Ê x œ M
1Î2
yz
yœ
69. M œ
œ
Mxz
M
, and z œ
3
1
; Mxy œ '0
21
81
3
œ 4 '0 ’ sin2 9 “
21
70. M œ '0
#
1Î#
1Î$
Mxy
M
œ
3
1
,
3
4
'11ÎÎ32 '02 z3# sin 9 d3 d9 d) œ '021 '11ÎÎ32 '02 3$ cos 9 sin 9 d3 d9 d) œ 4 '021 '11ÎÎ32
d) œ 4 '0 ˆ "# 38 ‰ d) œ
21
"
#
'021 d) œ 1
Ê zœ
Mxy
M
œ (1) ˆ 831 ‰ œ
3
8
$
$
$
%
%
cos 9 sin 9 d9 d)
, and x œ y œ 0, by symmetry
È
'01Î4 '0a 3# sin 9 d3 d9 d) œ a3 '021 '01Î4 sin 9 d9 d) œ a3 '021 2 #È2 d) œ 1a Š23 2‹ ;
1Î4
1Î4
21
a
21
21
a '
Mxy œ '0 '0 '0 3$ sin 9 cos 9 d3 d9 d) œ a4 '0 '0 sin 9 cos 9 d9 d) œ 16
d) œ 18a
0
21
œ
%
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
920
Chapter 15 Multiple Integrals
È2
‰ 2
œ Š 18a ‹ – $ 3 È — œ ˆ 3a
8 Š #
1a Š2 2‹
%
Mxy
M
Ê zœ
‹œ
3 Š2È2‹ a
16
, and x œ y œ 0, by symmetry
È
71. M œ '0
È
'04 '0 r dz r dr d) œ '021 '04 r$Î# dr d) œ 645 '021 d) œ 1285 1 ; Mxy œ '021 '04 '0 r z dz r dr d)
21
4
'021 d) œ 6431 Ê z œ MM œ 65 , and x œ y œ 0, by symmetry
œ "# '0 '0 r# dr d) œ 32
3
21
xy
1Î3
È
' È11 rr dz r dr d) œ ' 11ÎÎ33 '01 2rÈ1 r# dr d) œ ' 11ÎÎ33 ’ 23 a1 r# b$Î# “ " d)
!
È1 r
1
1
1Î3
1Î3
1 Î3
2 '
2
2
1
4
1
#
œ 3 c1Î3 d) œ ˆ 3 ‰ ˆ 3 ‰ œ 9 ; Myz œ ' 1Î3 '0 ' È1 r r cos ) dz dr d) œ 2 ' 1Î3 '0 r# È1 r# cos ) dr d)
72. M œ 'c1Î3 '0
1
#
#
#
#
1 Î3
œ 2 'c1Î3 ’ 18 sin" r "8 rÈ1 r# a1 2r# b“ cos ) d) œ
"
Myz
M
Ê xœ
œ
!
9È 3
32
1
8
' 11ÎÎ33 cos ) d) œ 18 csin )d 1Î13Î3 œ ˆ 18 ‰ Š2 † È#3 ‹ œ 1È8 3
, and y œ z œ 0, by symmetry
73. We orient the cone with its vertex at the origin and axis along the z-axis Ê 9 œ
which is through the vertex and parallel to the base of the cone Ê Ix œ '0
21
œ '0
. We use the the x-axis
1
'01 Šr$ sin# ) r% sin# ) 3r r3 ‹ dr d) œ '021 Š sin20 ) 10" ‹ d) œ 40) sin802) 10) ‘ #!1 œ #10 15 œ 14
21
%
74. Iz œ '0
21
œ
1
4
'0 'r ar# sin# ) z# b dz r dr d)
1
'0a
Èa cr
#
#
'
cÈa cr
#
#
r$ dz dr d) œ '0
21
#
'0a 2r$ Èa# r# dr d) œ 2 '021 ’Š r5
#
2a#
#
15 ‹ aa
“ d) œ 2 '0
!
2
15
a& d)
81 a&
15
75. Iz œ '0
21
'0a ' h r ax# y# b dz r dr d) œ '021 '0a
ˆh‰
a
œ '0 h Š a4
21
%
a&
5a ‹
76. (a) M œ '0
'01 '0r
21
d) œ
#
ha%
20
'021 d) œ 110ha
z dz r dr d) œ '0
21
21
'01 '0r
#
r# dz dr d) œ '0
21
"
2
hr
a
Šhr$
hr%
a ‹
dr d) œ '0
21
#
Ê zœ
"
#
%
h ’ r4
a
r&
5a “ !
d)
Ê zœ
5
14
z# dz r dr d)
, and x œ y œ 0, by symmetry; Iz œ '0
21
'01 r% dr d) œ 5" '021 d) œ 215 ; Mxy œ '021 '01 '0r
'021 '01 r' dr d) œ 14" '021 d) œ 17
21
1
21
œ '0 '0 r' dr d) œ 7" '0 d) œ 21
7
œ
hr
a
' h r$ dz dr d) œ '021 '0a
'01 "# r& dr d) œ 12" '021 d) œ 16 ; Mxy œ '021 '01 '0r
"
3
(b) M œ '0
h
'
%
'021 '01 r( dr d) œ 24" '021 d) œ 121
21
1
21
" '
œ "# '0 '0 r( dr d) œ 16
d) œ 18
0
œ
#
'01 '0r
#
zr$ dz dr d)
zr# dz dr d)
, and x œ y œ 0, by symmetry; Iz œ '0
21
'01 '0r
77. (a) M œ '0
#
r% dz dr d)
'01 'r 1 z dz r dr d) œ "# '021 '01 ar r$ b dr d) œ 8" '021 d) œ 14 ; Mxy œ '021 '01 'r 1 z# dz r dr d)
21
1
21
21
1
1
" '
œ 3" '0 '0 ar r% b dr d) œ 10
d) œ 15 Ê z œ 45 , and x œ y œ 0, by symmetry; Iz œ '0 '0 'r zr$ dz dr d)
0
21
1
21
1
" '
œ "# '0 '0 ar$ r& b dr d) œ 24
d) œ 12
0
21
1
1
21
1
1
21
1
M œ '0 '0 'r z# dz r dr d) œ 15 from part (a); Mxy œ '0 '0 'r z$ dz r dr d) œ 4" '0 '0 ar r& b dr d)
21
21
1
1
21
1
" '
œ 12
d) œ 16 Ê z œ 56 , and x œ y œ 0, by symmetry; Iz œ '0 '0 'r z# r$ dz dr d) œ "3 '0 '0 ar$ r' b dr d)
0
21
" '
1
œ 28
d) œ 14
0
21
(b)
21
$Î# a
r# b
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 15.7 Triple Integrals in Cylindrical and Spherical Coordinates
78. (a) M œ '0
'01 '0a 3% sin 9 d3 d9 d) œ a5 '021 '01 sin 9 d9 d) œ 2a5 '021 d) œ 415a ; Iz œ '021 '01 '0a
'021 '01 a1 cos# 9b sin 9 d9 d) œ a7 '021 ’ cos 9 cos3 9 “ 1 d) œ 4a#1 '021 d) œ 8a211
21
a(
7
œ
(b)
&
&
&
$
(
(
3' sin$ 9 d3 d9 d)
(
!
21
a
21
21
1
1
29 )
M œ '0 '0 '0 3$ sin# 9 d3 d9 d) œ a4 '0 '0 (1cos
d9 d) œ 18a '0 d) œ 1 4a ;
#
1
1
21
a
21
Iz œ '0 '0 '0 3& sin% 9 d3 d9 d) œ a6 '0 '0 sin% 9 d9 d)
1
21
21
21
1
1
œ a6 '0 Š’ sin 49 cos 9 “ 43 '0 sin# 9 d9‹ d) œ a8 '0 9# sin429 ‘ ! d) œ 116a '0 d) œ a 81
%
%
# %
'
$
'
'
'
' #
!
79. M œ '0
21
'0a '0
h
a
Èa cr
#
#
dz r dr d) œ '0
21
'0a
h
a
rÈa# r# dr d) œ
Èa cr
'021 ’ 3" aa# r# b$Î# “ a d)
h
a
!
h ' '
'0 3 d) œ 2ha3 1 ; Mxy œ '0 '0 '0
z dz r dr d) œ 2a
aa# r r$ b dr d)
0
0
21
h '
œ 2a
Š a# a4 ‹ d) œ a h4 1 Ê z œ Š 1a4h ‹ ˆ 2ha3 1 ‰ œ 83 h, and x œ y œ 0, by symmetry
0
œ
21 $
a
h
a
21
#
h
a
a
#
#
21
#
a
#
#
%
%
# #
# #
#
#
80. Let the base radius of the cone be a and the height h, and place the cone's axis of symmetry along the z-axis
with the vertex at the origin. Then M œ
œ
h#
#
'021 ’ r2
#
a
r%
4a# “ !
d) œ
h#
#
'021 Š a#
#
1a# h
3
and Mxy œ '0
a#
4‹
d) œ
21
h # a#
8
'0a ' h r z dz r dr d) œ "# '021 '0a Šh# r ha
#
#
ˆh‰
r$ ‹ dr d)
a
'021 d) œ h a4 1
# #
Ê zœ
Mxy
M
# #
œ Š h a4 1 ‹ ˆ 1a3# h ‰ œ
3
4
h, and
x œ y œ 0, by symmetry Ê the centroid is one fourth of the way from the base to the vertex
81. The density distribution function is linear so it has the form $ (3) œ k3 C, where 3 is the distance from the
center of the planet. Now, $ (R) œ 0 Ê kR C œ 0, and $ (3) œ k3 kR. It remains to determine the constant
k: M œ '0
21
œ'
Ê
21
'
'01 '0R (k3 kR) 3# sin 9 d3 d9 d) œ '021 '01 ’k 34
%
'
1
$
R
kR 33 “ sin 9 d9 d)
21
%
%
k Š R4 R3 ‹ sin 9 d9 d) œ 0 1k# R% c cos 9d 1! d) œ
0
0
$ (3) œ 13M
3 13M
R . At the center of the planet 3 œ 0 Ê
R%
R%
'0
21
!
%
6k R% d) œ k13R Ê k œ 13M
R%
‰R œ
$ (0) œ ˆ 13M
R%
3M
1R$
.
82. The mass of the plant's atmosphere to an altitude h above the surface of the planet is the triple integral
M(h) œ '0
'01 'Rh .! ecÐ3RÑ 3# sin 9 d3 d9 d) œ 'Rh '021 '01 .! ecÐ3RÑ 3# sin 9 d9 d) d3
h
21
h
21
h
1
œ 'R '0 .! ecÐ3RÑ 3# ( cos 9)‘ ! d) d3 œ 2 'R '0 .! ecR ec3 3# d) d3 œ 41.! ecR 'R ec3 3# d3
21
#
œ 41.! ecR ’ 3 ec
#
c3
œ 41.! ecR Š h ec
ch
23e c3
c#
2he ch
c#
h
2e c3
c$ “ R
2e ch
c$
(by parts)
R# e
c
cR
2Re cR
c#
2e cR
c$ ‹ .
The mass of the planet's atmosphere is therefore M œ lim
hÄ_
#
M(h) œ 41.! Š Rc
2R
c#
2
c$ ‹ .
83. (a) A plane perpendicular to the x-axis has the form x œ a in rectangular coordinates Ê r cos ) œ a Ê r œ
Ê r œ a sec ), in cylindrical coordinates.
(b) A plane perpendicular to the y-axis has the form y œ b in rectangular coordinates Ê r sin ) œ b Ê r œ
Ê r œ b csc ), in cylindrical coordinates.
84. ax by œ c Ê aar cos )b bar sin )b œ c Ê raa cos ) b sin )b œ c Ê r œ
c
a cos ) b sin ) .
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
921
a
cos )
b
sin )
922
Chapter 15 Multiple Integrals
85. The equation r œ fazb implies that the point ar, ), zb
œ afazb, ), zb will lie on the surface for all ). In particular
afazb, ) 1, zb lies on the surface whenever afazb, ), zb does
Ê the surface is symmetric with respect to the z-axis.
86. The equation 3 œ fa9b implies that the point a3, 9, )b œ afa9b, 9, )b lies on the surface for all ). In particular, if
afa9b, 9, )b lies on the surface, then afa9b, 9, ) 1b lies on the surface, so the surface is symmetric wiith respect to the
z-axis.
15.8 SUBSTITUTIONS IN MULTIPLE INTEGRALS
1. (a) x y œ u and 2x y œ v Ê 3x œ u v and y œ x u Ê x œ
` (xßy)
` (ußv)
œ
"
3
» 2
3
"
3
"
3
»œ
"
9
2
9
œ
"
3
(u v) and y œ
"
3
(2u v);
"
3
(u v) and x œ
"
3
(u 2v);
"
3
(b) The line segment y œ x from (!ß 0) to (1ß 1) is x y œ 0
Ê u œ 0; the line segment y œ 2x from (0ß 0) to
(1ß 2) is 2x y œ 0 Ê v œ 0; the line segment x œ 1
from (1ß 1) to ("ß 2) is (x y) (2x y) œ 3
Ê u v œ 3. The transformed region is sketched at the
right.
2. (a) x 2y œ u and x y œ v Ê 3y œ u v and x œ v y Ê y œ
` (xßy)
` (ußv)
œ»
"
3
1
3
2
3
3"
"
»œ9
2
9
œ 3"
(b) The triangular region in the xy-plane has vertices (0ß 0),
(2ß 0), and ˆ 23 ß 23 ‰ . The line segment y œ x from (0ß 0)
to ˆ 23 ß 23 ‰ is x y œ 0 Ê v œ 0; the line segment
y œ 0 from (0ß 0) to (#ß 0) Ê u œ v; the line segment
x 2y œ 2 from ˆ 23 ß 23 ‰ to (2ß 0) Ê u œ 2. The
transformed region is sketched at the right.
3. (a) 3x 2y œ u and x 4y œ v Ê 5x œ 2u v and y œ
` (xßy)
` (ußv)
œ»
2
5
1
10
15
3
10
»œ
6
50
1
50
œ
"
#
(u 3x) Ê x œ
"
5
(2u v) and y œ
"
10
(b) The x-axis y œ 0 Ê u œ 3v; the y-axis x œ 0
Ê v œ 2u; the line x y œ 1
"
Ê "5 (2u v) 10
(3v u) œ 1
Ê 2(2u v) (3v u) œ 10 Ê 3u v œ 10. The
transformed region is sketched at the right.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
"
10
(3v u);
Section 15.8 Substitutions in Multiple Integrals
4. (a) 2x 3y œ u and x y œ v Ê x œ u 3v and y œ v x Ê x œ u 3v and y œ u 2v;
" 3
` (xßy)
` (ußv) œ º 1 2 º œ 2 3 œ 1
(b) The line x œ 3 Ê u 3v œ 3 or u 3v œ 3;
x œ 0 Ê u 3v œ 0; y œ x Ê v œ 0; y œ x 1
Ê v œ 1. The transformed region is the parallelogram
sketched at the right.
5.
'04 'yÐÎy2Î2Ñ
"
#
œ
6.
1
ˆx y# ‰ dx dy œ ' ’ x2
0
4
#
y
xy 2
# “y
"
dy œ
2
"
#
'04 ’ˆ y# 1‰# ˆ y# ‰# ˆ y# 1‰ y ˆ y# ‰ y“ dy
'04 (y 1 y) dy œ "# '04 dy œ "# (4) œ 2
' ' a2x# xy y# b dx dy œ ' ' (x y)(2x y) dx dy
R
R
ßy)
" ''
œ ' ' uv ¹ `` (x
uv du dv;
(ußv) ¹ du dv œ 3
G
G
We find the boundaries of G from the boundaries of R,
shown in the accompanying figure:
xy-equations for
Corresponding uv-equations
Simplified
for the boundary of G
uv-equations
the boundary of R
y œ 2x 4
"
3
(2u v) œ (u v) 4
vœ4
y œ 2x 7
"
3
(2u v) œ 32 (u v) 7
vœ7
yœx2
"
3
yœx1
"
3
Ê
7.
"
3
2
3
(2u v) œ
1
3
(u v) 2
uœ2
(2u v) œ
1
3
(u v) 1
u œ 1
' ' uv du dv œ "3 ' ' uv dv du œ "3 ' u ’ v2# “ du œ
c1 4
c1
2
7
2
(
%
G
11
#
'c21 u du œ ˆ 11# ‰ ’ u2 “ #
#
"
‰
œ ˆ 11
4 (4 1) œ
' ' a3x# 14xy 8y# b dx dy
R
œ ' ' (3x 2y)(x 4y) dx dy
R
ßy)
œ ' ' uv ¹ `` (x
(ußv) ¹ du dv œ
G
"
10
' ' uv du dv;
G
We find the boundaries of G from the boundaries of R,
shown in the accompanying figure:
xy-equations for
the boundary of R
Corresponding uv-equations
Simplified
for the boundary of G
uv-equations
3
#
yœ x1
"
10
(3v u) œ
(2u v) 1
uœ2
y œ 3# x 3
"
10
3
(3v u) œ 10
(2u v) 3
uœ6
y œ 4" x
"
10
1
(3v u) œ 20
(2u v)
vœ0
"
10
(3v u) œ
vœ4
"
4
yœ x1
3
10
1
20
(2u v) 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
33
4
923
924
Chapter 15 Multiple Integrals
"
10
Ê
8.
' ' uv du dv œ
"
10
G
'26 '04 uv dv du œ 10" '26 u ’ v2 “ % du œ 45 '26 u du œ ˆ 54 ‰ ’ u2 “ ' œ ˆ 54 ‰ (18 2) œ 645
#
#
!
#
' ' 2(x y) dx dy œ ' ' 2v ¹ `` (x(ußßy)v) ¹ du dv œ ' ' 2v du dv; the region G is sketched in Exercise 4
R
G
G
3c3v
"
' ' 2v du dv œ ' '
2v du dv œ '0 2v(3 3v 3v) dv œ '0 6v dv œ c3v# d ! œ 3
0
c3v
1
Ê
1
1
G
9. x œ
v" uv#
œ v" u v" u œ 2u
v ;
v
u º
Ê v œ 1, and y œ 4x Ê v œ 2; xy œ 1 Ê u œ 1, and xy œ 9 Ê u œ 3; thus
and y œ uv Ê
u
v
y œ x Ê uv œ
u
v
œ v# and xy œ u# ;
y
x
` (xßy)
` (ußv)
œ J(uß v) œ º
' ' ŠÉ yx Èxy‹ dx dy œ ' ' (v u) ˆ 2uv ‰ dv du œ ' ' Š2u 2uv # ‹ dv du œ ' c2uv 2u# ln vd #" du
1
1
1
1
1
3
2
3
2
3
R
œ '1 a2u 2u# ln 2b du œ u# 23 u# ln 2‘ " œ 8 23 (26)(ln 2) œ 8
3
$
` (xßy)
` (ußv)
10. (a)
œ J(uß v) œ º
52
3
(ln 2)
" 0
œ u, and
v uº
the region G is sketched at the right
(b) x œ 1 Ê u œ 1, and x œ 2 Ê u œ 2; y œ 1 Ê uv œ 1 Ê v œ "u , and y œ 2 Ê uv œ 2 Ê v œ
'1 '1
2
œ
2
3
#
y
x
dy dx œ '1
2
'1Îu ˆ uvu ‰ u dv du œ '1 '1Îu uv dv du œ '1
2Îu
R
` (xßy)
` (ußv)
u ’ v2 “
2Îu
1Îu
du œ '1 u ˆ u2#
2
" ‰
2u#
; thus,
du
2
1
21
12.
#
#
I! œ ' ' ax# y# b dA œ '0
œ
2
'12 u ˆ u" ‰ du œ 3# cln ud #" œ 3# ln 2; '12 '12 yx dy dx œ '12 ’ x1 † y2 “ 2 dx œ 3# '12 dxx œ 3# cln xd #" œ 3# ln 2
11. x œ ar cos ) and y œ ar sin ) Ê
ab
4
2Îu
2
2
u
` (xßy)
` (rß))
œ J(rß )) œ º
#
#
sin 2)
4
b# )
2
21
b# sin 2)
“
4
!
œ
ab1 aa# b# b
4
È1cu#
1
a 0
œ ab; A œ ' ' dy dx œ ' ' ab du dv œ 'c1 'cÈ1cu# ab dv du
º
0 b
R
G
œ 2ab 'c1 È1u# du œ 2ab ’ u2 È1 u#
1
ar sin )
œ abr cos# ) abr sin# ) œ abr;
br cos ) º
'01 r# aa# cos# ) b# sin# )b kJ(rß ))k dr d) œ '021 '01 abr$ aa# cos# ) b# sin# )b dr d)
'021 aa# cos# ) b# sin# )b d) œ ab4 ’ a2) a
œ J(uß v) œ º
a cos )
b sin )
"
#
sin" u“
"
"
œ ab csin" 1 sin" (1)d œ ab 1# ˆ 1# ‰‘ œ ab1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 15.8 Substitutions in Multiple Integrals
13. The region of integration R in the xy-plane is
sketched in the figure at the right. The
boundaries of the image G are obtained as
follows, with G sketched at the right:
xy-equations for
Corresponding uv-equations
Simplified
for the boundary of G
uv-equations
the boundary of R
xœy
"
3
(u 2v) œ
x œ 2 2y
"
3
(u 2v) œ 2 32 (u v)
yœ0
0œ
Also, from Exercise 2,
` (xßy)
` (ußv)
1
3
(u v)
vœ0
uœ2
(u v)
1
3
vœu
œ J(uß v) œ 3" Ê
'02Î3 'y2
(x 2y) eÐy xÑ dx dy œ '0
2y
2
œ
"
3
'02 u cecv d !u du œ 3" '02 u a1 ecu b du œ 3" ’u au ecu b u#
œ
"
3
a3ec2 1b ¸ 0.4687
#
14. x œ u
` (xßy)
` (ußv)
v
#
#
ecu “ œ
!
"
3
'0u ue
v
¸ 3" ¸ dv du
c2 a2 ec2 b 2 ec2 1d
and y œ v Ê 2x y œ (2u v) v œ 2u and
" "#
v
º œ 1; next, u œ x #
0 "
and v œ y, so the boundaries of the region of
œ J(uß v) œ º
œx
y
#
integration R in the xy-plane are transformed to the
boundaries of G:
xy-equations for
Corresponding uv-equations
Simplified
for the boundary of G
uv-equations
œ
uœ0
the boundary of R
xœ
xœ
u
y
#
y
#
2
u
v
#
v
#
œ
v
#
v
#
2
uœ2
yœ0
vœ0
vœ0
yœ2
vœ2
vœ2
Ê '0
2
œ
"
4
15. x œ
'yÐÎy2Î2Ñ
ae
16
u
v
2
y$ (2x y) eÐ2xyÑ dx dy œ '0
#
% #
1b ’ v4 “
!
u
v
'02 v$ (2u) e4u
#
du dv œ '0 v$ ’ "4 e4u “ dv œ
2
#
#
!
"
4
'02 v$ ae16 1b dv
œe 1
16
v" uv#
œ v" u v" u œ 2u
v ;
v
u º
Ê v œ 1, and y œ 4x Ê v œ 2; xy œ 1 Ê u œ 1, and xy œ 4 Ê u œ 2; thus
and y œ uv Ê
y œ x Ê uv œ
2
y
x
œ v# and xy œ u# ;
` (xßy)
` (ußv)
œ J(uß v) œ º
'12 '1yÎyax2 y2 b dx dy '24 'y4ÎÎ4yax2 y2 b dx dy œ '12 '12 Š uv
2
2
'
‰
u2 v2 ‹ ˆ 2u
v du dv œ 1
2
'12 Š 2uv
3
3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
2u3 v‹ du dv
925
926
Chapter 15 Multiple Integrals
u
1 4
15
œ '1 ’ 2v
dv œ '1 ˆ 2v
3 2 u v“
3
2
#
4
2
"
16. x œ u2 v2 and y œ 2uv;
` (xßy)
` (ußv)
15v ‰
2
15
dv œ ’ 4v
2
œ J(uß v) œ º
2
15v2
4 “"
œ
225
16
2v
œ 4u2 4v2 œ 4au2 v2 b ;
2u º
2u
2v
y œ 2È1 x Ê y2 œ 4a1 xb Ê a2uvb2 œ 4a1 au2 v2 bb Ê u œ „ 1; y œ 0 Ê 2uv œ 0 Ê u œ 0 or v œ 0;
x œ 0 Ê u2 v2 œ 0 Ê u œ v or u œ v; This gives us four triangular regions, but only the one in the quadrant where
both u, v are positive maps into the region R in the xy-plane.
È
'01 '02 1 x Èx2 y2 dx dy œ '01 '0u Éau2 v2 b2 a2uvb2 † 4au2 v2 b dv du œ 4'01 '0u au2 v2 b2 dv du
2
u
112 1 6 ‘ 2
56
'2 5
œ 4'1 u4 v 23 u2 v3 15 v5 ‘0 du œ 112
15 1 u du œ 15 6 u " œ 45
17. (a) x œ u cos v and y œ u sin v Ê
` (xßy)
` (ußv)
ϼ
cos v u sin v
œ u cos# v u sin# v œ u
sin v
u cos v º
(b) x œ u sin v and y œ u cos v Ê
` (xßy)
` (ußv)
ϼ
sin v
u cos v
œ u sin# v u cos# v œ u
cos v u sin v º
18. (a) x œ u cos v, y œ u sin v, z œ w Ê
(b) x œ 2u 1, y œ 3v 4, z œ
â
â sin 9 cos )
â
19. â sin 9 sin )
â
â cos 9
œ (cos 9) º
3 cos 9 cos )
3 cos 9 sin )
3 sin 9
3 cos 9 cos )
3 cos 9 sin )
"
#
` (xßyßz)
` (ußvßw)
(w 4) Ê
â
â cos v
â
œ â sin v
â
â 0
` (xßyßz)
` (ußvßw)
u sin v
u cos v
0
â
â2
â
œ â0
â
â0
0
3
0
â
0â
â
0 â œ u cos# v u sin# v œ u
â
"â
â
0â
0 ââ œ (2)(3) ˆ #" ‰ œ 3
" â
# â
â
3 sin 9 sin ) â
â
3 sin 9 cos ) â
â
0
â
3 sin 9 sin )
sin 9 cos )
(3 sin 9) º
3 sin 9 cos ) º
sin 9 sin )
3 sin 9 sin )
3 sin 9 cos ) º
œ a3# cos 9b asin 9 cos 9 cos# ) sin 9 cos 9 sin# )b a3# sin 9b asin# 9 cos# ) sin# 9 sin# )b
œ 3# sin 9 cos# 9 3# sin$ 9 œ a3# sin 9b acos# 9 sin# 9b œ 3# sin 9
20. Let u œ gaxb Ê Jaxb œ
du
dx
œ gw axb Ê 'a faub du œ 'gaab fagaxbbgw axb dx in accordance with Theorem 7 in
gabb
b
Section 5.6. Note that gw axb represents the Jacobian of the transformation u œ gaxb or x œ g" aub.
21.
'03 '04 'y1Î2 ÐyÎ2Ñ ˆ 2x # y 3z ‰ dx dy dz œ '03 '04 ’ x2
#
œ '0 ’ (y 4 1)
3
#
y#
4
%
yz
3 “!
dz œ '0 ˆ 49
3
4z
3
xy
#
"ÐyÎ2Ñ
xz
3 “ yÎ2
4" ‰ dz œ '0 ˆ2
3
dy dz œ '0
4z ‰
3
3
'04 "# (y 1) y# 3z ‘ dy dz
dz œ ’2z
$
2z#
3 “!
œ 12
â
â
âa 0 0â
#
#
#
â
â
22. J(uß vß w) œ â 0 b 0 â œ abc; the transformation takes the ellipsoid region xa# by# cz# Ÿ 1 in xyz-space
â
â
â0 0 câ
into the spherical region u# v# w# Ÿ 1 in uvw-space ˆwhich has volume V œ 43 1‰
Ê V œ ' ' ' dx dy dz œ ' ' ' abc du dv dw œ
R
G
41abc
3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 15 Practice Exercises
927
â
â
âa 0 0â
â
â
23. J(uß vß w) œ â 0 b 0 â œ abc; for R and G as in Exercise 22, ' ' ' kxyzk dx dy dz
â
â
R
â0 0 câ
œ ' ' ' a# b# c# uvw dw dv du œ 8a# b# c#
G
œ
4a# b# c#
3
'01Î2 '01Î2
'01Î2 '01Î2 '01 (3 sin 9 cos ))(3 sin 9 sin ))(3 cos 9) a3# sin 9b d3 d9 d)
a # b # c#
3
sin ) cos ) sin$ 9 cos 9 d9 d) œ
'01Î2 sin ) cos ) d) œ a b6 c
# # #
â 1
â
â
24. u œ x, v œ xy, and w œ 3z Ê x œ u, y œ vu , and z œ "3 w Ê J(uß vß w) œ â uv#
â
â 0
0
"
u
0
0 ââ
0 ââ œ
" â
â
3
"
3u
;
' ' ' ax# y 3xyzb dx dy dz œ ' ' ' u# ˆ vu ‰ 3u ˆ vu ‰ ˆ w3 ‰‘ kJ(uß vß w)k du dv dw œ "3 ' ' ' ˆv vw
‰ du dv dw
u
0
0
1
3
D
2
2
G
œ
"
3
'0 '0 (v vw ln 2) dv dw œ 3" '03 (1 w ln 2) ’ v2 “ # dw œ 32 '03 (1 w ln 2) dw œ 32 ’w w2
œ
2
3
ˆ3
3
2
#
#
!
9
#
ln 2“
$
!
ln 2‰ œ 2 3 ln 2 œ 2 ln 8
25. The first moment about the xy-coordinate plane for the semi-ellipsoid,
x#
a#
y#
b#
z#
c#
œ 1 using the
transformation in Exercise 23 is, Mxy œ ' ' ' z dz dy dx œ ' ' ' cw kJ(uß vß w)k du dv dw
D
œ abc#
G
' ' ' w du dv dw œ aabc# b † aMxy of the hemisphere x# y# z# œ 1, z
G
the mass of the semi-ellipsoid is
#
2abc1
3
3 ‰
Ê z œ Š abc4 1 ‹ ˆ 2abc
1 œ
3
8
0b œ
abc# 1
4
;
c
26. A solid of revolution is symmetric about the axis of revolution, therefore, the height of the solid is solely a function of r.
That is, y œ faxb œ farb. Using cylindrical coordinates with x œ r cos ), y œ y and z œ r sin ), we have
V œ ' ' ' r dy d) dr œ 'a
G
b
'021 '0farb
r dy d) dr œ 'a
b
'021 c r y df0arb d) dr œ 'ab '021 r farb d) dr œ 'ab c r)farb d201 dr
'ab 21rfarbdr. In the last integral, r is a dummy or stand-in variable and as such it can be replaced by any variable name.
b
Choosing x instead of r we have V œ 'a 21xfaxbdx, which is the same result obtained using the shell method.
CHAPTER 15 PRACTICE EXERCISES
1.
'110 '01Îyyexy dx dy œ '110 cexy d !"Îy dy
10
œ '1 (e 1) dy œ 9e 9
'01 '0x eyÎx dy dx œ '01 x eyÎx ‘ !x
$
2.
$
œ '0 Šxex x‹ dx œ ’ "2 ex
1
#
#
dx
"
x#
# “!
œ
e2
#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
928
3.
Chapter 15 Multiple Integrals
È
È9
'03Î2 ' È99 4t4t t ds dt œ '03Î2 ctsd È dt
9 4t
3Î2
$Î# $Î#
œ '0 2tÈ9 4t# dt œ ’ "6 a9 4t# b “
#
#
œ 6" ˆ0$Î# 9$Î# ‰ œ
4.
!
œ
27
6
9
#
'01 'È2cy Èy xy dx dy œ '01 y ’ x2 “ 2cÈy
#
Èy
œ
"
#
dy
'01 y ˆ4 4Èy y y‰ dy
œ '0 ˆ2y 2y$Î# ‰ dy œ ’y#
1
'c02 '2x4 cb x4
"
4y&Î#
5 “!
"
5
œ
dy dx œ 'c2 ax# 2xb dx
#
5.
4t#
#
0
$
œ ’ x3 x# “
!
#
œ ˆ 38 4‰ œ
4
3
'04 'c(Èy c4 c4)/2y dx dy œ '04 ˆ y c2 4 È4 y‰ dy
4
2
œ ’ y2 2y 32 a4 yb3/2 “ œ 4 8
2
3
0
œ 4
6.
œ
16
3
† 43/2
4
3
'01 'yÈy Èx dx dy œ '01 23 x$Î# ‘ yÈy dy
œ
œ
2
3
2
3
'01 ˆy$Î% y$Î# ‰ dy œ 32 47 y(Î% 52 y&Î# ‘ "!
ˆ 47 25 ‰ œ
4
35
'01 'xx Èx dy dx œ '01 x1/2 ax x2 b dx œ '01 ˆx3/2 x5/2 ‰ dx
2
1
œ 25 x5/2 27 x7/2 ‘0 œ
7.
È9
'c33 '0Ð1Î2Ñ
x#
2
5
œ
2
7
y dy dx œ 'c3 ’ y2 “
3
#
œ 'c3 8" a9 x# b dx œ ’ 9x
8
3
œ ˆ 27
8
27 ‰
24
È
'03Î2 'È99 4y4y
#
#
4
35
ˆ 27
8
!
dx
$
x$
24 “ $
œ
27 ‰
24
È
Ð1Î2Ñ 9 x#
3Î2
27
6
œ
9
#
y dx dy œ '0 2yÈ94y# dy
3/2
œ "4 † 23 a94y# b3/2 º
œ
0
"
6
† 93/2 œ
27
6
œ
9
#
'02 '04 x 2x dy dx œ '02 c2xyd 04 x dx
2
2
œ '0 a2xa4 x2 bb dx œ '0 a8x 2x3 b dx
2
8.
2
œ ’4x2
È4 c y
'04 '0
2
x4
2 “!
œ 16
16
2
È4 c y
2x dx dy œ '0 cx2 d 0
4
œ '0 a4 yb dy œ ’ 4y
4
œ8
y2
2
4
dy
“ œ 16
0
16
2
œ8
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 15 Practice Exercises
9.
'01 '2y2 4 cos ax# b dx dy œ '02 '0x/2 4 cos ax# b dy dx œ '02 2x cos ax# b dx œ csin ax# bd #! œ sin 4
10.
'02 'y1Î2 ex
11.
'08 'È2x
y%
12.
'01 'È1y
21 sin a1x# b
x#
$
$
dx dy œ '0
1
#
"
1
'02x ex
dy dx œ '0
2
dy dx œ '0 2xex dx œ cex d ! œ e 1
1
#
'0y
$
"
y% 1
dx dy œ '0
1
4 c x#
#
dx dy œ
'02 y 4y 1 dy œ ln417
$
"
4
%
'0x 21 sinx a1x b dy dx œ '01 21x sin a1x# b dx œ c cos a1x# bd "! œ (1) (1) œ 2
$
#
#
13. A œ 'c2 '2x b 4 dy dx œ 'c2 ax# 2xb dx œ
0
15. V œ '0
1
0
4
'2Ècyy
$
$
x
"
12
6 c x#
16. V œ 'c3 'x
2
7 ‰
12
18. average value œ
È
'c11 'cÈ11ccxx
20.
'c11 'cÈ11ccyy
'01
"
ˆ1‰
4
È
#
#
2
y # b2
dx œ 'c3 a6x# x% x$ b dx œ
6 c x#
2
xy dy dx œ '0 ’ xy2 “ dx œ '0
1
"
#
1
!
È1 c x
a1 x #
dx dy œ '1 ˆÈy 2 y‰ dy œ
4
7x$
3 “
$
dx œ ’ 2x3
(2x)%
12
#
xy dy dx œ
dy dx œ '0
21
'01
4
1
'01 ’ xy2 “
#
21
dx œ
È1 c x
#
!
"
4
dx œ
2
1
'01 ax x$ b dx œ #"1
"
dr d) œ '0 1 " r# ‘ ! d) œ
21
2r
a1 r# b #
ln ax# y# 1b dx dy œ '0
x
2
'01 r ln ar# 1b dr d) œ '021 '12
"
#
"
#
'021 d) œ 1
ln u du d) œ
"
#
'021 cu ln u ud #" d)
'021 (2 ln 2 1) d) œ [ln (4) 1] 1
1Î4
#
1Î4
œ 'c1Î4 ’ 2 a1 " r# b “
22. (a)
"
#
1Î4
Ècos 2)
Ècos 2)
!
d) œ "#
1Î4
"
" '
"
' 11ÎÎ44 ˆ1 1 cos
‰
ˆ1 # cos
‰
) d)
2) d) œ #
1Î4
#
'c1Î4 Š1 sec# ) ‹ d) œ "# ) tan2 ) ‘ 1Î14Î4 œ 14 2
''
#
"
a1 x # y # b #
R
œ '0 ’ "#
1Î3
œ
(b)
"
7x%
12 “ !
125
4
21. ax# y# b ax# y# b œ 0 Ê r% r# cos 2) œ 0 Ê r# œ cos 2) so the integral is 'c1Î4 '0
œ
37
6
4
3
'01 '0
#
#
œ
2
1
19.
2%
12
x# dy dx œ 'c3 cx# yd x
17. average value œ '0
"
#
14. A œ '1
4
3
'x2 c x ax# y# b dy dx œ '01 ’x# y y3 “ 2cx dx œ '01 ’2x# (23x)
œ ˆ 23
œ
"
#
''
R
"
#
"
a1 x # y # b #
œ '0
1Î3
"
# a1 sec# )b “
’ È"2 tan"
1Î2
dx dy œ '0
u
È2 “
È$
!
d) œ
È2
4
œ
dx dy œ '0
1Î2
lim ’ "
bÄ_ #
'0sec
)
dr d) œ '0
1Î3
r
a1 r# b#
'01Î3 1 secsec) ) d); ”
#
"
#
#
"
# a1 b# b “d)
r
a1 r# b#
œ
"
#
'0
dr d) œ '0
1Î2
Î
1 2
d) œ
sec )
!
d)
u œ tan )
Ä
du œ sec# ) d) •
tan" É #3
'0_
’ 2 a1 " r# b “
"
#
È3
'0
du
2 u #
b
lim
bÄ_
’ 2 a1 " r# b “ d)
0
1
4
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
r
a1 r# b#
dr d)
929
930
23.
Chapter 15 Multiple Integrals
'01 '01 '01 cos (x y z) dx dy dz œ '01 '01 [sin (z y 1) sin (z y)] dy dz
1
œ '0 [ cos (z 21) cos (z 1) cos z cos (z 1)] dz œ 0
24.
'lnln67 '0ln 2 'lnln45 eÐxyzÑ dz dy dx œ 'lnln67 '0ln 2 eÐxyÑ dy dx œ 'lnln67 ex dx œ 1
25.
'01 '0x '0xby (2x y z) dz dy dx œ '01 '0x Š 3x#
26.
'1e '1x '0z 2yz dy dz dx œ '1e '1x "z dz dx œ '1e ln x dx œ cx ln x xd 1e œ 1
#
#
#
3y#
# ‹
dy dx œ '0 Š 3x#
1
%
x'
#‹
dx œ
8
35
$
27. V œ 2 '0
1Î2
28. V œ 4 '0
2
' 0cos y '0 2x dz dx dy œ 2 '01Î2 ' 0cos y
È4cx
'0
œ ’x a4 x# b
"
3
29. average œ
#
'04cx
$Î#
#
dz dy dx œ 4 '0
2
È4cx
'0
#
1 Î2
2x dx dy œ 2 '0 cos# y dy œ 2 ’ y2
a4 x# b dy dx œ 4 '0 a4 x# b
2
$Î#
1 Î2
sin 2y
4 “!
œ
1
#
dx
#
6xÈ4 x# 24 sin" x2 “ œ 24 sin" 1 œ 121
!
'01 '03 '01
30xzÈx# y dz dy dx œ
'01 '03 15xÈx# y dy dx œ 3" '03 '01 15xÈx# y dx dy
"
3
œ
"
3
'03 ’5 ax# yb$Î# “ " dy œ "3 '03 5(1 y)$Î# 5y$Î# ‘ dy œ "3 2(1 y)&Î# 2y&Î# ‘ $! œ "3 2(4)&Î# 2(3)&Î# 2‘
œ
"
3
2 ˆ31 3&Î# ‰‘
30. average œ
31. (a)
3
4 1 a$
È
È
!
'021 '01 '0a
È
3$ sin 9 d3 d9 d) œ
3a
161
'021 '01 sin 9 d9 d) œ 83a1 '021 d) œ 3a4
'cÈ22 'cÈ22ccyy 'Èx4bcyx cy 3 dz dx dy
#
#
#
#
#
#
(b)
'021 '01Î4 '02 33# sin 9 d3 d9 d)
(c)
'021 '0 2 'r
È
È4cr
#
3 dz r dr d) œ 3 '0
21
È2
'0
’r a4 r# b
"Î#
r# “ dr d) œ 3 '0 ’ "3 a4 r# b
21
$Î#
œ '0 ˆ2$Î# 2$Î# 4$Î# ‰ d) œ Š8 4È2‹'0 d) œ 21 Š8 4È2‹
21
21
'c11ÎÎ22 '01 ' rr
1Î2
21(r cos ))(r sin ))# dz r dr d) œ ' 1Î2 '0
#
#
32. (a)
'c11ÎÎ22 '01 ' rr
#
(b)
33. (a)
(b)
34. (a)
(c)
(d)
#
1Î2
21r$ cos ) sin# ) dz r dr d) œ 84 '0
1
' rr
#
#
$
r3 “
È#
!
d)
21r$ cos ) sin# ) dz r dr d)
'01 r' sin# ) cos ) dr d) œ 12'01Î2 sin# ) cos ) d) œ 4
'021 '01Î4 '0sec 9 3# sin 9 d3 d9 d)
'021 '01Î4 '0sec 9 3# sin 9 d3 d9 d) œ 3" '021 '01Î4 (sec 9)(sec 9 tan 9) d9 d) œ 3" '021 2" tan# 9‘ !1Î4 d) œ 6" '021 d) œ 13
È
È
1
r
1Î2
'01 '0 1cx '0 x y (6 4y) dz dy dx
(b) '0 '0 '0 (6 4r sin )) dz r dr d)
'01Î2 '11ÎÎ42 '0csc 9 (6 43 sin 9 sin )) a3# sin 9b d3 d9 d)
#
#
#
'01Î2 '01 '0r (6 4r sin )) dz r dr d) œ '01Î2 '01 a6r# 4r$ sin )b dr d) œ '01Î2 c2r$ r% sin )d "! d)
1Î2
1Î#
œ '0 (2 sin )) d) œ c2) cos )d ! œ 1 1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 15 Practice Exercises
35.
È
È3 È3cx
È4cx cy
'01 'È13ccxx '1
#
#
z# yx dz dy dx '1
#
#
'0
#
È4cx cy
'1
#
#
z# yx dz dy dx
36. (a) Bounded on the top and bottom by the sphere x# y# z# œ 4, on the right by the right circular
cylinder (x 1)# y# œ 1, on the left by the plane y œ 0
È
'01Î2 '02 cos 'cÈ44ccrr dz r dr d)
)
(b)
#
#
37. (a) V œ '0
21
È8cr
'02 '2
dz r dr d) œ '0
21
#
'02 ŠrÈ8 r# 2r‹ dr d) œ '021 ’ "3 a8 r# b$Î# r# “ # d)
!
œ '0 (4)$Î# 4 (8)$Î# ‘ d) œ '0
21
"
3
(b) V œ '0
21
œ
œ
Š2 3 2È8‹ d) œ
'0 '2 sec 9 3# sin 9 d3 d9 d) œ 83 '0 '0
1Î4
21
1Î4
4
3
Š4È2 5‹ '0 d) œ
21
81 Š4È2 5‹
3
Š2È2 sin 9 sec$ 9 sin 9‹ d9 d)
'021 '01Î4 Š2È2 sin 9 tan 9 sec# 9‹ d9 d) œ 83 '021 ’2È2 cos 9 "# tan# 9“ 1Î% d)
8
3
'0
21
32
5
È8
4
3
8
3
38. Iz œ '0
œ
21
"
3
!
21
"
#
Š2
2È2‹ d) œ
8
3
'
21
È
Š 5 #4 2 ‹
0
d) œ
81 Š4È2 5‹
3
'01Î3 '02 (3 sin 9)# a3# sin 9b d3 d9 d) œ '021 '01Î3 '02 3% sin$ 9 d3 d9 d)
'021 '01Î3 asin 9 cos# 9 sin 9b d9 d) œ 325 '021 ’ cos 9 cos3 9 “ 1Î$ d) œ 831
$
!
39. With the centers of the spheres at the origin, Iz œ '0
21
'01 'ab $(3 sin 9)# a3# sin 9b d3 d9 d)
'021 '01 sin$ 9 d9 d) œ $ ab 5 a b '021 '01 asin 9 cos# 9 sin 9b d9 d)
1
21
21
œ $ ab 5 a b '0 ’ cos 9 cos3 9 “ d) œ 4$ ab15 a b '0 d) œ 81$ ab15 a b
œ
$ ab& a& b
5
&
&
&
&
&
$
&
&
&
!
'01 '01ccos (3 sin 9)# a3# sin 9b d3 d9 d) œ '02 '0 '01ccos 3% sin$ 9 d3 d9 d)
21
21
1
1
œ "5 '0 '0 (1 cos 9)& sin$ 9 d9 d) œ '0 '0 (1 cos 9)' (1 cos 9) sin 9 d9 d);
40. Iz œ '0
21
)
1
u œ 1 cos 9
” du œ sin 9 d9 • Ä
œ
"
5
'021 2 56†2
$
41. M œ '1
2
&
d) œ
32
35
"
5
'021 '02 u' (2 u) du d) œ 5" '021 ’ 2u7
2
(
#
u)
8 “!
d) œ
"
5
'021 ˆ 7" 8" ‰ 2) d)
'021 d) œ 64351
'22Îx
y dy dx œ '1 ˆ2
2y c y#
2
2‰
x#
dx œ 1 Ê x œ y œ
42. M œ '0 'c2y dx dy œ '0 a4y y# b dy œ
4
2y c y#
4
32
3
4
# #
"
# ln 4
2y c y#
; Mx œ '0 'c2y y dx dy œ '0 a4y# y$ b dy œ ’ 4y3
4
y
My œ '0 'c2y x dx dy œ '0 ’ a2y#y b 2y# “ dy œ ’ 10
4
2
)
'22Îx dy dx œ '12 ˆ2 2x ‰ dx œ 2 ln 4; My œ '12 '22Îx x dy dx œ '12 x ˆ2 2x ‰ dx œ 1;
Mx œ '1
43. Io œ '0
1
&
%
y%
2 “!
4
œ 128
5 Ê xœ
$
My
M
œ 12
5 and y œ
'2x4 ax# y# b (3) dy dx œ 3 '02 Š4x# 643 14x3 ‹ dx œ 104
44. (a) Io œ
$
'c22 'c11 ax# y# b dy dx œ 'c22 ˆ2x# 23 ‰ dx œ 403
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
%
y%
4 “!
Mx
M
œ
œ2
64
3
;
931
932
Chapter 15 Multiple Integrals
(b) Ix œ 'ca 'cb y# dy dx œ 'ca
$
4ab ab# a# b
4a$ b
œ 4ab
3 3 œ
3
a
45. M œ $ '0
3
46. M œ '0
b
a
2b$
3
dx œ
4ab$
3
; Iy œ 'cb 'ca x# dx dy œ 'cb
b
a
b
2a$
3
dy œ
4a$ b
3
Ê Io œ Ix Iy
'02xÎ3 dy dx œ $ '03 2x3 dx œ 3$ ; Ix œ $ '03 '02xÎ3 y# dy dx œ 818$ '03 x$ dx œ ˆ 818$ ‰ Š 34 ‹ œ 2$
%
"3
'xx (x 1) dy dx œ '01 ax x$ b dx œ "4 ; Mx œ '01 'xx y(x 1) dy dx œ #" '01 ax$ x& x# x% b dx œ 120
;
1
x
1
1
x
2
8
13
My œ '0 'x x(x 1) dy dx œ '0 ax# x% b dx œ 15
Ê x œ 15
and y œ 30
; Ix œ '0 'x y# (x 1) dy dx
1
1
x
1
I
17
17
1
œ "3 '0 ax% x( x$ x' b dx œ 280
Ê Rx œ É M
œ É 70
; Iy œ '0 'x x# (x 1) dy dx œ '0 ax$ x& b dx œ 12
1
#
#
#
#
x
#
47. M œ 'c1 'c1 ˆx# y# 3" ‰ dy dx œ 'c1 ˆ2x# 34 ‰ dx œ 4; Mx œ 'c1 'c1 y ˆx# y# "3 ‰ dy dx œ 'c1 0 dx œ 0;
1
1
1
My œ ' ' x ˆx# y# "3 ‰ dy dx œ ' ˆ2x$ 34 x‰ dx œ 0
1
1
1
c1 c1
1
1
1
c1
48. Place the ?ABC with its vertices at A(0ß 0), B(bß 0) and C(aß h). The line through the points A and C is
yœ
h
a
x; the line through the points C and B is y œ
œ b$ '0 ˆ1 yh ‰ dy œ
h
1Î3
$ bh
#
49. M œ ' 1Î3 '0 r dr d) œ 9#
and y œ 0 by symmetry
50. M œ '0
1Î2
yœ
13
31
3
; Ix œ '0
h
(x b). Thus, M œ '0
h
'ayÐaÎh bÑyÎh b $ dx dy
'ayÐaÎh bÑyÎh b y# $ dx dy œ b$ '0h Šy# yh ‹ dy œ $1bh#
$
$
' 11ÎÎ33 d) œ 31; My œ ' 11ÎÎ33 '03 r# cos ) dr d) œ 9 ' 11ÎÎ33 cos ) d) œ 9È3
'13 r dr d) œ 4 '01Î2 d) œ 21; My œ '01Î2 '13
r# cos ) dr d) œ
26
3
'01Î2 cos ) d) œ 263
Ê xœ
Ê xœ
by symmetry
51. (a) M œ 2 '0
1Î2
'11bcos
)
1Î2
1Î2
My œ 'c1Î2 '1
1 cos )
1 cos 2) ‰
#
œ 'c1Î2 Šcos# ) cos$ )
Ê xœ
d) œ
81
4
;
(r cos )) r dr d)
1Î2
32 151
24
(b)
r dr d)
œ '0 ˆ2 cos )
œ
h
ab
151 32
61 48
cos% )
3 ‹
d)
, and
y œ 0 by symmetry
52. (a) M œ 'c! '0 r dr d) œ 'c!
d) œ a# !; My œ 'c! '0 (r cos )) r dr d) œ 'c!
!
2a sin !
Ê x œ 2a 3sin
œ0
! , and y œ 0 by symmetry; lim c x œ lim c
3!
!
(b) x œ
2a
51
!
a
and y œ 0
!
a#
#
!Ä1
!
a
a$ cos )
3
d) œ
2a$ sin !
3
!Ä1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
13
31
3È 3
1
, and
,
Chapter 15 Additional and Advanced Exercises
53. x œ u y and y œ v Ê x œ u v and y œ v
" "
Ê J(uß v) œ º
œ 1; the boundary of the
0 "º
image G is obtained from the boundary of R as
follows:
xy-equations for
Corresponding uv-equations
Simplified
the boundary of R
for the boundary of G
uv-equations
yœx
yœ0
Ê
vœuv
uœ0
vœ0
_
_ _
'0 '0 esx f(x yß y) dy dx œ '0 '0
x
vœ0
esÐuvÑ f(uß v) du dv
$s "t
!$ "#
54. If s œ !x " y and t œ # x $ y where (!$ "# )# œ ac b# , then x œ
"
(!$ "# )#
and J(sß t) œ
œ
"
Èac b#
$
º #
'021 '0_ rer
#
"
œ
! º
dr d) œ
"
!$ "#
"
#Èac b#
_ _
'c_
' _ e as t b È
#
Ê
'021 d) œ È
1
ac b#
#
"
ac b#
,yœ
,
ds dt
1
Èac b#
. Therefore,
# s !t
!$ "#
œ 1 Ê ac b# œ 1# .
CHAPTER 15 ADDITIONAL AND ADVANCED EXERCISES
6cx#
1. (a) V œ 'c3 'x
2
6cx#
(c) V œ 'c3 'x
2
6cx#
(b) V œ 'c3 'x
2
x# dy dx
6cx#
x# dy dx œ 'c3 'x
2
a6x# x% x$ b dx œ ’2x$
&
x
5
'0x
%
x
4
#
“
dz dy dx
#
$
œ
125
4
2. Place the sphere's center at the origin with the surface of the water at z œ 3. Then
9 œ 25 x# y# Ê x# y# œ 16 is the projection of the volume of water onto the xy-plane
Ê V œ '0
21
'04 'ccÈ325cr
dz r dr d) œ '0
21
#
'04 ŠrÈ25 r# 3r‹ dr d) œ '021 ’ "3 a25 r# b$Î# 3# r# “ % d)
21
21
œ '0 "3 (9)$Î# 24 3" (25)$Î# ‘ d) œ '0
3. Using cylindrical coordinates, V œ '0
21
œ '0 ˆ1
21
4. V œ 4 '0
1Î2
"
3
cos )
'01 'r
È2
#
œ 4 '0 Š
1Î2
"
3
"
4
r#
"
3
'01 '02crÐcos
sin )‰ d) œ )
1Î2
dz r dr d) œ 4 '0
2È 2
3 ‹
d) œ
!
26
3
"
3
d) œ
)
sin )
sin )Ñ
"
3
521
3
dz r dr d) œ '0
21
'01 a2r r# cos ) r# sin )b dr d)
#1
cos )‘ ! œ 21
'01 ŠrÈ2 r# r$ ‹ dr d) œ 4'01Î2 ’ "3 a2 r# b$Î# r4 “ " d)
%
!
'
1Î2
È
Š 8 327 ‹ 0
d) œ
1 Š8È2 7‹
6
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
933
934
Chapter 15 Multiple Integrals
5. The surfaces intersect when 3 x# y# œ 2x# 2y# Ê x# y# œ 1. Thus the volume is
V œ 4 '0
1
È1 c x
'0
6. V œ 8 '0
1Î2
œ
#
'2x3cx2ycy
#
#
#
#
1Î2
dz dy dx œ 4 '0
'01 '2r3
1Î2
dz r dr d) œ 4 '0
r#
#
'01 a3r 3r$ b dr d) œ 3'01Î2 d) œ 31#
'01Î2 '02 sin 9 3# sin 9 d3 d9 d) œ 643 '01Î2 '01Î2 sin% 9 d9 d)
'01Î2 ” sin 94cos 9 ¹1Î# 43 '01Î2 sin# 9 d9• d) œ 16 '01Î2 92 sin429 ‘ 1! Î# d) œ 41 '01Î2 d) œ 21#
$
64
3
!
7. (a) The radius of the hole is 1, and the
radius of the sphere is 2.
(b) V œ 2 '0
21
8. V œ '0
1
È4cz
È
'0 3 '1
È9cr
'03 sin '0
)
r dr dz d) œ '0
21
#
dz r dr d) œ '0
#
'03 sin
1
œ '0 ’ "3 a9 9 sin# )b
1
$Î#
)
È3
'0
a3 z# b dz d) œ 2È3 '0 d) œ 4È31
21
rÈ9 r# dr d) œ '0 ’ "3 a9 r# b
1
3" (9)$Î# “ d) œ 9'0 ’1 a1 sin# )b
1
œ '0 a1 cos ) sin# ) cos )b d) œ 9 ’) sin )
1
9. The surfaces intersect when x# y# œ
'01 'r r
1‰Î2
'12 '0r sin
) cos )
V œ 4 '0
1Î2
10. V œ '0
1Î2
œ
11.
#
#
'0
1Î2
15
4
'0_ ec
ax
ˆ
#
_
dx œ '0
1 Î2
dz r dr d) œ 4 '0
dz r dr d) œ '0
1Î2
sin ) cos ) d) œ
ecbx
x
x# y# 1
#
15
4
#
’ sin2 ) “
cxy
tÄ_
d)
1
Ê x# y# œ 1. Thus the volume in cylindrical coordinates is
$
#
%
"
r8 “ d) œ
!
"
#
'01Î2 d) œ 14
'12 r$ sin ) cos ) dr d) œ '0 Î2 ’ r4 “ # sin ) cos ) d)
1Î#
!
1
%
"
œ
b
tÄ_
!
!
œ 91
15
8
'ab exy dy dx œ 'ab '0_ exy dx dy œ 'ab Š
t
“
“ d) œ 9'0 a1 cos$ )b d)
'01 Š #r r# ‹ dr d) œ 4'01Î2 ’ r4
œ 'a lim ’ e y “ dy œ 'a lim Š "y
b
1
sin$ )
3 “!
$Î#
$Î# 3 sin )
ecyt
y ‹
dy œ 'a
b
"
y
lim
tÄ_
'0t exy dx‹ dy
dy œ cln yd ab œ ln ˆ ba ‰
12. (a) The region of integration is sketched at the right
Ê '0
a sin "
œ '0
"
È
'y cota "c y
#
#
ln ax# y# b dx dy
'0a r ln ar# b dr d);
u œ r#
” du œ 2r dr • Ä
"
#
'0" '0a ln u du d)
#
'0" [u ln u u] !a d)
"
œ "# '0 ’2a# ln a a# lim
œ
"
#
#
t ln t“ d) œ
a#
#
'0a cos " '0(tan ")x ln ax# y# b dy dx 'aacos " '0
#
tÄ0
(b)
'0" (2 ln a 1) d) œ a# " ˆln a "# ‰
Èa cx
#
ln ax# y# b dy dx
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 15 Additional and Advanced Exercises
13.
'0x '0u emÐxtÑ f(t) dt du œ '0x 't x emÐxtÑ f(t) du dt œ '0x (x t)emÐxtÑ f(t) dt; also
'0x '0v '0u emÐxtÑ f(t) dt du dv œ '0x 't x 't v emÐxtÑ f(t) du dv dt œ '0x 't x (v t)emÐxtÑ f(t) dv dt
x
x
x
œ '0 "2 (v t)# emÐxtÑ f(t)‘ t dt œ '0 (x # t) emÐxtÑ f(t) dt
#
14.
'01 f(x) Š'0x g(xy)f(y) dy‹ dx œ '01 '0x
œ '0
1
g(xy)f(x)f(y) dy dx
'y1 g(xy)f(x)f(y) dx dy œ '01 f(y) Œ'y1 g(xy)f(x) dx dy;
'01 '01 g akxykb f(x)f(y) dx dy œ '01 '0x g(xy)f(x)f(y) dy dx '01 'x1 g(yx)f(x)f(y) dy dx
1
1
1
1
œ '0 'y g(xy)f(x)f(y) dx dy '0 'x g(yx)f(x)f(y) dy dx
œ '0
1
'y1 g(xy)f(x)f(y) dx dy '01 'y1 g(xy)f(y)f(x) dx dy
ðóóóóóóóóóóóóñóóóóóóóóóóóóò
simply interchange x and y
variable names
œ 2'0
1
'y1 g(xy)f(x)f(y) dx dy, and the statement now follows.
15. Io (a) œ '0 '0
xÎa#
a
œ
a#
4
"
1#
ax# y# b dy dx œ '0 ’x# y
a
a# ; Iow (a) œ
"
#
dx œ '0 Š xa#
xÎa#
a
y$
3 “!
a "6 a$ œ 0 Ê a% œ
$
x$
3a' ‹
Ê a œ %É "3 œ
"
3
"
%
È
3
%
x
dx œ ’ 4a
#
a
x%
12a' “ !
. Since Iwwo (a) œ
"
#
#" a% 0, the
value of a does provide a minimum for the polar moment of inertia Io (a).
16. Io œ '0
2
'2x4 ax# y# b (3) dy dx œ 3 '02 Š4x# 14x3
17. M œ 'c) 'b sec ) r dr d) œ
)
a
'c Š a#
)
#
)
b#
#
$
64
3 ‹
dx œ 104
sec# )‹ d)
œ a# ) b# tan ) œ a# cos" ˆ ba ‰ b# Š
È a# b#
‹
b
œ a# cos" ˆ ba ‰ bÈa# b# ; Io œ 'c) 'b sec ) r$ dr d)
)
a
'c aa% b% sec% )b d)
œ "4 'c ca% b% a1 tan# )b asec# )bd d)
œ
)
"
4
)
)
)
)
b% tan$ )
“
3
)
œ
"
4
œ
%
$
a% )
b% tan )
)
b tan
#
#
6
" %
" $È #
" ˆ b ‰
a
# a cos
a # b
œ
%
’a ) b% tan )
2 ˆy# Î2‰
b# 6" b$ aa# b# b
18. M œ 'c2 '1cay#Î4b dx dy œ ' 2 Š1
2
œ 'c2 ’ x2 “
2
œ
19.
3
16
#
2c ˆy Î2‰
1
ˆ32
64
3
y#
4‹
dy œ ’y
Î4b
3
dy œ 'c2 32
a4 y# b dy œ
32 ‰
5
#
ay#
2
2
3 ‰ ˆ 32†8 ‰
œ ˆ 16
œ
15
48
15
'0a '0b emax ab x ßa y b dy dx œ '0a '0bxÎa eb x
# #
# #
# #
3
32
$Î#
#
y$
12 “ #
œ
8
3
; My œ '
' 2 y Î2
2 1 y Î4
ˆ
2
a
#
#
b
‰
x dx dy
'c22 a16 8y# y% b dy œ 163 ’16y 8y3
Ê xœ
My
M
dy dx '0
b
‰ ˆ 83 ‰ œ
œ ˆ 48
15
'0ayÎb ea y
# #
6
5
$
#
y&
5 “!
, and y œ 0 by symmetry
dx dy
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
935
936
Chapter 15 Multiple Integrals
# #
# #
# #
# #
"
"
œ '0 ˆ ba x‰ eb x dx '0 ˆ ba y‰ ea y dy œ ’ 2ab
eb x “ ’ 2ba
ea y “ œ
a
"
ab
œ
20.
b
b
!
!
"
#ab
# #
Šeb a 1‹
"
#ab
# #
Šea b 1‹
# #
Šea b 1‹
ßy)
'yy 'xx ``F(x
' y ` F(xßy) x
x ` y dx dy œ y ’ ` y “
"
!
a
#
"
"
!
!
" ßy)
dy œ 'y ’ ` F(x
`y
y"
"
!
x!
` F(x! ßy)
`y “
dx œ cF(x" ß y) F(x! ß y)d yy!"
œ F(x" ß y" ) F(x! ß y" ) F(x" ß y! ) F(x! ß y! )
21. (a) (i)
(ii)
(iii)
(iv)
Fubini's Theorem
Treating G(y) as a constant
Algebraic rearrangement
The definite integral is a constant number
(b)
'0ln 2 '01Î2 ex cos y dy dx œ Œ'0ln 2 ex dx Œ'01Î2 cos y dy œ aeln 2 e0 b ˆsin 1# sin 0‰ œ (1)(1) œ 1
(c)
'12 'c11 yx
#
dx dy œ Œ'1
2
"
y#
dy Œ'c1 x dx œ ’ y" “ ’ x2 “
#
1
#
"
"
"
œ ˆ "# 1‰ ˆ "# "# ‰ œ 0
22. (a) ™ f œ xi yj Ê Du f œ u" x u# y; the area of the region of integration is
Ê average œ 2'0
1
#
œ 2 ’u" Š x2
"
area
(b) average œ
_ _
23. (a) I# œ '0
œ "#
'0
'01Î2
_
e
x$
3‹
'01cx (u" x u# y) dy dx œ 2 '01 u" x(1 x) "# u# (1 x)# ‘ dx
ˆ "# u# ‰
"
(1x)$
3 “!
' ' (u" x u# y) dA œ
R
ˆx# y# ‰
lim
bÄ_
1Î2
dx dy œ '0
#
_
21
œ 2 ˆ 6" u" 6" u# ‰ œ
"
#
R
È
r#
1Î2
b r dr d) œ '0 ” lim
bÄ_
'01Î2 d) œ 14
"Î# y#
e
Ê Iœ
_
$
Èh
'cÈhh 'cÈhhccxx ah x# y# b dy dx œ '021 '0
#
#
'0b re
È
$
È
r#
dr• d)
È1
#
(2y) dy œ 2 '0 ey dy œ 2 Š
'0R kr# (1 sin )) dr d) œ kR3 '021 (1 sin )) d) œ kR3
È
(u" u# )
R
25. For a height h in the bowl the volume of water is V œ
œ
"
3
M
u# ' '
' ' x dA area
y dA œ u" Š My ‹ u# ˆ MMx ‰ œ u" x u# y
u"
area
'0_ ae
aecb 1b d) œ
(b) > ˆ "# ‰ œ '0 t"Î# et dt œ '0 ay# b
24. Q œ '0
"
#
#
È1
# ‹
c) cos )d #!1 œ
'cÈhh 'cÈhhccxx 'xhby
œ È1, where y œ Èt
21kR$
3
#
#
#
#
dz dy dx
ah r# b r dr d) œ '0 ’ hr2 r4 “
21
#
%
Èh
!
d) œ '0
21
h#
4
d) œ
h# 1
#
.
Since the top of the bowl has area 101, then we calibrate the bowl by comparing it to a right circular cylinder
whose cross sectional area is 101 from z œ 0 to z œ 10. If such a cylinder contains
to a depth w then we have 101w œ
rain, w œ 3 and h œ È60.
h# 1
#
Ê wœ
h#
20
h# 1
#
cubic inches of water
. So for 1 inch of rain, w œ 1 and h œ È20; for 3 inches of
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Chapter 15 Additional and Advanced Exercises
26. (a) An equation for the satellite dish in standard position
is z œ "# x# "# y# . Since the axis is tilted 30°, a unit
vector v œ 0i aj bk normal to the plane of the
È3
#
È
#3
water level satisfies b œ v † k œ cos ˆ 16 ‰ œ
Ê a œ È1 b# œ "# Ê v œ "# j
Ê "# (y 1)
Ê zœ
"
È3
È3
#
y Š "#
k
ˆz "# ‰ œ 0
"
È3 ‹
is an equation of the plane of the water level. Therefore
' Èx byby c È dz dy dx, where R is the interior of the ellipse
the volume of water is V œ ' '
1
1
2
3
#
1
2
R
1
2
1
3
#
È 3 Ê 3 4 Š È 3 1‹
2
x# y#
È
2
and " œ
y1
2
È3
Ê 43
3
œ 0. When x œ 0, then y œ ! or y œ " , where ! œ
2
È3
Ê Vœ
3
#
(b) x œ 0 Ê z œ
"
#
y# and
!
œ y; y œ 1 Ê
dz
dy
"Î#
Š yb1c È3 cy ‹
' 'c
"
4 Š È 1‹
2
2
3
#
2
3
"Î#
Š yb1c È3 cy# ‹
2
3
2
yb
' Èb
1
1
2
x#
1
2
1
2
4
2
#
c È3
1
1 dz dx dy
y#
œ 1 Ê the tangent line has slope 1 or a 45° slant
dz
dy
Ê at 45° and thereafter, the dish will not hold water.
27. The cylinder is given by x# y# œ 1 from z œ 1 to _ Ê
œ '0
21
_
'0 '1
1
z
ar# z# b&Î#
'0 '0 '
21
dz r dr d) œ a lim
Ä_
1
' ' ' z ar# z# b&Î# dV
D
a
rz
1 ar# z# b&Î#
dz dr d)
œ a lim
Ä_
'021 '01 ’ˆ "3 ‰
œ a lim
Ä_
'021 ’ 3" ar# a# b"Î# 3" ar# 1b"Î# “ " d) œ a lim
' 21 ’ 3" a1 a# b"Î# 3" ˆ2"Î# ‰ 3" aa# b"Î# 3" “ d)
Ä_ 0
œ a lim
21 ’ 3" a1 a# b
Ä_
a
r
“
ar# z# b$Î# 1
"Î#
3" Š
'021 '01 ’ˆ 3" ‰
dr d) œ a lim
Ä_
È2
# ‹
ˆ "3 ‰
r
ar# a# b$Î#
!
r
“
ar# 1b$Î#
dr d)
È2
# “.
3" ˆ "a ‰ 3" “ œ 21 ’ 3" ˆ 3" ‰
28. Let's see?
The length of the "unit" line segment is: L œ 2'0 dx œ 2.
1
The area of the unit circle is: A œ 4'0
1
È1 c x
'0
The volume of the unit sphere is: V œ 8'0
1
2
dy dx œ 1.
È1 c x
'0
2
È1 c x c y
'0
2
2
dz dy dx œ 43 1.
Therefore, the hypervolume of the unit 4-sphere should be:
Vhyper œ 16'0
1
È1cx
'0
2
È1cx cy
'0
2
2
È1cx cy cz
'0
2
2
2
dw dz dy dx.
Mathematica is able to handle this integral, but we'll use the brute force approach.
Vhyper œ 16'0
1
È1cx
œ 16'0
'0
œ 16'0
'0
œ 16'0
'0
1
1
1
È1cx
È1cx
È1cx
'0
2
2
2
2
È1cx cy
'0
È1cx cy
'0
2
2
È1cx cy cz
'0
2
2
2
dw dz dy dx œ 16'0
1
2
z2
È 1 x2 y2 É 1
1 c x2 c y2
dz dy dx œ –
È1cx
'0
dz œ
a1 x2 y2 b'1/2 È1 cos2 ) sin ) d) dy dx œ 16'0
0
1
4 a1
1
2
È1cx cy
'0
2
2
È 1 x 2 y 2 z2
z
È1 x2 y2 œ cos )
È1 x2 y2 sin
È1cx
'0
2
) d)
—
a1 x2 y2 b'1/2 sin2 ) d) dy dx
0
1
1 x3
$
‘ dx œ 83 1' a1 x2 b3/2 dx œ ”
0
1
0
x œ cos )
œ 83 1'1/2 sin4 ) d)
dx œ sin ) d) •
2 ) ‰2
œ 83 1'1/2 ˆ 1 cos
d) œ 23 1'1/2 a1 2 cos 2) cos2 2)bd) œ 23 1'1/2 ˆ #3 2 cos 2)
2
0
dz dy dx
3/2
x2 y2 b dy dx œ 41'0 ŠÈ1 x2 x2 È1 x2 3" a1 x2 b ‹ dx
œ 41'0 È1 x2 a1 x2 b
1
2
0
0
cos 4) ‰
d)
2
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
œ
12
2
937
938
Chapter 15 Multiple Integrals
NOTES:
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
CHAPTER 16 INTEGRATION IN VECTOR FIELDS
16.1 LINE INTEGRALS
1. r œ ti a" tbj Ê x œ t and y œ 1 t Ê y œ 1 x Ê (c)
2. r œ i j tk Ê x œ 1, y œ 1, and z œ t Ê (e)
3. r œ a2 cos tbi a2 sin tbj Ê x œ 2 cos t and y œ 2 sin t Ê x# y# œ 4 Ê (g)
4. r œ ti Ê x œ t, y œ 0, and z œ 0 Ê (a)
5. r œ ti tj tk Ê x œ t, y œ t, and z œ t Ê (d)
6. r œ tj a2 2tbk Ê y œ t and z œ 2 2t Ê z œ 2 2y Ê (b)
7. r œ at# 1b j 2tk Ê y œ t# 1 and z œ 2t Ê y œ
z#
4
1 Ê (f)
8. r œ a2 cos tbi a2 sin tbk Ê x œ 2 cos t and z œ 2 sin t Ê x# z# œ 4 Ê (h)
9. ratb œ ti a1 tbj , 0 Ÿ t Ÿ 1 Ê
œ i j Ê ¸ ddtr ¸ œ È2 j ; x œ t and y œ 1 t Ê x y œ t (" t) œ 1
dr
dt
Ê 'C faxß yß zb ds œ '0 fatß 1 tß 0b ¸ ddtr ¸ dt œ '0 (1) ŠÈ2‹ dt œ ’È2 t“ œ È2
1
"
1
!
10. r(t) œ ti (1 t)j k , 0 Ÿ t Ÿ 1 Ê
œ t (1 t) 1 2 œ 2t 2 Ê
dr
dt
œ i j Ê ¸ ddtr ¸ œ È2; x œ t, y œ 1 t, and z œ 1 Ê x y z 2
'C f(xß yß z) ds œ '01 (2t 2) È2 dt œ È2 ct# 2td "! œ È2
11. r(t) œ 2ti tj (2 2t)k , 0 Ÿ t Ÿ 1 Ê
dr
dt
œ 2i j 2k Ê ¸ ddtr ¸ œ È4 1 4 œ 3; xy y z
œ (2t)t t (2 2t) Ê 'C f(xß yß z) ds œ '0 a2t# t 2b 3 dt œ 3 23 t$ "# t# 2t‘ ! œ 3 ˆ 23
1
12. r(t) œ (4 cos t)i (4 sin t)j 3tk , 21 Ÿ t Ÿ 21 Ê
Ê ¸ ddtr ¸ œ È16 sin#
1
œ c20td ##
1 œ 801
"
dr
dt
dr
dt
Ê ¸ ddtr ¸ œ È1 9 4 œ È14 ; x y z œ (1 t) (2 3t) (3 2t) œ 6 6t Ê
œ i 3 j 2k
'C f(xß yß z) ds
œ '0 (6 6t) È14 dt œ 6È14 ’t t2 “ œ Š6È14‹ ˆ "# ‰ œ 3È14
"
!
14. r(t) œ ti tj tk , 1 Ÿ t Ÿ _ Ê
_
dr
dt
13
#
'C f(xß yß z) ds œ 'c2211 (4)(5) dt
13. r(t) œ (i 2j 3k) t(i 3j 2k) œ (1 t)i (2 3t)j (3 2t)k , 0 Ÿ t Ÿ 1 Ê
#
2‰ œ
œ (4 sin t)i (4 cos t)j 3k
t 16 cos# t 9 œ 5; Èx# y# œ È16 cos# t 16 sin# t œ 4 Ê
1
"
#
œ i j k Ê ¸ ddtr ¸ œ È3 ;
È3
x # y# z#
_
Ê 'C f(xß yß z) ds œ '1 Š 3t#3 ‹ È3 dt œ 1t ‘ " œ lim ˆ b" 1‰ œ 1
œ
È3
t# t# t#
œ
È3
3t#
È
bÄ_
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
940
Chapter 16 Integration in Vector Fields
15. C" : r(t) œ ti t# j , 0 Ÿ t Ÿ 1 Ê
œ i 2tj Ê ¸ ddtr ¸ œ È1 4t# ; x Èy z# œ t Èt# 0 œ t ktk œ 2t
dr
dt
$Î#
0 Ê 'C f(xß yß z) ds œ '0 2tÈ1 4t# dt œ ’ "6 a" 4t# b “ œ
"
1
since t
!
"
C# : r(t) œ i j tk, 0 Ÿ t Ÿ 1 Ê
dr
dt
1
"
#
"
5
6
#
16. C" : r(t) œ tk , 0 Ÿ t Ÿ 1 Ê
dr
dt
(5)$Î#
"
6
"
6
œ
Š5È5 1‹ ;
œ k Ê ¸ ddtr ¸ œ 1; x Èy z# œ 1 È1 t# œ 2 t#
Ê 'C f(xß yß z) ds œ '0 a2 t# b (1) dt œ 2t "3 t$ ‘ ! œ 2
œ 'C f(xß yß z) ds 'C f(xß yß z) ds œ
"
6
È5
"
3
œ
5
3
; therefore 'C f(xß yß z) ds
3
#
œ k Ê ¸ ddtr ¸ œ 1; x Èy z# œ 0 È0 t# œ t#
Ê 'C f(xß yß z) ds œ '0 at# b (1) dt œ ’ t3 “ œ 3" ;
1
"
$
!
"
C# : r(t) œ tj k, 0 Ÿ t Ÿ 1 Ê
œ j Ê ¸ ddtr ¸ œ 1; x Èy z# œ 0 Èt 1 œ Èt 1
dr
dt
"
Ê 'C f(xß yß z) ds œ '0 ˆÈt 1‰ (1) dt œ 23 t$Î# t‘ ! œ
1
#
C$ : r(t) œ ti j k , 0 Ÿ t Ÿ 1 Ê
dr
dt
#
œ "6
17. r(t) œ ti tj tk , 0 a Ÿ t Ÿ b Ê
Ê
"
!
$
dr
dt
1 œ 3" ;
œ i Ê ¸ ddtr ¸ œ 1; x Èy z# œ t È1 1 œ t
Ê 'C f(xß yß z) ds œ '0 (t)(1) dt œ ’ t2 “ œ
1
2
3
"
#
Ê
'C f(xß yß z) ds œ 'C
"
œ i j k Ê ¸ ddtr ¸ œ È3 ;
f ds 'C f ds 'C f ds œ 3" ˆ 3" ‰
#
xyz
x # y # z#
œ
'C f(xß yß z) ds œ 'ab ˆ 1t ‰ È3 dt œ ’È3 ln ktk “ b œ È3 ln ˆ ba ‰ , since 0 a Ÿ b
$
ttt
t# t# t#
œ
"
#
1
t
a
18. r(t) œ aa cos tb j aa sin tb k , 0 Ÿ t Ÿ 21 Ê
dr
dt
œ (a sin t) j (a cos t) k Ê ¸ ddtr ¸ œ Èa# sin# t a# cos# t œ kak ;
21
1
kak sin t, 0 Ÿ t Ÿ 1
Èx# z# œ È0 a# sin# t œ œ
Ê 'C f(xß yß z) ds œ '0 kak# sin t dt '1 kak# sin t dt
kak sin t, 1 Ÿ t Ÿ 21
1
#1
œ ca# cos td ! ca# cos td 1 œ ca# (1) a# d ca# a# (1)d œ 4a#
Ê 'C x ds œ '0 t
È5
2
4
È5
2 dt
È5
2
'04 t dt œ ’ È45 t2 “ 4 œ 4È5
19. (a) ratb œ ti "# tj , 0 Ÿ t Ÿ 4 Ê
dr
dt
œ i "# j Ê ¸ ddtr ¸ œ
(b) ratb œ ti t j , 0 Ÿ t Ÿ 2 Ê
dr
dt
œ i 2tj Ê ¸ ddtr ¸ œ È1 4t2 Ê 'C x ds œ '0 t È1 4t2 dt
2
3Î2 2
1
œ ’ 12
a1 4t2 b
“ œ
!
œ
2
17È17 "
12
20. (a) ratb œ ti 4tj , 0 Ÿ t Ÿ 1 Ê
dr
dt
œ i 4j Ê ¸ ddtr ¸ œ È17 Ê 'C Èx 2y ds œ '0 Èt 2a4tb È17 dt
1
œ È17'0 È9t dt œ 3È17'0 Èt dt œ ’2È17 t2Î3 “ œ 2È17
1
1
1
!
(b) C" : ratb œ ti , 0 Ÿ t Ÿ 1 Ê
'C Èx 2y ds œ 'C
1
œ i Ê ¸ ddtr ¸ œ 1; C2 : ratb œ i tj, 0 Ÿ t Ÿ 1 Ê
dr
dt
C2
œ '0 Èt dt '0 È1 2t dt œ
1
2
21. ratb œ 4ti 3tj , 1 Ÿ t Ÿ 2 Ê
dr
dt
2
16t
œ 15'c1 t e16t dt œ ’ 15
“
32 e
2
2
dr
dt
œ j Ê ¸ ddtr ¸ œ 1
Èx 2y ds ' Èx 2y ds œ ' Èt 2a0b dt ' È1 2atb dt
1
2
c1
23 t2Î3 ‘ 1
!
2
0
0
2
’ 13 a1 2tb2Î3 “ œ
!
2
3
È
Š5 3 5
31 ‹ œ
5È 5 1
3
œ 4i 3j Ê ¸ ddtr ¸ œ 5 Ê 'C y ex ds œ 'c1 a3tb ea4tb † 5dt
2
64
œ 15
32 e
15 16
32 e
œ
15 16
32 ae
2
2
e64 b
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
!
Section 16.1 Line Integrals
22. ratb œ acos tbi asin tbj , 0 Ÿ t Ÿ 21 Ê
941
œ asin tbi acos tbj Ê ¸ ddtr ¸ œ Èsin2 t cos2 t œ 1 Ê 'C ax y 3b ds
dr
dt
œ '0 acos t sin t 3b † 1 dt œ csin t cos t 3td 201 œ 61
21
23. ratb œ t2 i t3 j , 1 Ÿ t Ÿ 2 Ê
œ '1
2
œ '1Î2
œ 2ti 3t2 j Ê ¸ ddtr ¸ œ Éa2tb2 a3t2 b2 œ tÈ4 9t2 Ê 'C
3Î2
1
a4 9t2 b “ œ
† tÈ4 9t2 dt œ '1 t È4 9t2 dt œ ’ 27
ˆt2 ‰2
2
2
at3 b4Î3
ŸtŸ1Ê
1
2
dr
dt
1
25. C" : ratb œ ti t2 j , 0 Ÿ t Ÿ 1 Ê
Ê
dr
dt
ds
œ 3t2 i 4t3 j Ê ¸ ddtr ¸ œ Éa3t2 b2 a4t3 b2 œ t2 È9 16t2 Ê 'C
1
a9 16t2 b
† t2 È9 16t2 dt œ '1Î2 t È9 16t2 dt œ ’ 48
Èt4
t3
x2
y4Î3
80È10 13È13
27
1
24. ratb œ t3 i t4 j ,
1
dr
dt
3 Î2
1
“
1Î2
œ
Èy
x
ds
125 13È13
48
œ i 2tj Ê ¸ ddtr ¸ œ È1 4t2 ; C2 : ratb œ a1 tbi a1 tbj, 0 Ÿ t Ÿ 1
dr
dt
œ i j Ê ¸ ddtr ¸ œ È2 Ê 'C ˆx Èy‰ds œ 'C ˆx Èy‰ds 'C ˆx Èy‰ds
1
2
œ '0 Št Èt2 ‹È1 4t2 dt '0 Ša1 tb È1 t‹ È2dt œ '0 2tÈ1 4t2 dt '0 Š1 t È1 t‹ È2dt
1
1
œ ’ 16 a1 4t2 b
1
3 Î2 1
1
0
0
Î
“ È2’t "# t2 23 a1 tb3 2 “ œ
26. C" : ratb œ ti , 0 Ÿ t Ÿ 1 Ê
5È 5 1
6
7È 2
6
1
œ
5È 5 7È 2 1
6
œ i Ê ¸ ddtr ¸ œ 1; C2 : ratb œ i tj, 0 Ÿ t Ÿ 1 Ê ddtr œ j Ê ¸ ddtr ¸ œ 1;
C3 : ratb œ a1 tbi j, 0 Ÿ t Ÿ 1 Ê ddtr œ i Ê ¸ ddtr ¸ œ 1; C4 : ratb œ a1 tbj, 0 Ÿ t Ÿ 1 Ê ddtr œ j Ê ¸ ddtr ¸ œ 1;
Ê 'C
œ '0
1
1
x2 y2 1 ds
dt
t2 1
'0
œ ctan1 td 0
1
1
œ 'C
1
x2 y2 1 ds
1
dt
t2 2
dr
dt
'0
1
x#
#
2
dt
a1 tb2 2
1
t
1
È2 ’tan Š È2 ‹“
27. r(x) œ xi yj œ xi
'C
1
0
'0
1
x2 y2 1 ds
1
'C
œ '0 (2x)È1 x# dx œ ’ 23 a1 x# b
2
4
1
x2 y2 1 ds
1
0
ctan1 a1 tbd 0 œ
1
1
2
2
1
1
È2 tan Š È2 ‹
#
“ œ
!
'C
dr ¸
œ i xj Ê ¸ dx
œ È1 x# ; f(xß y) œ f Šxß x# ‹ œ
dr
dx
$Î# #
1
x2 y2 1 ds
dt
a1 tb2 1
1
1 t 1
È2 ’tan Š È2 ‹“
j, 0 Ÿ x Ÿ 2 Ê
3
2
3
ˆ5$Î# 1‰ œ
#
Š x# ‹
10È5 2
3
28. r(t) œ a1 tbi #1 a1 tb2 j, 0 Ÿ t Ÿ 1 Ê ¸ ddtr ¸ œ É1 a1 tb# ; f(xß y) œ f Ša1 tbß #1 a1 tb2 ‹ œ
Ê
'C f ds œ '01 a1 tb
œ 0 ˆ "#
4
1
4 a1 tb
#
É1 a1 tb
" ‰
#0
œ
œ 2x Ê 'C f ds
x$
É1 a1 tb# dt œ ' Ša1 tb 14 a1 tb4 ‹ dt œ ’ "# a1 tb2
0
1
a1 tb 14 a1 tb4
É1 a1 tb#
1
20 a1
tb5 “
"
!
11
#0
29. r(t) œ (2 cos t) i (2 sin t) j , 0 Ÿ t Ÿ
1
#
Ê
dr
dt
œ (2 sin t) i (2 cos t) j Ê ¸ ddtr ¸ œ 2; f(xß y) œ f(2 cos tß 2 sin t)
œ 2 cos t 2 sin t Ê 'C f ds œ '0 (2 cos t 2 sin t)(2) dt œ c4 sin t 4 cos td !
1Î2
30. r(t) œ (2 sin t) i (2 cos t) j , 0 Ÿ t Ÿ
œ 4 sin# t 2 cos t Ê
1Î#
1
4
Ê
dr
dt
œ 4 (4) œ 8
œ (2 cos t) i (2 sin t) j Ê ¸ ddtr ¸ œ 2; f(xß y) œ f(2 sin tß 2 cos t)
'C f ds œ '01Î4 a4 sin# t 2 cos t b (2) dt œ c4t 2 sin 2t 4 sin td 01Î% œ 1 2Š1 È2‹
31. y œ x2 , 0 Ÿ x Ÿ 2 Ê ratb œ ti t2 j , 0 Ÿ t Ÿ 2 Ê
dr
dt
œ i 2tj Ê ¸ ddtr ¸ œ È1 4t2 Ê A œ 'C fax, yb ds
3 Î2
œ 'C ˆx Èy‰ds œ '0 Št Èt2 ‹È1 4t2 dt œ '0 2tÈ1 4t2 dt œ ’ 16 a1 4t2 b “ œ
2
2
2
0
17È17 1
6
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
942
Chapter 16 Integration in Vector Fields
32. 2x 3y œ 6, 0 Ÿ x Ÿ 6 Ê ratb œ ti ˆ2 23 t‰j , 0 Ÿ t Ÿ 6 Ê
œ 'C a4 3x 2ybds œ '0 ˆ4 3t 2ˆ2 23 t‰‰
6
33. r(t) œ at# 1b j 2tk , 0 Ÿ t Ÿ 1 Ê
dr
dt
È13
3
dt œ
È13
3
dr
dt
È13
3
œ i 23 j Ê ¸ ddtr ¸ œ
Ê A œ 'C fax, yb ds
'06 ˆ8 35 t‰dt œ È313 8t 65 t2 ‘ 60 œ 26È13
œ 2tj 2k Ê ¸ ddtr ¸ œ 2Èt# 1; M œ 'C $ (xß yß z) ds œ '0 $ (t) Š2Èt# 1‹ dt
1
3/2
œ '0 ˆ 3# t‰ Š2Èt# 1‹ dt œ ’at# 1b “ œ 2$Î# 1 œ 2È2 1
"
1
!
34. r(t) œ at# 1b j 2tk , 1 Ÿ t Ÿ 1 Ê ddtr œ 2tj 2k
Ê ¸ dr ¸ œ 2Èt# 1; M œ ' $ (xß yß z) ds
dt
C
œ 'c1
1
ˆ15Èat#
1b 2‰ Š2Èt# 1‹ dt
œ 'c1 30 at# 1b dt œ ’30 Š t3 t‹“
1
$
"
"
œ 60 ˆ 3" 1‰ œ 80;
Mxz œ 'C y$ (xß yß z) ds œ 'c1 at# 1b c30 at# 1bd dt
1
œ 'c1 30 at% 1b dt œ ’30 Š t5 t‹“
1
&
œ 48 Ê y œ
Mxz
M
"
"
œ 60 ˆ 5" 1‰
48
œ 80
œ 53 ; Myz œ 'C x$ (xß yß z) ds œ 'C 0 $ ds œ 0 Ê x œ 0; z œ 0 by symmetry (since $ is
independent of z) Ê (xß yß z) œ ˆ!ß 35 ß 0‰
35. r(t) œ È2t i È2t j a4 t# b k , 0 Ÿ t Ÿ 1 Ê
dr
dt
œ È2i È2j 2tk Ê ¸ ddtr ¸ œ È2 2 4t# œ 2È1 t# ;
(a) M œ 'C $ ds œ '0 (3t) Š2È1 t# ‹ dt œ ’2 a1 t# b
1
$Î# "
“ œ 2 ˆ2$Î# 1‰ œ 4È2 2
!
(b) M œ 'C $ ds œ '0 a1b Š2È1 t# ‹ dt œ ’tÈ1 t# ln Št È1 t# ‹“ œ ’È2 ln Š1 È2‹“ a0 ln 1b
"
1
!
œ È2 ln Š1 È2‹
36. r(t) œ ti 2tj 23 t$Î# k , 0 Ÿ t Ÿ 2 Ê
dr
dt
œ i 2j t"Î# k Ê ¸ ddtr ¸ œ È1 4 t œ È5 t;
#
M œ 'C $ ds œ '0 ˆ3È5 t‰ ˆÈ5 t‰ dt œ '0 3(5 t) dt œ 32 (5 t)# ‘ ! œ
2
2
3
#
a7# 5# b œ
Myz œ 'C x$ ds œ '0 t[3(5 t)] dt œ '0 a15t 3t# b dt œ "25 t# t$ ‘ ! œ 30 8 œ 38;
2
2
2
2
#
œ '0 ˆ10t$Î# 2t&Î# ‰ dt œ 4t&Î# 47 t(Î# ‘ ! œ 4(2)&Î# 47 (2)(Î# œ 16È2
2
œ
38
36
œ
19
18
,yœ
Mxz
M
œ
76
36
œ
19
9
, and z œ
(24) œ 36;
#
Mxz œ 'C y$ ds œ '0 2t[3(5 t)] dt œ 2 '0 a15t 3t# b dt œ 76; Mxy œ 'C z$ ds œ '0
2
3
#
Mxy
M
œ
144È2
7†36
37. Let x œ a cos t and y œ a sin t, 0 Ÿ t Ÿ 21. Then
dx
dt
œ
4
7
32
7
È2 œ
dz
dt
œ0
2 $Î#
[3(5
3 t
144
7
t)] dt
È2 Ê x œ
Myz
M
È2
œ a sin t,
dy
dt
œ a cos t,
‰ Š dy
ˆ dz ‰ dt œ a dt; Iz œ ' ax# y# b $ ds œ ' aa# sin# t a# cos# tb a$ dt
Ê Êˆ dx
dt
dt ‹ dt
C
0
#
#
21
#
œ '0 a$ $ dt œ 21$ a$ .
21
38. r(t) œ tj (2 2t)k , 0 Ÿ t Ÿ 1 Ê
dr
dt
œ j 2k Ê ¸ ddtr ¸ œ È5; M œ 'C $ ds œ '0 $ È5 dt œ $ È5;
1
"
Ix œ 'C ay# z# b $ ds œ '0 ct# (2 2t)# d $ È5 dt œ '0 a5t# 8t 4b $ È5 dt œ $ È5 53 t$ 4t# 4t‘ ! œ
1
1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
5
3
$ È5 ;
Section 16.1 Line Integrals
"
Iy œ 'C ax# z# b $ ds œ '0 c0# (2 2t)# d $ È5 dt œ '0 a4t# 8t 4b $ È5 dt œ $ È5 43 t$ 4t# 4t‘ ! œ
1
1
Iz œ 'C ax# y# b $ ds œ '0 a0# t# b $ È5 dt œ $ È5 ’ t3 “ œ
1
"
$
!
39. r(t) œ (cos t)i (sin t)j tk , 0 Ÿ t Ÿ 21 Ê
"
3
4
3
$ È5 ;
$ È5
œ ( sin t)i (cos t)j k Ê ¸ ddtr ¸ œ Èsin# t cos# t 1 œ È2;
dr
dt
(a) Iz œ 'C ax# y# b $ ds œ '0 acos# t sin# tb $ È2 dt œ 21$ È2
21
(b) Iz œ 'C ax# y# b $ ds œ '0 $ È2 dt œ 41$ È2
41
40. r(t) œ (t cos t)i (t sin t)j
2È2 $Î#
k,
3 t
0ŸtŸ1 Ê
dr
dt
œ (cos t t sin t)i (sin t t cos t)j È2t k
"
Ê ¸ ddtr ¸ œ È(t 1)# œ t 1 for 0 Ÿ t Ÿ 1; M œ 'C $ ds œ '0 (t 1) dt œ "2 (t 1)# ‘ ! œ
1
Mxy œ 'C z$ ds œ '
È
Š 2 3 2 t$Î# ‹ (t
0
œ
2È 2
3
ˆ 27 52 ‰ œ
1
2È 2
3
ˆ 24
‰
35 œ
1) dt œ
16È2
35
Ê zœ
2È 2
3
'0 ˆt&Î# t$Î# ‰ dt œ
Mxy
M
œ Š 1635 2 ‹ ˆ 23 ‰ œ
È
32È2
105
œ '0 at# cos# t t# sin# tb (t 1) dt œ '0 at$ t# b dt œ ’ t4 t3 “ œ
1
2È 2
3
1
1
%
"
$
!
"
4
"
#
a2# 1# b œ
3
#
;
27 t(Î# 25 t&Î# ‘ "
!
; Iz œ 'C ax# y# b $ ds
"
3
œ
7
12
41. $ (xß yß z) œ 2 z and r(t) œ (cos t)j (sin t)k , 0 Ÿ t Ÿ 1 Ê M œ 21 2 as found in Example 3 of the text;
also ¸ ddtr ¸ œ 1; Ix œ 'C ay# z# b $ ds œ '0 acos# t sin# tb (2 sin t) dt œ '0 (2 sin t) dt œ 21 2
1
42. r(t) œ ti
2È2 $Î#
j
3 t
t#
#
k, 0 Ÿ t Ÿ 2 Ê
1
dr
dt
œ i È2 t"Î# j tk Ê ¸ ddtr ¸ œ È1 2t t# œ È(1 t)# œ 1 t for
0 Ÿ t Ÿ 2; M œ 'C $ ds œ '0 ˆ t"1 ‰ (1 t) dt œ '0 dt œ 2; Myz œ 'C x$ ds œ '0 t ˆ t"1 ‰ (1 t) dt œ ’ t2 “ œ 2;
2
Mxz œ 'C y$ ds œ '
2È2 $Î#
3 t
0
yœ
Mxz
M
œ
16
15
2
, and z œ
Mxy
M
œ
2
dt œ
#
3
#
È
’ 4152 t&Î# “
!
œ
2
2
$
œ '0 ˆt# 89 t$ ‰ dt œ ’ t3 29 t% “ œ
$
; Mxy œ 'C z$ ds œ '0
2 #
t
#
dt œ
#
$ #
’ t6 “
!
; Ix œ 'C ay# z# b $ ds œ '0 ˆ 98 t$ "4 t% ‰ dt œ ’ 92 t%
Iy œ 'C ax# z# b $ ds œ '0 ˆt# 4" t% ‰ dt œ ’ t3
2
32
15
2
#
!
8
3
32
9
œ
#
t&
20 “ !
œ
8
3
32
20
œ
64
15
œ
#
t&
20 “ !
œ
; Iz œ 'C ax# y# b $ ds
56
9
43-46. Example CAS commands:
Maple:
f := (x,y,z) -> sqrt(1+30*x^2+10*y);
g := t -> t;
h := t -> t^2;
k := t -> 3*t^2;
a,b := 0,2;
ds := ( D(g)^2 + D(h)^2 + D(k)^2 )^(1/2):
'ds' = ds(t)*'dt';
F := f(g,h,k):
'F(t)' = F(t);
Int( f, s=C..NULL ) = Int( simplify(F(t)*ds(t)), t=a..b );
`` = value(rhs(%));
#
!
%
3
# (a)
# (b)
# (c)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Ê xœ
Myz
M
œ
32
9
32
20
œ 1,
232
45
;
943
944
Chapter 16 Integration in Vector Fields
Mathematica: (functions and domains may vary)
Clear[x, y, z, r, t, f]
f[x_,y_,z_]:= Sqrt[1 30x2 10y]
{a,b}= {0, 2};
x[t_]:= t
y[t_]:= t2
z[t_]:= 3t2
r[t_]:= {x[t], y[t], z[t]}
v[t_]:= D[r[t], t]
mag[vector_]:=Sqrt[vector.vector]
Integrate[f[x[t],y[t],z[t]] mag[v[t]], {t, a, b}]
N[%]
16.2 VECTOR FIELDS, WORK, CIRCULATION, AND FLUX
1. f(xß yß z) œ ax# y# z# b
`f
`y
#
#
"Î#
# $Î#
œ y ax y z b
and
`f
`y
œ
y
x # y# z#
and
`f
`z
"
#
2. f(xß yß z) œ ln Èx# y# z# œ
similarly,
`f
`x
Ê
`f
`z
#
$Î#
# $Î#
#
œ z ax y z b
ln ax# y# z# b Ê
œ
3. g(xß yß z) œ ez ln ax# y# b Ê
œ #" ax# y# z# b
z
x # y # z#
`g
`x
Ê ™fœ
œ x# 2x
y# ,
`g
`y
`f
`x
(2x) œ x ax# y# z# b
Ê ™fœ
œ
"
#
$Î#
; similarly,
xi yj zk
ax# y# z# b$Î#
Š x# y"# z# ‹ (2x) œ
x
x# y# z#
;
xi yj zk
x # y# z#
œ x# 2y
y# and
`g
`z
œ ez
z
Ê ™ g œ Š x#2xy# ‹ i Š x# 2y
y# ‹ j e k
`g
`x
4. g(xß yß z) œ xy yz xz Ê
œ y z,
`g
`y
œ x z, and
`g
`z
œ y x Ê ™ g œ (y z)i (B z)j (x y)k
5. kFk inversely proportional to the square of the distance from (xß y) to the origin Ê È(M(xß y))# (N(xß y))#
œ
k
x# y#
y
x
È x # y# i È x# y# j
Then M(xß y) œ Èx#ax
and N(xß y) œ Èx#ay
y#
y#
ky
k
kx
a œ x# y# Ê F œ # # $Î# i # # $Î# j , for any constant
ax y b
ax y b
, k 0; F points toward the origin Ê F is in the direction of n œ
Ê F œ an , for some constant a 0.
Ê È(M(xß y))# (N(xß y))# œ a Ê
k0
6. Given x# y# œ a# b# , let x œ Èa# b# cos t and y œ Èa# b# sin t. Then
r œ ŠÈa# b# cos t‹ i ŠÈa# b# sin t‹ j traces the circle in a clockwise direction as t goes from 0 to 21
Ê v œ ŠÈa# b# sin t‹ i ŠÈa# b# cos t‹ j is tangent to the circle in a clockwise direction. Thus, let
F œ v Ê F œ yi xj and F(0ß 0) œ 0 .
7. Substitute the parametric representations for r(t) œ x(t)i y(t)j z(t)k representing each path into the vector
field F , and calculate 'C F †
dr
dt
.
(a) F œ 3ti 2tj 4tk and
dr
dt
œijk Ê F†
(b) F œ 3t# i 2tj 4t% k and
œ
7
3
2œ
dr
dt
dr
dt
œ 9t Ê
œ i 2tj 4t$ k Ê F †
dr
dt
'01 9t dt œ 9#
œ 7t# 16t( Ê
'01 a7t# 16t( b dt œ 37 t$ 2t) ‘ "!
13
3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 16.2 Vector Fields, Work, Circulation, and Flux
(c) r" œ ti tj and r# œ i j tk ; F" œ 3ti 2tj and
F# œ 3i 2j 4tk and
œ k Ê F# †
d r#
dt
d r#
dt
d r"
dt
œ i j Ê F" †
'01 4t dt œ 2
œ 4t Ê
Ê
d r"
dt
5
#
'01 5t dt œ #5 ;
œ 5t Ê
2œ
9
#
8. Substitute the parametric representation for r(t) œ x(t)i y(t)j z(t)k representing each path into the vector
field F, and calculate 'C F †
dr
dt
.
" ‰
(a) F œ ˆ t#
1 j and
dr
dt
œijkÊF†
" ‰
(b) F œ ˆ t#
1 j and
dr
dt
œ i 2tj 4t$ k Ê F †
dr
dt
"
t# 1
œ
dr
dt
" ‰
(c) r" œ ti tj and r# œ i j tk ; F" œ ˆ t#
1 j
Ê F# †
d r#
dt
œ 0 Ê '0
1
"
t# 1
dt œ
Ê '0
1
œ
2t
t# 1
and ddtr"
"
t# 1
Ê '0
1
"
dt œ ctan" td ! œ
2t
t# 1
1
4
"
dt œ cln at# 1bd ! œ ln 2
œ i j Ê F" †
d r"
dt
œ
"
t# 1
; F# œ
"
#
j and
d r#
dt
œk
1
4
9. Substitute the parametric representation for r(t) œ x(t)i y(t)j z(t)k representing each path into the vector
field F, and calculate 'C F †
dr
dt
.
'01 ˆ2Èt 2t‰ dt œ 43 t$Î# t# ‘ "! œ "3
1
"
F œ t# i 2tj tk and ddtr œ i 2tj 4t$ k Ê F † ddtr œ 4t% 3t# Ê '0 a4t% 3t# b dt œ 45 t& t$ ‘ ! œ "5
1
r" œ ti tj and r# œ i j tk ; F" œ 2tj Èt k and ddtr œ i j Ê F" † ddtr œ 2t Ê '0 2t dt œ 1;
1
F# œ Èti 2j k and ddtr œ k Ê F# † ddtr œ 1 Ê '0 dt œ 1 Ê 1 1 œ 0
(a) F œ Èti 2tj Ètk and
(b)
(c)
dr
dt
œijk Ê F†
œ 2Èt 2t Ê
dr
dt
"
#
"
#
10. Substitute the parametric representation for r(t) œ x(t)i y(t)j z(t)k representing each path into the vector
field F, and calculate 'C F †
dr
dt
.
œ 3t# Ê '0 3t# dt œ 1
1
(a) F œ t# i t# j t# k and
dr
dt
œijk Ê F†
(b) F œ t$ i t' j t& k and
dr
dt
œ i 2tj 4t$ k Ê F †
%
œ ’ t4
t)
4
"
94 t* “ œ
!
dr
dt
œ t$ 2t( 4t) Ê '0 at$ 2t( 4t) b dt
1
17
18
(c) r" œ ti tj and r# œ i j tk ; F" œ t# i and
F# œ i tj tk and
dr
dt
d r#
dt
œ k Ê F# †
d r#
dt
d r"
dt
œ i j Ê F" †
œ t Ê '0 t dt œ
1
"
#
Ê
d r"
dt
"
3
œ t# Ê '0 t# dt œ
1
"
#
œ
"
3
;
5
6
11. Substitute the parametric representation for r(t) œ x(t)i y(t)j z(t)k representing each path into the vector
field F, and calculate 'C F †
dr
dt
.
(a) F œ a3t# 3tb i 3tj k and
†
(b) F œ a3t# 3tb i 3t% j k
Ê F†
Ê
dr
dt œ i j k Ê F
and ddtr œ i 2tj 4t$ k
dr
dt
œ 3t# 1 Ê
œ 6t& 4t$ 3t# 3t
'0 a6t& 4t$ 3t# 3tb dt œ t' t% t$ 3# t# ‘ "! œ 3#
1
(c) r" œ ti tj and r# œ i j tk ; F" œ a3t# 3tb i k and
Ê
dr
dt
'01 a3t# 1b dt œ ct$ td "! œ 2
d r"
dt
œ i j Ê F" †
d r"
dt
œ 3t# 3t
œ k Ê F# †
d r#
dt
œ1 Ê
'0 a3t# 3tb dt œ t$ 32 t# ‘ "! œ "# ; F# œ 3tj k and ddtr
1
Ê "# 1 œ
#
'01 dt œ 1
1
2
12. Substitute the parametric representation for r(t) œ x(t)i y(t)j z(t)k representing each path into the vector
field F, and calculate 'C F †
dr
dt
.
(a) F œ 2ti 2tj 2tk and
dr
dt
œijk Ê F†
dr
dt
œ 6t Ê
'01 6t dt œ c3t# d "! œ 3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
945
946
Chapter 16 Integration in Vector Fields
(b) F œ at# t% b i at% tb j at t# b k and
Ê '0 a6t& 5t% 3t# b dt œ ct' t&
1
dr
dt œ i
"
t$ d ! œ 3
2tj 4t$ k Ê F †
(c) r" œ ti tj and r# œ i j tk ; F" œ ti tj 2tk and
F# œ (1 t)i (t 1)j 2k and
d r#
dt
œ k Ê F# †
d r#
dt
dr
dt
œ 6t& 5t% 3t#
œ i j Ê F" †
dr"
dt
œ 2t Ê '0 2t dt œ ";
1
d r"
dt
œ 2 Ê '0 2 dt œ 2 Ê " 2 œ 3
1
13. x œ t, y œ 2t 1, 0 Ÿ t Ÿ 3 Ê dx œ dt Ê 'C ax yb dx œ '0 at a2t 1bb dt œ '0 at 1b dt œ "# t2 t‘ ! œ 15
2
3
14. x œ t, y œ t2 , 1 Ÿ t Ÿ 2 Ê dy œ 2t dt Ê 'C
x
y
dy œ '1
2
t
t2 a2tb dt
3
3
œ '1 2 dt œ c2td21 œ 2
2
15. C1 : x œ t, y œ 0, 0 Ÿ t Ÿ 3 Ê dy œ 0; C2 : x œ 3, y œ t, 0 Ÿ t Ÿ 3 Ê dy œ dt Ê 'C ax2 y2 b dy
œ 'C ax2 y2 b dx 'C ax2 y2 b dx œ '0 at2 02 b † 0 '0 a32 t2 b dt œ '0 a9 t2 bdt œ 9t 13 t3 ‘ ! œ 36
3
1
3
3
3
2
16. C1 : x œ t, y œ 3t, 0 Ÿ t Ÿ 1 Ê dx œ dt; C2 : x œ 1 t, y œ 3, 0 Ÿ t Ÿ 1 Ê dx œ dt; C3 : x œ 0, y œ 3 t, 0 Ÿ t Ÿ 3
Ê dx œ 0 Ê 'C Èx y dx œ 'C Èx y dx 'C Èx y dx 'C Èx y dx
1
2
3
œ '0 Èt 3t dt '0 Èa1 tb 3 a1bdt '0 È0 a3 tb † 0 œ '0 2Èt dt '0 È4 t dt
1
1
3
1
œ 43 t2Î3 ‘ ! ’ 23 a4 tb2Î3 “ œ
1
4
3
!
Š2È3
16
3 ‹
1
1
œ 2È3 4
17. ratb œ ti j t2 k , 0 Ÿ t Ÿ 1 Ê dx œ dt, dy œ 0, dz œ 2t dt
(a)
(b)
(c)
'C ax y zb dx œ '01 at 1 t2 b dt œ 12 t2 t 13 t3 ‘ 1! œ 56
'C ax y zb dy œ '01 at 1 t2 b † 0 œ 0
'C ax y zb dz œ '01 at 1 t2 b 2t dt œ '01 a2t2 2t 2t3 b dt œ
1
œ 23 t3 t2 12 t4 ‘ ! œ 56
18. ratb œ acos tbi asin tbj acos tbk , 0 Ÿ t Ÿ 1 Ê dx œ sin t dt, dy œ cos t dt, dz œ sin t dt
(a)
'C x z dx œ '01 acos tb acos tbasin tbdt œ '01 cos2 t sin tdt œ ’ 13 acos tb3 “ 1 œ 23
(b)
'C x z dy œ '01 acos tb acos tbacos tbdt œ '01 cos3 t dt œ '01 a1 sin2 tb cos t dt œ ’ 13 asin tb3 sin t“ 1 œ 0
(c)
!
'C x y z dz œ '0 acos tbasin tb acos tbasin tbdt œ '0
1
1
1
œ 18 '0 a1 cos 4tb dt œ 18 t 32
sin 4t‘ ! œ 18
1
1
cos t sin t dt œ
2
2
14
'0
1
19. r œ ti t# j tk , 0 Ÿ t Ÿ 1, and F œ xyi yj yzk Ê F œ t$ i t# j t$ k and
Ê F†
dr
dt
œ 2t$ Ê work œ '0 2t$ dt œ
1
sin 2t dt œ
dr
dt
2
41
œ i 2tj k
"
#
20. r œ (cos t)i (sin t)j 6t k , 0 Ÿ t Ÿ 21, and F œ 2yi 3xj (x y)k
Ê F œ (2 sin t)i (3 cos t)j (cos t sin t)k and
œ 3 cos# t 2sin2 t
œ 32 t
3
4
"
6
sin 2t t
cos t
sin 2t
2
"
6
"
6
dr
dt
œ ( sin t)i (cos t)j 6" k Ê F †
sin t Ê work œ '0 ˆ3 cos# t 2 sin2 t
sin t
"
6
cos
#1
t‘ !
21
"
6
cos t
"
6
dr
dt
sin t‰ dt
œ1
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
'
1
!
1 cos 4t
2
0
dt
Section 16.2 Vector Fields, Work, Circulation, and Flux
21. r œ (sin t)i (cos t)j tk , 0 Ÿ t Ÿ 21, and F œ zi xj yk Ê F œ ti (sin t)j (cos t)k and
dr
dt
œ (cos t)i (sin t)j k Ê F †
œ cos t t sin t
t
2
sin 2t
4
dr
dt
œ t cos t sin# t cos t Ê work œ '0 at cos t sin# t cos tb dt
21
#1
sin t‘ ! œ 1
22. r œ (sin t)i (cos t)j 6t k , 0 Ÿ t Ÿ 21, and F œ 6zi y# j 12xk Ê F œ ti acos# tbj (12 sin t)k and
dr
dt
œ (cos t)i (sin t)j 6" k Ê F †
dr
dt
œ t cos t sin t cos# t 2 sin t
Ê work œ '0 at cos t sin t cos# t 2 sin tb dt œ cos t t sin t
21
1
3
#1
cos$ t 2 cos t‘ ! œ 0
23. x œ t and y œ x# œ t# Ê r œ ti t# j , 1 Ÿ t Ÿ 2, and F œ xyi (x y)j Ê F œ t$ i at t# b j and
dr
dt
œ i 2tj Ê F †
dr
dt
œ t$ a2t# 2t$ b œ 3t$ 2t# Ê 'C xy dx (x y) dy œ 'C F †
#
œ 34 t% 32 t$ ‘ " œ ˆ12
16 ‰
3
ˆ 34 23 ‰ œ
45
4
18
3
œ
dr
dt
dt œ 'c" a3t$ 2t# b dt
#
69
4
24. Along (0ß 0) to (1ß 0): r œ ti , 0 Ÿ t Ÿ 1, and F œ (x y)i (x y)j Ê F œ ti tj and
dr
dt
œi Ê F†
dr
dt
œ t;
Along (1ß 0) to (0ß 1): r œ (1 t)i tj , 0 Ÿ t Ÿ 1, and F œ (x y)i (x y)j Ê F œ (1 2t)i j and
dr
dr
dt œ i j Ê F † dt œ 2t;
Along (0ß 1) to (0ß 0): r œ (1 t)j , 0 Ÿ t Ÿ 1, and F œ (x y)i (x y)j Ê F œ (t 1)i (1 t)j and
dr
dt
œ j Ê F †
dr
dt
œ t 1 Ê 'C (x y) dx (x y) dy œ '0 t dt '0 2t dt '0 (t 1) dt œ '0 (4t 1) dt
1
1
1
1
dr
dy
œ 2yi j and F †
"
œ c2t# td ! œ 2 1 œ 1
25. r œ xi yj œ y# i yj , 2
Ê
1, and F œ x# i yj œ y% i yj Ê
y
dr
dy
œ 2y& y
4‰
3
63
39
'C F † T ds œ '2c1 F † dydr dy œ '2c1 a2y& yb dy œ 3" y' "# y# ‘ "
œ ˆ 3" #" ‰ ˆ 64
3 # œ # 3 œ #
#
26. r œ (cos t)i (sin t)j , 0 Ÿ t Ÿ
ÊF†
dr
dt
1
#
, and F œ yi xj Ê F œ (sin t)i (cos t)j and
œ sin# t cos# t œ 1 Ê
'C F † dr œ '0
1Î2
dr
dt
œ ( sin t)i (cos t)j
(1) dt œ 1#
27. r œ (i j) t(i 2j) œ (1 t)i (1 2t)j , 0 Ÿ t Ÿ 1, and F œ xyi (y x)j Ê F œ a1 3t 2t# b i tj and
dr
dt
œ i 2j Ê F †
dr
dt
œ 1 5t 2t# Ê work œ 'C F †
dr
dt
dt œ '0 a1 5t 2t# b dt œ t 25 t# 23 t$ ‘ ! œ
1
"
28. r œ (2 cos t)i (2 sin t)j , 0 Ÿ t Ÿ 21, and F œ ™ f œ 2(x y)i 2(x y)j
Ê F œ 4(cos t sin t)i 4(cos t sin t)j and ddtr œ (2 sin t)i (2 cos t)j Ê F †
25
6
dr
dt
œ 8 asin t cos t sin# tb 8 acos# t cos t sin tb œ 8 acos# t sin# tb œ 8 cos 2t Ê work œ 'C ™ f † dr
œ 'C F †
dr
dt
dt œ '0 8 cos 2t dt œ c4 sin 2td #!1 œ 0
21
29. (a) r œ (cos t)i (sin t)j , 0 Ÿ t Ÿ 21, F" œ xi yj , and F# œ yi xj Ê
F" œ (cos t)i (sin t)j , and F# œ ( sin t)i (cos t)j Ê F" †
dr
dt
dr
dt
œ ( sin t)i (cos t)j ,
œ 0 and F# †
dr
dt
œ sin# t cos# t œ 1
Ê Circ" œ '0 0 dt œ 0 and Circ# œ '0 dt œ 21; n œ (cos t)i (sin t)j Ê F" † n œ cos# t sin# t œ 1 and
21
21
F# † n œ 0 Ê Flux" œ '0 dt œ 21 and Flux# œ '0 0 dt œ 0
21
21
(b) r œ (cos t)i (4 sin t)j , 0 Ÿ t Ÿ 21 Ê
F# œ (4 sin t)i (cos t)j Ê F" †
dr
dt
dr
dt
œ ( sin t)i (4 cos t)j , F" œ (cos t)i (4 sin t)j , and
œ 15 sin t cos t and F# †
dr
dt
œ 4 Ê Circ" œ '0 15 sin t cos t dt
21
œ "25 sin# t‘ ! œ 0 and Circ# œ '0 4 dt œ 81; n œ Š È417 cos t‹ i Š È"17 sin t‹ j Ê F" † n
#1
21
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
947
948
Chapter 16 Integration in Vector Fields
œ
4
È17
cos# t
sin# t and F# † n œ È1517 sin t cos t Ê Flux" œ '0 (F" † n) kvk dt œ '0 Š È417 ‹ È17 dt
21
4
È17
21
# ‘
œ 81 and Flux# œ '0 (F# † n) kvk dt œ '0 Š È1517 sin t cos t‹ È17 dt œ 15
2 sin t ! œ 0
21
21
#1
30. r œ (a cos t)i (a sin t)j , 0 Ÿ t Ÿ 21, F" œ 2xi 3yj , and F# œ 2xi (x y)j Ê
œ (a sin t)i (a cos t)j ,
dr
dt
F" œ (2a cos t)i (3a sin t)j , and F# œ (2a cos t)i (a cos t a sin t)j Ê n kvk œ (a cos t)i (a sin t)j ,
F" † n kvk œ 2a# cos# t 3a# sin# t, and F# † n kvk œ 2a# cos# t a# sin t cos t a# sin# t
Ê Flux" œ '0 a2a# cos# t 3a# sin# tb dt œ 2a# 2t
21
sin 2t ‘ #1
4
!
Flux# œ '0 a2a# cos# t a# sin t cos t a# sin# tb dt œ 2a# 2t
21
31. F" œ (a cos t)i (a sin t)j ,
d r"
dt
sin 2t ‘ #1
4
!
œ 1a# , and
a#
#
#1
3a# 2t
œ (a sin t)i (a cos t)j Ê F" †
sin 2t ‘ #1
4
!
d r"
dt
csin# td ! a# 2t
sin 2t ‘ #1
4
!
œ 1a#
œ 0 Ê Circ" œ 0; M" œ a cos t,
N" œ a sin t, dx œ a sin t dt, dy œ a cos t dt Ê Flux" œ 'C M" dy N" dx œ '0 aa# cos# t a# sin# tb dt
œ '0 a# dt œ a# 1;
1
1
F # œ ti ,
d r#
dt
œ i Ê F# †
d r#
dt
œ t Ê Circ# œ 'ca t dt œ 0; M# œ t, N# œ 0, dx œ dt, dy œ 0 Ê Flux#
a
œ 'C M# dy N# dx œ 'ca 0 dt œ 0; therefore, Circ œ Circ" Circ# œ 0 and Flux œ Flux" Flux# œ a# 1
a
32. F" œ aa# cos# tb i aa# sin# tb j ,
d r"
dt
œ (a sin t)i (a cos t)j Ê F" †
d r"
dt
œ a$ sin t cos# t a$ cos t sin# t
Ê Circ" œ '0 aa$ sin t cos# t a$ cos t sin# tb dt œ 2a3 ; M" œ a# cos# t, N" œ a# sin# t, dy œ a cos t dt,
1
$
dx œ a sin t dt Ê Flux" œ 'C M" dy N" dx œ '0 aa$ cos$ t a$ sin$ tb dt œ
1
F # œ t# i ,
d r#
dt
œ i Ê F# †
d r#
dt
œ t# Ê Circ# œ 'ca t# dt œ
a
2a$
3
4
3
a$ ;
; M# œ t# , N# œ 0, dy œ 0, dx œ dt
Ê Flux# œ 'C M# dy N# dx œ 0; therefore, Circ œ Circ" Circ# œ 0 and Flux œ Flux" Flux# œ
33. F" œ (a sin t)i (a cos t)j ,
d r"
dt
œ (a sin t)i (a cos t)j Ê F" †
d r"
dt
4
3
a$
œ a# sin# t a# cos# t œ a#
Ê Circ" œ '0 a# dt œ a# 1 ; M" œ a sin t, N" œ a cos t, dx œ a sin t dt, dy œ a cos t dt
1
Ê Flux" œ 'C M" dy N" dx œ '0 aa# sin t cos t a# sin t cos tb dt œ 0; F# œ tj ,
1
dr#
dt
œ i Ê F# †
d r#
dt
œ0
Ê Circ# œ 0; M# œ 0, N# œ t, dx œ dt, dy œ 0 Ê Flux# œ 'C M# dy N# dx œ 'ca t dt œ 0; therefore,
a
Circ œ Circ" Circ# œ a# 1 and Flux œ Flux" Flux# œ 0
34. F" œ aa# sin# tb i aa# cos# tb j ,
d r"
dt
œ (a sin t)i (a cos t)j Ê F" †
Ê Circ" œ '0 aa$ sin$ t a$ cos$ tb dt œ
1
4
3
d r"
dt
œ a$ sin$ t a$ cos$ t
a$ ; M" œ a# sin# t, N" œ a# cos# t, dy œ a cos t dt, dx œ a sin t dt
Ê Flux" œ 'C M" dy N" dx œ '0 aa$ cos t sin# t a$ sin t cos# tb dt œ
1
2
3
a$ ; F# œ t# j ,
d r#
dt
œ i Ê F# †
d r#
dt
œ0
Ê Circ# œ 0; M# œ 0, N# œ t# , dy œ 0, dx œ dt Ê Flux# œ 'C M# dy N# dx œ 'ca t# dt œ 23 a$ ; therefore,
a
Circ œ Circ" Circ# œ
4
3
a$ and Flux œ Flux" Flux# œ 0
35. (a) r œ (cos t)i (sin t)j , 0 Ÿ t Ÿ 1, and F œ (x y)i ax# y# b j Ê
F œ (cos t sin t)i acos# t sin# tb j Ê F †
dr
dt
(b) r œ (1 2t)i , 0 Ÿ t Ÿ 1, and F œ (x y)i
F†
dr
dt
œ 4t 2 Ê 'C F † T ds œ '0 (4t
1
œ (sin t)i (cos t)j and
œ sin t cos t sin# t cos t Ê 'C F † T ds
œ '0 a sin t cos t sin# t cos tb dt œ 2" sin# t
1
dr
dt
sin 2t
1
‘1
4 sin t ! œ #
ax# y# b j Ê ddtr œ 2i and F œ (1
"
2) dt œ c2t# 2td ! œ 0
t
#
2t)i (1 2t)# j Ê
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 16.2 Vector Fields, Work, Circulation, and Flux
(c) r" œ (1 t)i tj , 0 Ÿ t Ÿ 1, and F œ (x y)i ax# y# b j Ê
Ê F†
d r"
dt
œ (2t 1) a1 2t 2t# b œ 2t# Ê Flow" œ 'C F †
d r"
dt
"
#
#
0 Ÿ t Ÿ 1, and F œ (x y)i ax y b j Ê
œ i a2t# 2t 1b j Ê F †
"
œ t# 23 t$ ‘ ! œ
"
3
d r#
dt
œ i j and F œ (1 2t)i a1 2t 2t# b j
d r"
dt
œ '0 2t# dt œ
1
#
2
3
; r# œ ti (t 1)j ,
#
œ i j and F œ i at t 2t 1b j
œ 1 a2t# 2t 1b œ 2t 2t# Ê Flow# œ 'C F †
dr #
dt
949
#
Ê Flow œ Flow" Flow# œ
2
3
"
3
dr #
dt
œ '0 a2t 2t# b dt
1
œ1
36. From (1ß 0) to (0ß 1): r" œ (1 t)i tj , 0 Ÿ t Ÿ 1, and F œ (x y)i ax# y# b j Ê
d r"
dt
œ i j ,
F œ i a1 2t 2t# b j , and n" kv" k œ i j Ê F † n" kv" k œ 2t 2t# Ê Flux" œ '0 a2t 2t# b dt
1
"
œ t# 23 t$ ‘ ! œ
"
3
;
From (0ß 1) to (1ß 0): r# œ ti (1 t)j , 0 Ÿ t Ÿ 1, and F œ (x y)i ax# y# b j Ê
d r#
dt
œ i j ,
#
F œ (1 2t)i a1 2t 2t b j , and n# kv# k œ i j Ê F † n# kv# k œ (2t 1) a1 2t 2t# b œ 2 4t 2t#
Ê Flux# œ '0 a2 4t 2t# b dt œ 2t 2t# 23 t$ ‘ ! œ 23 ;
1
"
From (1ß 0) to (1ß 0): r$ œ (1 2t)i , 0 Ÿ t Ÿ 1, and F œ (x y)i ax# y# b j Ê
#
d r$
dt
œ 2i ,
#
F œ (1 2t)i a1 4t 4t b j , and n$ kv$ k œ 2j Ê F † n$ kv$ k œ 2 a1 4t 4t b
Ê Flux$ œ 2 '0 a1 4t 4t# b dt œ 2 t 2t# 43 t$ ‘ ! œ
1
"
37. (a) y œ 2x, 0 Ÿ x Ÿ 2 Ê ratb œ ti 2tj , 0 Ÿ t Ÿ 2 Ê
œ 4t2 8t2 œ 12t2 Ê Flow œ 'C F †
dr
dt
2
3
2
3
œ
œ Ša2tb2 i 2atba2tbj‹ † ai 2jb
2
dr
dt
œ Šat2 b i 2atbat2 bj‹ † ai 2tjb
dr
dt
œ i 2tj Ê F †
2
dt œ '0 5t4 dt œ ct5 d ! œ 32
2
dr
dt
2
œ Šˆ "# t3 ‰ i 2atbˆ "# t3 ‰j‹ † ai 3t2 jb œ 14 t6 32 t6 œ 74 t6 Ê Flow œ 'C F †
2
dr
dt
dr
dt
œ i 3t2 j
dt œ '0 74 t6 dt œ 14 t7 ‘ !
2
2
œ 32
38. (a) C1 : ratb œ a1 tbi j , 0 Ÿ t Ÿ 2 Ê
C4 : ratb œ i at 1bj , 0 Ÿ t Ÿ 2 Ê
Ê Flow œ 'C F †
dr
dt
dt œ 'C F †
1
dr
dt
œ i Ê F †
dr
dt
C2 : ratb œ i a1 tbj , 0 Ÿ t Ÿ 2 Ê
C3 : ratb œ at 1bi j , 0 Ÿ t Ÿ 2 Ê
dr
dt
dr
dt
dr
dt
dr
dt
œ j Ê F †
œiÊF†
œjÊF†
dt 'C F †
dr
dt
2
dr
dt
dr
dt
œ aa1bi aa1 tb 2a1bbjb † aib œ 1;
dr
dt
œ aa1 tbi aa1b 2a1 tbbjb † ajb œ 2t 1;
œ aa1bi aat 1b 2a1bbjb † aib œ 1;
œ aat 1bi aa1b 2at 1bbjb † ajb œ 2t 1;
dt 'C F †
3
dr
dt
dt 'C F †
4
dr
dt
dt
œ '0 a1b dt '0 a2t 1b dt '0 a1b dt '0 a2t 1b dt œ ctd 2! ct2 td ! ctd !2 ct2 td !
2
2
2
2
2
œ 2 2 2 2 œ 0
(b) x2 y2 œ 4 Ê ratb œ a2cos tbi a2sin tbj , 0 Ÿ t Ÿ 21 Ê
ÊF†
dr
dt
dr
dt
2
œ a2sin tbi a2cos tbj
œ aa2sin tbi a2cos t 2a2sin tbbjb † aa2sin tbi a2cos tbjb œ 4sin2 t 4cos2 t 8sin t cos t
œ 4cos 2t 4sin 2t Ê Flow œ 'C F †
dr
dt
(c) answers will vary, one possible path is:
C1 : ratb œ ti , 0 Ÿ t Ÿ 1 Ê ddtr œ i Ê F †
C2 : ratb œ a1 tbi tj , 0 Ÿ t Ÿ 1 Ê
C3 : ratb œ a1 tbj , 0 Ÿ t Ÿ 1 Ê
dr
dt
"
3
2
(c) answers will vary, one possible path is y œ 12 x3 , 0 Ÿ x Ÿ 2 Ê ratb œ ti "# t3 j , 0 Ÿ t Ÿ 2 Ê
ÊF†
dr
dt
œ i 2tj Ê F †
dr
dt
"
3
dt œ '0 12t2 dt œ c4t3 d ! œ 32
dr
dt
(b) y œ x2 , 0 Ÿ x Ÿ 2 Ê ratb œ ti t2 j , 0 Ÿ t Ÿ 2 Ê
œ t4 4t4 œ 5t4 Ê Flow œ 'C F †
Ê Flux œ Flux" Flux# Flux$ œ
2
3
dr
dt
dt œ '0 a4cos 2t 4sin 2tb dt œ c2sin 2t 2cos 2td 2!1 œ 0
21
dr
dt
œ aa0bi at 2a1bbjb † aib œ 0;
œ i j Ê F †
œ j Ê F †
dr
dt
dr
dt
œ ati aa1 tb 2tbjb † ai jb œ 1;
œ aa1 tbi a0 2a1 tbbjb † ajb œ 2t 1;
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
950
Chapter 16 Integration in Vector Fields
Ê Flow œ 'C F †
dt œ 'C F †
dr
dt
1
dr
dt
dt 'C F †
2
dr
dt
dt 'C F †
3
dr
dt
dt œ '0 a0b dt '0 a1b dt '0 a2t 1b dt
1
1
1
1
œ 0 ctd 1! ct2 td ! œ 1 a1b œ 0
39. F œ Èx#y y# i
j on x# y# œ 4;
x
È x# y#
at (2ß 0), F œ j ; at (0ß 2), F œ i ; at (2ß 0),
È
F œ j ; at (!ß 2), F œ i ; at ŠÈ2ß È2‹ , F œ #3 i #" j ;
at ŠÈ2ß È2‹ , F œ
Fœ
È3
#
È3
#
i #" j ; at ŠÈ2ß È2‹ ,
i #" j ; at ŠÈ2ß È2‹ , F œ
È3
#
i #" j
40. F œ xi yj on x# y# œ 1; at (1ß 0), F œ i ;
at (1ß 0), F œ i ; at (0ß 1), F œ j ; at (0ß 1),
F œ j ; at Š "# ß
at Š "# ß
È3
# ‹,
at Š "# ß
È3
# ‹,
at Š "# ß
È3
# ‹,
Fœ
"
#
F œ "# i
Fœ
È3
# ‹,
"
#
i
i
È3
#
È3
#
È3
#
j;
j;
j;
F œ "# i
È3
#
j.
41. (a) G œ P(xß y)i Q(xß y)j is to have a magnitude Èa# b# and to be tangent to x# y# œ a# b# in a
counterclockwise direction. Thus x# y# œ a# b# Ê 2x 2yyw œ 0 Ê yw œ xy is the slope of the tangent
line at any point on the circle Ê yw œ ba at (aß b). Let v œ bi aj Ê kvk œ Èa# b# , with v in a
counterclockwise direction and tangent to the circle. Then let P(xß y) œ y and Q(xß y) œ x
Ê G œ yi xj Ê for (aß b) on x# y# œ a# b# we have G œ bi aj and kGk œ Èa# b# .
(b) G œ ˆÈx# y# ‰ F œ ŠÈa# b# ‹ F .
42. (a) From Exercise 41, part a, yi xj is a vector tangent to the circle and pointing in a counterclockwise
direction Ê yi xj is a vector tangent to the circle pointing in a clockwise direction Ê G œ Èyxi #xjy#
is a unit vector tangent to the circle and pointing in a clockwise direction.
(b) G œ F
43. The slope of the line through (xß y) and the origin is
pointing away from the origin Ê F œ
xi yj
È x# y#
y
x
Ê v œ xi yj is a vector parallel to that line and
is the unit vector pointing toward the origin.
44. (a) From Exercise 43, Èxxi #yjy# is a unit vector through (xß y) pointing toward the origin and we want
kFk to have magnitude Èx# y# Ê F œ Èx# y# Š Èxxi #yjy# ‹ œ xi yj .
(b) We want kFk œ
C
È x# y#
where C Á 0 is a constant Ê F œ
C
È x# y#
yj
Š Èxxi #yjy# ‹ œ C Š xx#i
y# ‹.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 16.2 Vector Fields, Work, Circulation, and Flux
951
45. Yes. The work and area have the same numerical value because work œ 'C F † dr œ 'C yi † dr
œ 'b [f(t)i] † i
a
df
dt
j‘ dt
[On the path, y equals f(t)]
œ 'a f(t) dt œ Area under the curve
b
46. r œ xi yj œ xi f(x)j Ê
from the origin Ê F †
'C
Ê
dr
dx
dr
dx
œ
F † T ds œ 'C F †
[because f(t) 0]
œ i f w (x)j ; F œ
k†y†f (x)
È x# y#
w
kx
È x# y#
dx œ 'a k
b
dr
dx
d
dx
k
È x# y#
œ
(xi yj) has constant magnitude k and points away
kx k†f(x)†f (x)
Èx# [f(x)]#
w
œk
d
dx
Èx# [f(x)]# , by the chain rule
Èx# [f(x)]# dx œ k Èx# [f(x)]# ‘ b
a
œ k ˆÈb# [f(b)]# Èa# [f(a)]# ‰ , as claimed.
47. F œ 4t$ i 8t# j 2k and
48. F œ 12t# j 9t# k and
dr
dt
œ i 2tj Ê F †
dr
dt
œ 3j 4k Ê F †
49. F œ (cos t sin t)i (cos t)k and
dr
dt
dr
dt
œ 12t$ Ê Flow œ '0 12t$ dt œ c3t% d ! œ 48
2
œ 72t# Ê Flow œ '0 72t# dt œ c24t$ d ! œ 24
1
œ ( sin t)i (cos t)k Ê F †
dr
dt
#
dr
dt
"
œ sin t cos t 1
Ê Flow œ '0 ( sin t cos t 1) dt œ 2" cos# t t‘ ! œ ˆ #" 1‰ ˆ #" 0‰ œ 1
1
1
50. F œ (2 sin t)i (2 cos t)j 2k and
dr
dt
œ (2 sin t)i (2 cos t)j 2k Ê F †
dr
dt
œ 4 sin# t 4 cos# t 4 œ 0
Ê Flow œ 0
1
#
51. C" : r œ (cos t)i (sin t)j tk , 0 Ÿ t Ÿ
Ê F†
dr
dt
Ê F œ (2 cos t)i 2tj (2 sin t)k and
1Î2
C# : r œ j
1
#
1Î#
( sin 2t 2t cos t 2 sin t) dt œ 2" cos 2t 2t sin t 2 cos t 2 cos t‘ !
(1 t)k , 0 Ÿ t Ÿ 1 Ê F œ 1(1 t)j 2k and
Ê Flow# œ '0 1 dt œ
1
c1td "!
Ê Flow$ œ '0 2t dt œ
1
œx
dx
dt
y
dy
dt
z
"
ct# d !
dz
dt
dr
dt
œ 1# k Ê F †
dr
dt
œ 1 1;
œ 1
œ 1 ;
C$ : r œ ti (1 t)j , 0 Ÿ t Ÿ 1 Ê F œ 2ti 2(1 t)k and
dr
dt
œ ( sin t)i (cos t)j k
œ 2 cos t sin t 2t cos t 2 sin t œ sin 2t 2t cos t 2 sin t
Ê Flow" œ '0
52. F †
dr
dt
dr
dt
œij Ê F†
dr
dt
œ 2t
œ 1 Ê Circulation œ (1 1) 1 1 œ 0
œ
` f dx
` x dt
` f dy
` y dt
by the chain rule Ê Circulation œ 'C F †
dr
dt
` f dz
` z dt
dt œ 'a
, where f(xß yß z) œ
b
d
dt afaratbbb
"
#
ax# y# x# b Ê F †
dr
dt
œ
d
dt afaratbbb
dt œ farabbb faraabb. Since C is an entire ellipse,
rabb œ raab, thus the Circulation œ 0.
53. Let x œ t be the parameter Ê y œ x# œ t# and z œ x œ t Ê r œ ti t# j tk , 0 Ÿ t Ÿ 1 from (0ß 0ß 0) to (1ß 1ß 1)
Ê
œ
dr
dt
œ i 2tj k and F œ xyi yj yzk œ t$ i t# j t$ k Ê F †
œ t$ 2t$ t$ œ 2t$ Ê Flow œ '0 2t$ dt
1
"
#
54. (a) F œ ™ axy# z$ b Ê F †
œ 'a
(b)
dr
dt
b
d
dt afaratbbb
dr
dt
œ
` f dx
` x dt
` f dy
` y dt
` z dz
` z dt
œ
df
dt
, where f(xß yß z) œ xy# z$ Ê )C F †
dr
dt
dt œ farabbb faraabb œ 0 since C is an entire ellipse.
Ð2ß1ß 1Ñ
'C F † ddtr œ 'Ð1ß1ß1Ñ
d
dt
Ð#ß"ß"Ñ
axy# z$ b dt œ cxy# z$ d Ð"ß"ß"Ñ œ (2)(1)# (1)$ (1)(1)# (1)$ œ 2 1 œ 3
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
dt
952
Chapter 16 Integration in Vector Fields
55-60. Example CAS commands:
Maple:
with( LinearAlgebra );#55
F := r -> < r[1]*r[2]^6 | 3*r[1]*(r[1]*r[2]^5+2) >;
r := t -> < 2*cos(t) | sin(t) >;
a,b := 0,2*Pi;
dr := map(diff,r(t),t);
# (a)
F(r(t));
# (b)
q1 := simplify( F(r(t)) . dr ) assuming t::real;
# (c)
q2 := Int( q1, t=a..b );
value( q2 );
Mathematica: (functions and bounds will vary):
Exercises 55 and 56 use vectors in 2 dimensions
Clear[x, y, t, f, r, v]
f[x_, y_]:= {x y6 , 3x (x y5 2)}
{a, b}={0, 21};
x[t_]:= 2 Cos[t]
y[t_]:= Sin[t]
r[t_]:={x[t], y[t]}
v[t_]:= r'[t]
integrand= f[x[t], y[t]] . v[t] //Simplify
Integrate[integrand,{t, a, b}]
N[%]
If the integration takes too long or cannot be done, use NIntegrate to integrate numerically. This is suggested for exercises
57 - 60 that use vectors in 3 dimensions. Be certain to leave spaces between variables to be multiplied.
Clear[x, y, z, t, f, r, v]
f[x_, y_, z_]:= {y y z Cos[x y z], x2 x z Cos[x y z], z x y Cos[x y z]}
{a, b}={0, 21};
x[t_]:= 2 Cos[t]
y[t_]:= 3 Sin[t]
z[t_]:= 1
r[t_]:={x[t], y[t], z[t]}
v[t_]:= r'[t]
integrand= f[x[t], y[t],z[t]] . v[t] //Simplify
NIntegrate[integrand,{t, a, b}]
16.3 PATH INDEPENDENCE, POTENTIAL FUNCTIONS, AND CONSERVATIVE FIELDS
1.
`P
`y
œxœ
`N
`z
2.
`P
`y
œ x cos z œ
3.
`P
`y
œ 1 Á 1 œ
5.
`N
`x
œ0Á1œ
6.
`P
`y
œ0œ
`N
`z
,
,
`M
`z
`N
`z
œyœ
,
`N
`z
`M
`y
`M
`z
`M
`z
`P
`x
,
`N
`x
`M
`y
œzœ
œ y cos z œ
`P
`x
,
`N
`x
Ê Conservative
œ sin z œ
Ê Not Conservative
`M
`y
4.
Ê Conservative
`N
`x
œ 1 Á 1 œ
`M
`y
Ê Not Conservative
Ê Not Conservative
œ0œ
`P
`x
,
`N
`x
œ ex sin y œ
`M
`y
Ê Conservative
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 16.3 Path Independence, Potential Functions, and Conservative Fields
7.
`f
`x
`f
`z
Ê
8.
`f
`x
`f
`y
œ 2x Ê f(xß yß z) œ x# g(yß z) Ê
`f
`z
œ xe
`f
`x
h(z) Ê
`f
`z
œ 2xe
y2z
`f
`y œ
y2z
`f
`y
œ y sin z Ê f(xß yß z) œ xy sin z g(yß z) Ê
`f
`z
œ
Ê f(xß yß z) œ
z
y # z#
"
#
œ
y
1 x# y#
`g
`y
œ
z
È 1 y # z#
Ê
`f
`z
œ
y
È 1 y # z#
`g
`y
œxz Ê
œ z Ê g(yß z) œ zy h(z)
w
`g
`y
`g
`y
œ xey2z Ê
œ 0 Ê f(xß yß z)
w
Ê h (z) œ 0 Ê h(z) œ C Ê f(xß yß z) œ xey2z C
œ x sin z
`g
`y
`g
`y
œ x sin z Ê
œ 0 Ê g(yß z) œ h(z)
w
`f
`x
ln ay# z# b g(xß y) Ê
"
#
`g
`x œ
#
œ
ln x sec# (x y) Ê g(xß y)
ln ay# z b (x ln x x) tan (x y) h(y)
y)
Ê f(xß yß z) œ tan" (xy) g(yß z) Ê
Ê
h(z)
œ xy cos z h (z) œ xy cos z Ê h (z) œ 0 Ê h(z) œ C Ê f(xß yß z)
Ê `` yf œ y# y z# sec# (x y) hw (y) œ sec# (x
œ "# ln ay# z# b (x ln x x) tan (x y) C
`f
`x
3y#
#
2z# C
w
œ (x ln x x) tan (x y) h(y) Ê f(xß yß z) œ
12.
`g
`y
xey2z
h (z) œ 2xe
œ xy sin z C
`f
`z
œx
h(z) Ê f(xß yß z) œ x#
œ x y h (z) œ x y Ê h (z) œ 0 Ê h(z) œ C Ê f(xß yß z)
w
Ê f(xß yß z) œ xy sin z h(z) Ê
11.
`f
`y
3y
#
#
3y#
#
w
œ ey2z Ê f(xß yß z) œ xey2z g(yß z) Ê
y2z
10.
œ 3y Ê g(yß z) œ
œ y z Ê f(xß yß z) œ (y z)x g(yß z) Ê
œ (y z)x zy C
`f
`x
`g
`y
œ hw (z) œ 4z Ê h(z) œ 2z# C Ê f(xß yß z) œ x#
Ê f(xß yß z) œ (y z)x zy h(z) Ê
9.
œ
953
`f
`y
y
y# z#
œ
Ê hw (y) œ 0 Ê h(y) œ C Ê f(xß yß z)
x
1 x# y#
`g
`y
œ
x
1 x# y#
z
È1 y# z#
Ê g(yß z) œ sin" (yz) h(z) Ê f(xß yß z) œ tan" (xy) sin" (yz) h(z)
hw (z) œ
y
È 1 y # z#
"
z
Ê hw (z) œ
"
z
Ê h(z) œ ln kzk C
Ê f(xß yß z) œ tan" (xy) sin" (yz) ln kzk C
13. Let F(xß yß z) œ 2xi 2yj 2zk Ê
exact;
Ê
`f
`x
`f
`z
`P
`y
`N
`z
`M
`P `N
`M
`z œ 0 œ `x , `x œ 0 œ `y
`g
`f
#
` y œ ` y œ 2y Ê g(yß z) œ y
œ0œ
#
œ 2x Ê f(xß yß z) œ x g(yß z) Ê
œ f(2ß 3ß 6) f(!ß !ß !) œ 2# 3# (6)# œ 49
exact;
`f
`x
`N
`z
œxœ
œ yz Ê f(xß yß z) œ xyz g(yß z) Ê
œ xyz h(z) Ê
Ê
`P
`y
Ð3ß5ß0Ñ
'Ð1ß1ß2Ñ
`f
`z
w
,
`f
`y
`M
`z
œyœ
œ xz
`g
`y
`P
`x
,
`N
`x
œzœ
œ xz Ê
`g
`y
h(z) Ê f(xß yß z) œ x# y# œ h(z)
'Ð0Ð2ß0ß3ß0ßÑ 6Ñ 2x dx 2y dy 2z dz
œ hw (z) œ 2z Ê h(z) œ z# C Ê f(xß yß z) œ x# y# z# C Ê
14. Let F(xß yß z) œ yzi xzj xyk Ê
Ê M dx N dy P dz is
,
`M
`y
Ê M dx N dy P dz is
œ 0 Ê g(yß z) œ h(z) Ê f(xß yß z)
w
œ xy h (z) œ xy Ê h (z) œ 0 Ê h(z) œ C Ê f(xß yß z) œ xyz C
yz dx xz dy xy dz œ f(3ß 5ß 0) f(1ß 1ß 2) œ 0 2 œ 2
15. Let F(xß yß z) œ 2xyi ax# z# b j 2yzk Ê
Ê M dx N dy P dz is exact;
`f
`x
`P
`y
œ 2z œ
`N
`z
,
`M
`z
œ0œ
`P
`x
œ 2xy Ê f(xß yß z) œ x# y g(yß z) Ê
Ê g(yß z) œ yz# h(z) Ê f(xß yß z) œ x# y yz# h(z) Ê
`f
`z
,
`N
`x
`f
`y
w
œ 2x œ
œ x#
`g
`y
`M
`y
œ x# z# Ê
`g
`y
œ z#
œ 2yz h (z) œ 2yz Ê hw (z) œ 0 Ê h(z) œ C
Ê f(xß yß z) œ x# y yz# C Ê 'Ð0ß0ß0Ñ 2xy dx ax# z# b dy 2yz dz œ f("ß #ß $) f(!ß !ß !) œ 2 2(3)# œ 16
Ð1ß2ß3Ñ
16. Let F(xß yß z) œ 2xi y# j ˆ 1 4 z# ‰ k Ê
Ê M dx N dy P dz is exact;
`f
`x
`P
`y
œ0œ
`N
`z
,
`M
`z
œ0œ
`P
`x
,
`N
`x
œ 2x Ê f(xß yß z) œ x# g(yß z) Ê
œ0œ
`f
`y
œ
`M
`y
`g
`y
$
œ y# Ê g(yß z) œ y3 h(z)
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
954
Chapter 16 Integration in Vector Fields
Ê f(xß yß z) œ x#
y$
3
`f
`z
h(z) Ê
œ hw (z) œ 1 4 z# Ê h(z) œ 4 tan" z C Ê f(xß yß z)
œ x#
y$
3
4 tan" z C Ê 'Ð0ß0ß0Ñ 2x dx y# dy
œ ˆ9
27
3
4 † 14 ‰ (! ! 0) œ 1
Ð3ß3ß1Ñ
17. Let F(xß yß z) œ (sin y cos x)i (cos y sin x)j k Ê
Ê M dx N dy P dz is exact;
`g
`y
œ cos y sin x Ê
`f
`x
4
1 z#
`P
`y
dz œ f(3ß 3ß 1) f(!ß !ß !)
œ0œ
`N
`z
`M
`z
,
`P
`x
œ0œ
,
`N
`x
œ cos y cos x œ
`f
`y
œ sin y cos x Ê f(xß yß z) œ sin y sin x g(yß z) Ê
`f
`z
œ 0 Ê g(yß z) œ h(z) Ê f(xß yß z) œ sin y sin x h(z) Ê
`M
`y
œ cos y sin x
`g
`y
œ hw (z) œ 1 Ê h(z) œ z C
Ê f(xß yß z) œ sin y sin x z C Ê 'Ð1ß0ß0Ñ sin y cos x dx cos y sin x dy dz œ f(0ß 1ß 1) f(1ß !ß !)
Ð0ß1ß1Ñ
œ (0 1) (0 0) œ 1
18. Let F(xß yß z) œ (2 cos y)i Š "y 2x sin y‹ j ˆ "z ‰ k Ê
Ê M dx N dy P dz is exact;
"
y
œ
`g
`y
2x sin y Ê
"
y
œ
`f
`x
`P
`y
`N
`z
œ0œ
`M
`z
,
œ0œ
`P
`x
œ 2 cos y Ê f(xß yß z) œ 2x cos y g(yß z) Ê
,
`f
`y
`N
`x
œ 2 sin y œ
œ 2x sin y
`f
`z
Ê g(yß z) œ ln kyk h(z) Ê f(xß yß z) œ 2x cos y ln kyk h(z) Ê
`M
`y
`g
`y
œ hw (z) œ
"
z
Ê h(z) œ ln kzk C Ê f(xß yß z) œ 2x cos y ln kyk ln kzk C
Ê 'Ð0ß2ß1Ñ
Ð1ß1Î2ß2Ñ
2 cos y dx Š "y 2x sin y‹ dy
œ ˆ2 † 0 ln
1
#
"
z
dz œ f ˆ1ß 1# ß 2‰ f(!ß #ß ")
ln 2‰ (0 † cos 2 ln 2 ln 1) œ ln
#
`P
`y
19. Let F(xß yß z) œ 3x# i Š zy ‹ j (2z ln y)k Ê
Ê M dx N dy P dz is exact;
`f
`x
œ
2z
y
1
#
`N
`z
œ
`M
`z
,
œ0œ
`P
`x
`f
`y
œ 3x# Ê f(xß yß z) œ x$ g(yß z) Ê
Ê f(xß yß z) œ x$ z# ln y h(z) Ê
œ x$ z# ln y C Ê 'Ð1ß1ß1Ñ 3x# dx
Ð1ß2ß3Ñ
`N
`x
,
œ0œ
œ
`g
`y
œ
`M
`y
z#
y
Ê g(yß z) œ z# ln y h(z)
`f
`z
œ 2z ln y hw (z) œ 2z ln y Ê hw (z) œ 0 Ê h(z) œ C Ê f(xß yß z)
z#
y
dy 2z ln y dz œ f(1ß 2ß 3) f("ß "ß ")
œ (1 9 ln 2 C) (1 0 C) œ 9 ln 2
#
`P
`y
20. Let F(xß yß z) œ (2x ln y yz)i Š xy xz‹ j (xy)k Ê
Ê M dx N dy P dz is exact;
x#
y
œ
xz Ê
`g
`y
`f
`x
œ x œ
`N
`z
,
`M
`z
œ y œ
`P
`x
,
`N
`x
œ 2x ln y yz Ê f(xß yß z) œ x# ln y xyz g(yß z) Ê
`f
`z
œ 0 Ê g(yß z) œ h(z) Ê f(xß yß z) œ x# ln y xyz h(z) Ê
œ
2x
y
`f
`y
œ
zœ
x#
y
`M
`y
xz
`g
`y
œ xy hw (z) œ xy Ê hw (z) œ 0
Ê h(z) œ C Ê f(xß yß z) œ x# ln y xyz C Ê 'Ð1ß2ß1Ñ (2x ln y yz) dx Š xy xz‹ dy xy dz
Ð2ß1ß1Ñ
#
œ f(2ß 1ß 1) f("ß 2ß 1) œ (4 ln 1 2 C) (ln 2 2 C) œ ln 2
21. Let F(xß yß z) œ Š "y ‹ i Š 1z
x
y# ‹ j
Ê M dx N dy P dz is exact;
Ê
`g
`y
œ
"
z
Ê g(yß z) œ
Ê f(xß yß z) œ
x
y
y
z
y
z
ˆ zy# ‰ k Ê
`f
`x
œ
"
y
Ð2ß2ß2Ñ
"
y
œ z"# œ
Ê f(xß yß z) œ
h(z) Ê f(xß yß z) œ
C Ê 'Ð1ß1ß1Ñ
`P
`y
x
y
dx Š 1z
y
z
x
y# ‹
x
y
`N
`z
`M
`z
,
œ0œ
`f
`y
zy#
g(yß z) Ê
`f
`z
h(z) Ê
dy
y
z#
œ
`P
`x
,
`N
`x
œ y1# œ
œ yx#
`g
`y
œ
"
z
Ê
`f
`x
œ
`P
`y
2xi 2yj 2zk
x # y # z#
œ 4yz
œ
3%
2x
x # y # z#
`N
`z
,
`M
`z
Šand let 3# œ x# y# z# Ê
œ 4xz
œ
3%
#
`P
`x
#
,
`N
`x
œ 4xy
œ
3%
#
`3
`x
dz œ f(2ß 2ß 2) f("ß 1ß 1) œ ˆ 2#
`M
`y
Ê f(xß yß z) œ ln ax y z b g(yß z) Ê
œ
x
3
,
`3
`y
œ
y
3
,
`3
`z
œ 3z ‹
Ê M dx N dy P dz is exact;
`f
`y
œ
2y
x # y # z#
x
y#
hw (z) œ zy# Ê hw (z) œ 0 Ê h(z) œ C
œ0
22. Let F(xß yß z) œ
`M
`y
`g
`y
œ
2y
x # y # z#
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
2
#
C‰ ˆ "1
"
1
C‰
Section 16.3 Path Independence, Potential Functions, and Conservative Fields
Ê
œ
`g
`y œ 0
2z
x # y # z#
`f
`z
Ê g(yß z) œ h(z) Ê f(xß yß z) œ ln ax# y# z# b h(z) Ê
Ð2ß2ß2Ñ
Ê 'Ð 1ß 1ß 1Ñ
œ
955
hw (z)
2z
x # y# z#
Ê hw (z) œ 0 Ê h(z) œ C Ê f(xß yß z) œ ln ax# y# z# b C
2x dx 2y dy 2z dz
x # y # z#
œ f(2ß 2ß 2) f("ß 1ß 1) œ ln 12 ln 3 œ ln 4
23. r œ (i j k) t(i 2j 2k) œ (1 t)i (1 2t)j (1 2t)k, 0 Ÿ t Ÿ 1 Ê dx œ dt, dy œ 2 dt, dz œ 2 dt
Ð2ß3ß 1Ñ
Ê 'Ð1ß1ß1Ñ y dx x dy 4 dz œ '0 (2t 1) dt (t 1)(2 dt) 4(2) dt œ '0 (4t 5) dt œ c2t# 5td ! œ 3
1
1
24. r œ t(3j 4k), 0 Ÿ t Ÿ 1 Ê dx œ 0, dy œ 3 dt, dz œ 4 dt Ê
' 000304
Ð ß ß Ñ
Ð ß ß Ñ
"
#
x# dx yz dy Š y# ‹ dz
œ '0 a12t# b (3 dt) Š 9t# ‹ (4 dt) œ '0 54t# dt œ c18t# d ! œ 18
1
25.
`P
`y
1
#
œ0œ
`N
`z
,
`M
`z
œ 2z œ
`P
`x
,
`N
`x
,
`M
`z
"
`M
`y
œ0œ
Ê M dx N dy P dz is exact Ê F is conservative
Ê path independence
26.
`P
`y
œ ˆÈ
yz
x # y# z# ‰
œ
$
`N
`z
œ ˆÈ
xz
$
x # y# z# ‰
œ
`P
`x
,
`N
`x
œ ˆÈ
xy
x # y# z# ‰
$
œ
`M
`y
Ê M dx N dy P dz is exact Ê F is conservative Ê path independence
27.
`P
`y
`f
`x
œ0œ
œ
2x
y
`N
`z
,
œ0œ
Ê f(xß y) œ
Ê f(xß y) œ
28.
`M
`z
x#
y
"
y
`N
`z
,
`M
`z
#
x
y
`P
`x
`N
`x
,
œ 2x
y# œ
œ xy# gw (y) œ
C Ê F œ ™ Šx
`P
`x
`N
`x
#
œ cos z œ
`f
`x
œ ex ln y Ê f(xß yß z) œ ex ln y g(yß z) Ê
,
œ
ex
y
1 x#
y#
"
y#
Ê gw (y) œ
Ê g(y) œ "y C
1
y ‹
`P
`y
œ0œ
Ê F is conservative Ê there exists an f so that F œ ™ f;
#
`f
`y
g(y) Ê
`M
`y
œ
`M
`y
Ê F is conservative Ê there exists an f so that F œ ™ f;
`f
`y
œ
ex
y
œ y sin z h(z) Ê f(xß yß z) œ e ln y y sin z h(z) Ê
x
`g
ex
`y œ y
`f
`z œ y
x
`g
`y
sin z Ê
œ sin z Ê g(yß z)
w
cos z h (z) œ y cos z Ê hw (z) œ 0
Ê h(z) œ C Ê f(xß yß z) œ ex ln y y sin z C Ê F œ ™ ae ln y y sin zb
29.
`P
`y
`f
`x
œ0œ
`N
`z
,
`M
`z
#
`P
`x
œ x y Ê f(xß yß z) œ
Ê f(xß yß z) œ
œ
œ0œ
"
3
x$ xy
" $
3 x xy
" $
z
3 y ze
(a) work œ 'A F †
B
dr
dt
,
"
3
"
3
`N
`x
œ1œ
`M
`y
Ê F is conservative Ê there exists an f so that F œ ™ f;
`f
`y
$
x xy g(yß z) Ê
y$ h(z) Ê
œx
`g
`y
œ y# x Ê
`f
`z
`g
`y
z
œ y# Ê g(yß z) œ
"
3
y$ h(z)
œ hw (z) œ zez Ê h(z) œ zez e C Ê f(xß yß z)
ez C Ê F œ ™ ˆ "3 x$ xy 3" y$ zez ez ‰
dt œ 'A F † dr œ 3" x$ xy 3" y$ zez ez ‘ Ð"ß!ß!Ñ œ ˆ 3" 0 0 e e‰ ˆ 3" 0 0 1‰
B
Ð"ß!ß"Ñ
œ1
(b) work œ 'A F † dr œ "3 x$ xy 3" y$ zez ez ‘ Ð"ß!ß!Ñ œ 1
B
Ð"ß!ß"Ñ
(c) work œ 'A F † dr œ "3 x$ xy 3" y$ zez ez ‘ Ð"ß!ß!Ñ œ 1
B
Ð"ß!ß"Ñ
Note: Since F is conservative, 'A F † dr is independent of the path from (1ß 0ß 0) to (1ß 0ß 1).
B
30.
`P
`y
œ xeyz xyzeyz cos y œ
that F œ ™ f;
`f
`x
œe
yz
`N
`z
,
`M
`z
œ yeyz œ
`P
`x
,
`N
`x
œ zeyz œ
Ê f(xß yß z) œ xe g(yß z) Ê
yz
`f
`y
`M
`y
œ xze
Ê g(yß z) œ z sin y h(z) Ê f(xß yß z) œ xe z sin y h(z) Ê
yz
Ê F is conservative Ê there exists an f so
yz
`f
`z
`g
`y
œ xzeyz z cos y Ê
w
`g
`y
œ z cos y
œ xye sin y h (z) œ xyeyz sin y
yz
Ê hw (z) œ 0 Ê h(z) œ C Ê f(xß yß z) œ xeyz z sin y C Ê F œ ™ axeyz z sin yb
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
956
Chapter 16 Integration in Vector Fields
(a) work œ 'A F † dr œ cxeyz z sin yd Ð"ß!ß"Ñ
B
Ð"ß1Î#ß!Ñ
œ (1 0) (1 0) œ 0
(b) work œ 'A F † dr œ cxeyz z sin yd Ð"ß!ß"Ñ
B
Ð"ß1Î#ß!Ñ
(c) work œ 'A F † dr œ cxeyz z sin yd Ð"ß!ß"Ñ
B
Ð"ß1Î#ß!Ñ
œ0
œ0
Note: Since F is conservative, 'A F † dr is independent of the path from (1ß 0ß 1) to ˆ1ß 1# ß 0‰ .
B
31. (a) F œ ™ ax$ y# b Ê F œ 3x# y# i 2x$ yj ; let C" be the path from (1ß 1) to (0ß 0) Ê x œ t 1 and
y œ t 1, 0 Ÿ t Ÿ 1 Ê F œ 3(t 1)# (t 1)# i 2(t 1)$ (t 1)j œ 3(t 1)% i 2(t 1)% j
and r" œ (t 1)i (t 1)j Ê dr" œ dt i dt j Ê
'C
"
F † dr" œ '0 c3(t 1)% 2(t 1)% d dt
1
1
"
œ '0 5(t 1)% dt œ c(t 1)& d ! œ 1; let C# be the path from (0ß 0) to (1ß 1) Ê x œ t and y œ t,
1
0 Ÿ t Ÿ 1 Ê F œ 3t% i 2t% j and r# œ ti tj Ê dr# œ dt i dt j Ê 'C F † dr# œ '0 a3t% 2t% b dt
1
œ '0 5t% dt œ 1
Ê 'C F † dr œ 'C F † dr" 'C
"
#
#
F † dr# œ 2
Ð1ß1Ñ
(b) Since f(xß y) œ x$ y# is a potential function for F, 'Ð 1ß1Ñ F † dr œ f(1ß 1) f(1ß 1) œ 2
32.
`P
`y
`f
`x
œ0œ
`N
`z
,
`M
`z
œ0œ
`P
`x
,
`N
`x
œ 2x sin y œ
#
œ 2x cos y Ê f(xß yß z) œ x cos y g(yß z) Ê
#
Ê f(xß yß z) œ x cos y h(z) Ê
(a)
(b)
(c)
(d)
`M
`y
`f
`z
Ê F is conservative Ê there exists an f so that F œ ™ f;
`f
`y
œ x# sin y
w
`g
`y
œ x# sin y Ê
`g
`y
œ 0 Ê g(yß z) œ h(z)
#
œ h (z) œ 0 Ê h(z) œ C Ê f(xß yß z) œ x cos y C Ê F œ ™ ax# cos yb
'C 2x cos y dx x# sin y dy œ cx# cos yd Ð!ß"Ñ
Ð"ß!Ñ œ 0 1 œ 1
'C 2x cos y dx x# sin y dy œ cx# cos yd Ð"ß!Ñ
Ð"ß1Ñ œ 1 (1) œ 2
'C 2x cos y dx x# sin y dy œ cx# cos yd Ð"ß!Ñ
Ð"ß!Ñ œ 1 1 œ 0
'C 2x cos y dx x# sin y dy œ cx# cos yd Ð"ß!Ñ
Ð"ß!Ñ œ 1 1 œ 0
33. (a) If the differential form is exact, then
all x, and
`N
`x
œ
`M
`y
`P
`y
œ
`N
`z
Ê 2ay œ cy for all y Ê 2a œ c,
`M
`z
œ
`P
`x
Ê 2cx œ 2cx for
Ê by œ 2ay for all y Ê b œ 2a and c œ 2a
(b) F œ ™ f Ê the differential form with a œ 1 in part (a) is exact Ê b œ 2 and c œ 2
34. F œ ™ f Ê g(xß yß z) œ 'Ð0ß0ß0Ñ F † dr œ 'Ð0ß0ß0Ñ ™ f † dr œ f(xß yß z) f(0ß 0ß 0) Ê
ÐxßyßzÑ
`g
`z
œ
`f
`z
ÐxßyßzÑ
`g
`x
œ
`f
`x
0,
`g
`y
œ
`f
`y
0, and
0 Ê ™ g œ ™ f œ F, as claimed
35. The path will not matter; the work along any path will be the same because the field is conservative.
36. The field is not conservative, for otherwise the work would be the same along C" and C# .
37. Let the coordinates of points A and B be axA , yA , zA b and axB , yB , zB b, respectively. The force F œ ai bj ck is
conservative because all the partial derivatives of M, N, and P are zero. Therefore, the potential function is
fax, y, zb œ ax by cz C, and the work done by the force in moving a particle along any path from A to B is
faBb faAb œ f axB , yB , zB b faxA , yA , zA b œ aaxB byB czB Cb aaxA byA czA Cb
Ä
œ aaxB xA b bayB yA b cazB zA b œ F † BA
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 16.4 Green's Theorem in the Plane
38. (a) Let GmM œ C Ê F œ C ’
`P
`y
Ê
œ
3yzC
ax# y# z# b&Î#
`f
`x
œ
xC
ax# y# z# b$Î#
yC
Ê `` gy œ
ax# y# z# b$Î#
some f;
œ
œ
`N
`z
,
x
ax# y# z# b$Î#
`M
`z
œ
i
y
ax# y# z# b$Î#
3xzC
ax# y# z# b&Î#
Ê f(xß yß z) œ
œ
,
C
ax# y# z# b"Î#
0 Ê g(yß z) œ h(z) Ê
Ê h(z) œ C" Ê f(xß yß z) œ
`P
`x
C
ax# y# z# b"Î#
`f
`z
œ
j
`N
`x
œ
z
ax# y# z# b$Î#
3xyC
ax# y# z# b&Î#
g(yß z) Ê
`f
`y
k“
`M
`y
œ
œ
hw (z) œ
zC
ax# y# z# b$Î#
Ê F œ ™ f for
yC
ax# y# z# b$Î#
C" . Let C" œ 0 Ê f(xß yß z) œ
`g
`y
zC
ax# y# z# b$Î#
GmM
ax# y# z# b"Î#
is a potential
function for F.
(b) If s is the distance of (xß yß z) from the origin, then s œ Èx# y# z# . The work done by the gravitational field
F is work œ 'P F † dr œ ’ Èx#GmM
“
y # z#
P#
T#
"
T"
œ
GmM
s#
GmM
s"
œ GmM Š s"#
"
s" ‹ ,
as claimed.
16.4 GREEN'S THEOREM IN THE PLANE
1. M œ y œ a sin t, N œ x œ a cos t, dx œ a sin t dt, dy œ a cos t dt Ê
`N
`y
`M
`x
œ 0,
`M
`y
œ 1,
`N
`x
œ 1, and
œ 0;
Equation (3):
)C M dy N dx œ '021 [(a sin t)(a cos t) (a cos t)(a sin t)] dt œ '021 0 dt œ 0;
' ' Š ``Mx ``Ny ‹ dx dy œ ' ' 0 dx dy œ 0, Flux
R
R
Equation (4):
)C M dx N dy œ '021 [(a sin t)(a sin t) (a cos t)(a cos t)] dt œ '021 a# dt œ 21a# ;
Èa c x
' ' Š ``Nx ``My ‹ dx dy œ ' '
ca cc
a
R
#
œ 2a
ˆ 1#
1‰
#
#
#
2 dy dx œ 'ca 4Èa# x# dx œ 4 ’ x2 Èa# x#
a
sin" xa “
a
ca
#
œ 2a 1, Circulation
2. M œ y œ a sin t, N œ 0, dx œ a sin t dt, dy œ a cos t dt Ê
Equation (3):
a#
#
)C M dy N dx œ '0
21
`M
`x
œ 0,
`M
`y
œ 1,
`N
`x
œ 0, and
`N
`y
œ 0;
#1
a# sin t cos t dt œ a# 2" sin# t‘ ! œ 0; ' ' 0 dx dy œ 0, Flux
R
21
#1
Equation (4): )C M dx N dy œ '0 aa# sin# tb dt œ a# 2t sin4 2t ‘ ! œ 1a# ; ' ' Š ``Nx ``My ‹ dx dy
œ ' ' 1 dx dy œ '0
21
R
'0
a
r dr d) œ '0
21
R
a#
#
d) œ 1a# , Circulation
3. M œ 2x œ 2a cos t, N œ 3y œ 3a sin t, dx œ a sin t dt, dy œ a cos t dt Ê
`N
`y
`M
`x
œ 2,
`M
`y
`N
`x
œ 0,
œ 0, and
œ 3;
Equation (3):
)C M dy N dx œ '021 [(2a cos t)(a cos t) (3a sin t)(a sin t)] dt
œ '0 a2a# cos# t 3a# sin# tb dt œ 2a# 2t
21
sin 2t ‘ #1
4
!
3a# 2t
sin 2t ‘ #1
4
!
œ 21a# 31a# œ 1a# ;
' ' Š ``Mx ``Ny ‹ œ ' ' 1 dx dy œ ' ' r dr d) œ ' a## d) œ 1a# , Flux
0
0
0
21
R
a
21
R
Equation (4):
)C M dx N dy œ '021 [(2a cos t)(a sin t) (3a sin t)(a cos t)] dt
#1
œ '0 a2a# sin t cos t 3a# sin t cos tb dt œ 5a# 12 sin# t‘ ! œ 0; ' ' 0 dx dy œ 0, Circulation
21
R
4. M œ x# y œ a$ cos# t, N œ xy# œ a$ cos t sin# t, dx œ a sin t dt, dy œ a cos t dt
Ê ``Mx œ 2xy, ``My œ x2 , ``Nx œ y# , and ``Ny œ 2xy;
Equation (3):
)C M dy N dx œ '021 aa% cos$ t sin t a% cos t sin$ tb œ ’ a4
%
cos% t
a%
4
sin% t“
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
#1
!
œ 0;
957
958
Chapter 16 Integration in Vector Fields
' ' Š ``Mx ``Ny ‹ dx dy œ ' ' (2xy 2xy) dx dy œ 0, Flux
R
R
21
21
Equation (4): )C M dx N dy œ '0 aa% cos# t sin# t a% cos# t sin# tb dt œ '0 a2a% cos# t sin# tb dt
21
41
%1
œ '0 "# a% sin# 2t dt œ a4 '0 sin# u du œ a4 u2 sin42u ‘ ! œ 1#a ; ' ' Š ``Nx ``My ‹ dx dy œ ' ' ay# x# b dx dy
%
%
R
21
a
21
œ '0 '0 r# † r dr d) œ '0 a4
%
`M
`x
5. M œ x y, N œ y x Ê
Circ œ ' '
%
d) œ
1 a%
#
, Circulation
œ 1,
`M
`y
œ 1,
`N
`x
`N
`y
œ 1,
R
œ 1 Ê Flux œ ' ' 2 dx dy œ '0
1
R
'01 2 dx dy œ 2;
[1 (1)] dx dy œ 0
R
`M
`x
6. M œ x# 4y, N œ x y# Ê
`M
`y
œ 2x,
œ 4,
`N
`x
œ 1,
`N
`y
œ 2y Ê Flux œ ' ' (2x 2y) dx dy
R
1
1
1
1
"
"
œ '0 '0 (2x 2y) dx dy œ '0 cx# 2xyd ! dy œ '0 (1 2y) dy œ cy y# d ! œ 2; Circ œ ' '
œ '0
1
'01 3 dx dy œ 3
`M
`x
7. M œ y# x# , N œ x# y# Ê
œ '0
3
œ 2x,
'0 (2x 2y) dy dx œ '0 a2x
x
3
#
`M
`y
œ 2y,
`N
`x
#
x b dx œ
"
3
œ 2x,
$
x$ ‘ !
`N
`y
œ 2y Ê Flux œ ' ' (2x 2y) dx dy
R
œ 9; Circ œ ' ' (2x 2y) dx dy
R
3
x
3
œ '0 '0 (2x 2y) dy dx œ '0 x# dx œ 9
8. M œ x y, N œ ax# y# b Ê
`M
`x
`M
`y
œ 1,
œ 1,
`N
`x
œ 2x,
1
x
1
œ '0 '0 (1 2y) dy dx œ '0 ax x# b dx œ "6 ; Circ œ ' '
œ '0 a2x xb dx œ
1
#
R
œ '0
œ '0
1
Èx
'x
2
`M
`x
`M
`y
œ y,
œ x 2y,
`N
`x
œ 1,
`N
`y
œ 1 Ê Flux œ ' ' ay a1bb dy dx
R
' ' a1 ax 2ybb dy dx
ay 1b dy dx œ '0 ˆ "# x Èx "# x4 x# ‰ dx œ 11
60 ; Circ œ
2
7
a1 x 2yb dy dx œ '0 ˆÈx x3Î2 x x# x3 x4 ‰ dx œ 60
Circ œ ' '
R
1
`M
`x
œ 1,
`M
`y
œ 3,
`N
`x
œ 2,
`N
`y
œ 1 Ê Flux œ ' ' a1 a1bb dy dx œ 0
R
È2
È2
2 x Î2
a2 3b dy dx œ 'cÈ2 'È 2 c x Î2 a1b dy dx œ È22 'cÈ2 È2 x2 dx œ 1È2
Èa
2b
a
11. M œ x3 y2 , N œ "# x4 y Ê
2
R
1
x
(2x 1) dx dy œ '0 '0 (2x 1) dy dx
1
10. M œ x 3y, N œ 2x y Ê
œ '0
œ 2y Ê Flux œ ' ' (1 2y) dx dy
R
Èx
'x
`N
`y
7
6
9. M œ xy y2 , N œ x y Ê
1
(1 4) dx dy
R
`M
`x
œ 3x2 y2 ,
2b
`M
`y
œ 2x3 y,
`N
`x
œ 2x3 y,
`N
`y
œ "# x4 Ê Flux œ ' ' ˆ3x2 y2 "# x4 ‰ dy dx
'xx x ˆ3x2 y2 "# x4 ‰ dy dx œ '02 ˆ3x5 72 x6 3x7 x8 ‰ dx œ 649 ; Circ œ ' '
2
R
a2x3 y 2x3 yb dy dx œ 0
R
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 16.4 Green's Theorem in the Plane
12. M œ
x
1 y2 ,
È1 y2
œ 'c1 'È1 y2
1
`M
`x
N œ tan1 y Ê
2
1 y2
1
`M
1 y2 , ` y
œ
2x y
, `N
a 1 y 2 b2 ` x
œ
œ 0,
`N
`y
œ
Ê Flux œ ' ' Š 1 1 y2
1
1 y2
R
1
1 y2 ‹
dx dy
dx dy œ 'c1 4 1 1 y2y dx œ 41È2 41 ; Circ œ ' ' Š0 Š a12xy2yb2 ‹‹ dy dx
È
1
2
R
È1 y2
y
œ 'c1 'È1 y2 Š a1 2x
‹ dy dx œ 'c1 a0b dx œ 0
y 2 b2
1
1
`M
`x
13. M œ x ex sin y, N œ x ex cos y Ê
Ècos 2)
1Î4
Ê Flux œ ' ' dx dy œ 'c1Î4 '0
R
œ 1 ex sin y,
Î
`M
`y
œ ex cos y,
1 4
1Î%
Ècos 2)
1Î4
R
R
y
x
, N œ ln ax# y# b Ê
Ê Flux œ ' ' Š x#yy#
R
Circ œ ' ' Š x# 2x
y#
x
x# y# ‹
R
`M
`x
15. M œ xy, N œ y# Ê
œ '0 Š 3x#
1
#
3x%
# ‹
2y
x# y# ‹
dx œ
`M
`x
dx dy œ '0
1
dx dy œ '0
1
`M
`y
œ y,
y
x# y#
œ
Ê Flux œ ' ' (x sin y) dx dy œ '0
R
œ 0,
'0
1Î2
1Î2
R
`M
`x
, N œ ex tan " y Ê
R
`N
`y
"
1 y#
1
3cx
œ 'c1 'x b 1
%
#
Ê
œ '0
'0
x$
2y
x# y#
'x
#
`N
`x
,
`M
`y
2xy dy dx œ '
#
1
2
0 3
x
œ 6xy# ,
`N
`x
"!
2
33
20. M œ 4x 2y, N œ 2x 4y Ê
dx œ
`M
`y
œ 2,
x dy dx œ '0 ax# x$ b dx œ 1"#
`N
`y
œ cos y,
œ x sin y
#
#
œ 3y
"
1 y#
,
`N
`y
œ
"
1 y#
`N
`x
œ
ex
y
#1
!
'0aÐ1
cos )Ñ
(3r sin )) r dr d)
œ 4a$ a4a$ b œ 0
Ê Circ œ ' ' ’ ey Š1
x
R
ex
y ‹“
dx dy œ ' ' (1) dx dy
R
œ 8xy# Ê work œ )C 2xy$ dx 4x# y# dy œ ' ' a8xy# 6xy# b dx dy
R
`N
`x
œ 2 Ê work œ )C (4x 2y) dx (2x 4y) dy
œ ' ' [2 (2)] dx dy œ 4 ' ' dx dy œ 4(Area of the circle) œ 4(1 † 4) œ 161
R
#
'01Î2 2 cos y dx dy œ '01Î2 1 cos y dy œ c1 sin yd 1Î#
œ1
!
R
ex
y
'xx 3y dy dx
1
21
œ1
œ
1
1
dy dx œ 'c1 ca3 x# b ax% 1bd dx œ 'c1 ax% x# 2b dx œ 44
15
19. M œ 2xy$ , N œ 4x# y# Ê
1
`M
`y
`N
`y
(x sin y) dx dy œ '0 Š 18 sin y‹ dy œ 18 ;
$
ex
y
,
R
x
1Î2
œ '0 a$ (1 cos ))$ (sin )) d) œ ’ a4 (1 cos ))% “
18. M œ y ex ln y, N œ
2x
x# y#
1
dx dy œ ' ' 3y dx dy œ '0
"
1 y# ‹
21
œ
œ 2y Ê Flux œ ' ' (y 2y) dy dx œ '0
œ cos y,
Circ œ ' ' [cos y ( cos y)] dx dy œ '0
Ê Flux œ ' ' Š3y
`N
`x
"
#
#
`M
`y
œ 0,
,
Î
1 4
#
R
`M
`x
x
x# y#
;
r dr d) œ ' 1Î4 ˆ "# cos 2)‰ d) œ
'12 ˆ r sinr ) ‰ r dr d) œ '01 sin ) d) œ 2;
; Circ œ ' ' x dy dx œ '0
"
5
1Î2
x
1 y#
œ
1
16. M œ sin y, N œ x cos y Ê
17. M œ 3xy
`M
`y
"
#
œ ex sin y
'12 ˆ r cosr ) ‰ r dr d) œ '01 cos ) d) œ 0
`N
`x
œ x,
,
`N
`y
œ 1 ex cos y,
r dr d) œ ' 1Î4 ˆ "# cos 2)‰ d) œ 4" sin 2)‘ 1Î% œ
Circ œ ' ' a1 ex cos y ex cos yb dx dy œ ' ' dx dy œ 'c1Î4 '0
14. M œ tan"
`N
`x
R
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
959
960
Chapter 16 Integration in Vector Fields
`M
`y
21. M œ y# , N œ x# Ê
œ '0
1
1cx
'0
œ 2y,
œ 2x Ê )C y# dx x# dy œ ' ' (2x 2y) dy dx
`N
`x
R
(2x 2y) dy dx œ '0 a3x 4x 1b dx œ cx 2x# xd ! œ 1 2 1 œ 0
1
22. M œ 3y, N œ 2x Ê
`M
`y
œ 3,
#
`N
`x
œ 2 Ê )C 3y dx 2x dy œ ' ' a2 3b dx dy œ '0
`M
`y
œ 6,
1
R
1
œ '0 sin x dx œ 2
23. M œ 6y x, N œ y 2x Ê
"
$
'0sin x a1bdy dx
œ 2 Ê )C (6y x) dx (y 2x) dy œ ' ' (2 6) dy dx
`N
`x
R
œ 4(Area of the circle) œ 161
24. M œ 2x y# , N œ 2xy 3y Ê
`M
`y
œ 2y,
`N
`x
œ 2y Ê
)C a2x y# b dx (2xy 3y) dy œ ' ' (2y 2y) dx dy œ 0
R
25. M œ x œ a cos t, N œ y œ a sin t Ê dx œ a sin t dt, dy œ a cos t dt Ê Area œ
œ
'0
21
"
#
'0
21
"
#
aa# cos# t a# sin# tb dt œ
'021 aab cos# t ab sin# tb dt œ "# '021 ab dt œ 1ab
"
#
)C
"
#
)C x dy y dx
x dy y dx
a# dt œ 1a#
26. M œ x œ a cos t, N œ y œ b sin t Ê dx œ a sin t dt, dy œ b cos t dt Ê Area œ
œ
"
#
)C x dy y dx
41
3 '
œ "# '0 a3 sin# t cos# tb acos# t sin# tb dt œ "# '0 a3 sin# t cos# tb dt œ 38 '0 sin# 2t dt œ 16
sin# u du
0
27. M œ x œ cos$ t, N œ y œ sin$ t Ê dx œ 3 cos# t sin t dt, dy œ 3 sin# t cos t dt Ê Area œ
21
œ
3
16
u2
21
sin 2u ‘ %1
4
!
œ
3
8
21
"
#
1
28. C1 : M œ x œ t, N œ y œ 0 Ê dx œ dt, dy œ 0; C2 : M œ x œ a21 tb sina21 tb œ 21 t sin t, N œ y
œ 1 cosa21 tb œ 1 cos t Ê dx œ acos t 1b dt, dy œ sin t dt
Ê Area œ
œ
"
#
"
#
)C x dy y dx œ "# )C
"
x dy y dx
"
#
)C
2
x dy y dx
'021 a0bdt "# '021 ca21 t sin tbasin tb a1 cos tb acos t 1bd dt œ "# '021 a2 cos t t sin t 2 21 sin tb dt
œ 12 c3 sin t t cos t 2t 21 cos td20 1 œ 31
29. (a) M œ f(x), N œ g(y) Ê
(b) M œ ky, N œ hx Ê
`M
`y
`M
`y
œ 0,
œ k,
`N
`x
`N
`x
œ 0 Ê )C f(x) dx g(y) dy œ ' ' Š ``Nx
R
œh
`M
`y ‹
dx dy œ ' ' 0 dx dy œ 0
R
Ê )C ky dx hx dy œ ' ' Š ``Nx ``My ‹ dx dy
œ ' ' (h k) dx dy œ (h k)(Area of the region)
R
R
30. M œ xy# , N œ x# y 2x Ê
`M
`y
œ 2xy,
`N
`x
œ 2xy 2 Ê )C xy# dx ax# y 2xb dy œ ' ' Š ``Nx
œ ' ' (2xy 2 2xy) dx dy œ 2 ' ' dx dy œ 2 times the area of the square
R
R
R
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
`M
`y ‹
dx dy
Section 16.4 Green's Theorem in the Plane
961
31. The integral is 0 for any simple closed plane curve C. The reasoning: By the tangential form of Green's
Theorem, with M œ 4x$ y and N œ x% , )C 4x$ y dx x% dy œ ' ' ’ ``x ax% b
`
`y
R
œ ' ' ðóóñóóò
a4x$ 4x$ b dx dy œ 0.
a4x$ yb“ dx dy
R
0
32. The integral is 0 for any simple closed curve C. The reasoning: By the normal form of Green's theorem, with
`
`
$
$
M œ x$ and N œ y$ , )C y$ dy x$ dx œ ' ' ”ðñò
` x ay b ï
` y ax b • dx dy œ 0.
R
0
`M
`x
33. Let M œ x and N œ 0 Ê
œ 1 and
`N
`y
œ0 Ê
0
)C M dy N dx œ ' ' Š ``Mx ``Ny ‹ dx dy
œ ' ' (1 0) dx dy Ê Area of R œ ' ' dx dy œ )C x dy; similarly, M œ y and N œ 0 Ê
R
`N
`x
R
œ 0 Ê )C M dx N dy œ ' ' Š ``Nx
R
œ ' ' dx dy œ Area of R
Ê )C x dy
R
`M
`y ‹
`M
`y
œ 1 and
dy dx Ê )C y dx œ ' ' (0 1) dy dx Ê )C y dx
R
R
34.
'ab f(x) dx œ Area of R œ )C y dx, from Exercise 33
35. Let $ (xß y) œ 1 Ê x œ
My
M
' ' x $ (xßy) dA
œ 'R '
$ (xßy) dA
' ' x dA
œ 'R '
R
dA
' ' x dA
œ
Ê Ax œ ' ' x dA œ ' ' (x 0) dx dy
R
A
R
R
R
œ )C x#
dy, Ax œ ' ' x dA œ ' ' (0 x) dx dy œ ) xy dx, and Ax œ ' ' x dA œ ' ' ˆ 23 x "3 x‰ dx dy
œ)
#
#
"
C 3
R
R
"
3
"
#
x dy xy dx Ê
)C x
C
dy œ )C xy dx œ
#
"
3
)C x
dy xy dx œ Ax
36. If $ (xß y) œ 1, then Iy œ ' ' x# $ (xß y) dA œ ' ' x# dA œ ' ' ax# 0b dy dx œ
R
R
R
R
#
R
"
3
)C
x$ dy,
' ' x# dA œ ' ' a0 x# b dy dx œ ) x# y dx, and ' ' x# dA œ ' ' ˆ 34 x# 4" x# ‰ dy dx
C
R
R
œ)
"
C 4
37. M œ
38. M œ
`f
`y
"
4
ellipse
$
"
4
#
x dy x y dx œ
, N œ `` xf Ê
`M
`y
"
4
œ
x# y "3 y$ , N œ x Ê
"
4
)C x
` #f
` y#
`M
`y
,
R
$
#
dy x y dx Ê
`N
`x
œ
1
4
"
3
œ `` xf# Ê )C
#
x# y# ,
`N
`x
)C x
`f
`y
R
$
dy œ )C x# y dx œ
dx
`f
`x
œ 1 Ê Curl œ
"
4
)C
dy œ ' ' Š `` xf#
#
R
`N
`x
`M
`y
x$ dy x# y dx œ Iy
` #f
` y# ‹
dx dy œ 0 for such curves C
œ 1 ˆ "4 x# y# ‰ 0 in the interior of the
x# y# œ 1 Ê work œ 'C F † dr œ ' ' ˆ1 4" x# y# ‰ dx dy will be maximized on the region
R
R œ {(xß y) | curl F}
0 or over the region enclosed by 1 œ
2y
39. (a) ™ f œ Š x# 2x
y# ‹ i Š x# y# ‹ j Ê M œ
2x
x# y#
,Nœ
"
4
x# y#
2y
x# y#
; since M, N are discontinuous at (0ß 0), we
compute 'C ™ f † n ds directly since Green's Theorem does not apply. Let x œ a cos t, y œ a sin t Ê dx œ a sin t dt,
dy œ a cos t dt, M œ
2
a
cos t, N œ
2
a
sin t, 0 Ÿ t Ÿ 21, so 'C ™ f † n ds œ 'C M dy N dx
œ '0 ˆ 2a cos t‰aa cos tb ˆ 2a sin t‰aa sin tb ‘dt œ '0 2acos2 t sin2 tbdt œ 41. Note that this holds for any
21
21
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
962
Chapter 16 Integration in Vector Fields
a 0, so 'C ™ f † n ds œ 41 for any circle C centered at a0, 0b traversed counterclockwise and 'C ™ f † n ds œ 41
if C is traversed clockwise.
(b) If K does not enclose the point (0ß 0) we may apply Green's Theorem: 'C ™ f † n ds œ 'C M dy N dx
œ ' ' Š ``Mx
R
`N
`y ‹
dx dy œ ' ' Š ax2 y2 b2
2 ˆy 2 x 2 ‰
R
2 ˆx 2 y 2 ‰
‹
ax2 y2 b2
dx dy œ ' ' 0 dx dy œ 0. If K does enclose the point
R
(0ß 0) we proceed as follows:
Choose a small enough so that the circle C centered at (0ß 0) of radius a lies entirely within K. Green's Theorem
applies to the region R that lies between K and C. Thus, as before, 0 œ ' ' Š ``Mx
R
`N
`y ‹
dx dy
œ 'K M dy N dx 'C M dy N dx where K is traversed counterclockwise and C is traversed clockwise.
Hence by part (a) 0 œ ’ ' M dy N dx “ 41 Ê 41 œ
K
'K ™ f † n ds œ œ 0
'K M dy N dx
œ 'K ™ f † n ds. We have shown:
if (0ß 0) lies inside K
if (0ß 0) lies outside K
41
40. Assume a particle has a closed trajectory in R and let C" be the path Ê C" encloses a simply connected region
R" Ê C" is a simple closed curve. Then the flux over R" is )C F † n ds œ 0, since the velocity vectors F are
"
tangent to C" . But 0 œ )C F † n ds œ )C M dy N dx œ ' ' Š ``Mx
"
"
R"
`N
`y ‹
dx dy Ê Mx Ny œ 0, which is a
contradiction. Therefore, C" cannot be a closed trajectory.
41.
'gg yy
#Ð Ñ
"Ð Ñ
`N
`x
dx dy œ N(g# (y)ß y) N(g" (y)ß y) Ê
'cd 'gg yy ˆ ``Nx dx‰ dy œ 'cd [N(g# (y)ß y) N(g" (y)ß y)] dy
#Ð Ñ
"Ð Ñ
œ 'c N(g# (y)ß y) dy 'c N(g" (y)ß y) dy œ 'c N(g# (y)ß y) dy 'd N(g" (y)ß y) dy œ 'C N dy 'C N dy
d
œ )C dy
d
Ê
)C N dy œ ' '
R
d
c
#
`N
`x
"
dx dy
42. The curl of a conservative two-dimensional field is zero. The reasoning: A two-dimensional field F œ Mi Nj
can be considered to be the restriction to the xy-plane of a three-dimensional field whose k component is zero,
and whose i and j components are independent of z. For such a field to be conservative, we must have
`N
`M
`N
`M
` x œ ` y by the component test in Section 16.3 Ê curl F œ ` x ` y œ 0.
43-46. Example CAS commands:
Maple:
with( plots );#43
M := (x,y) -> 2*x-y;
N := (x,y) -> x+3*y;
C := x^2 + 4*y^2 = 4;
implicitplot( C, x=-2..2, y=-2..2, scaling=constrained, title="#43(a) (Section 16.4)" );
curlF_k := D[1](N) - D[2](M):
# (b)
'curlF_k' = curlF_k(x,y);
top,bot := solve( C, y );
# (c)
left,right := -2, 2;
q1 := Int( Int( curlF_k(x,y), y=bot..top ), x=left..right );
value( q1 );
Mathematica: (functions and bounds will vary)
The ImplicitPlot command will be useful for 43 and 44, but is not needed for 43 and 44. In 44, the equation of the line
from (0, 4) to (2, 0) must be determined first.
Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.
Section 16.5 Surfaces and Area
963
Clear[x, y, f]
<
Source Exif Data:
File Type : PDF
File Type Extension : pdf
MIME Type : application/pdf
PDF Version : 1.6
Linearized : Yes
Encryption : Standard V4.4 (128-bit)
User Access : Print, Extract, Print high-res
Author : Bill Ardis
Create Date : 2009:12:21 14:39:58+05:30
Modify Date : 2010:01:08 09:49:17-05:00
Has XFA : No
XMP Toolkit : Adobe XMP Core 4.2.1-c041 52.342996, 2008/05/07-21:37:19
Producer : Acrobat Distiller 6.0 (Windows)
Metadata Date : 2010:01:08 09:49:17-05:00
Creator Tool : PScript5.dll Version 5.2.2
Document ID : uuid:c26874ad-773c-4fae-b29e-24bff859293f
Instance ID : uuid:8dd97f60-b29a-cd44-94f4-126d983e4937
Format : application/pdf
Title : ISM_T12_PRE_VII
Creator : Bill Ardis
Page Count : 439
EXIF Metadata provided by EXIF.tools