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Math Tournament Guide also available at khaacademy.github.io from Kha Le Table of Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Derivative Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sin & cos cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Everything About Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e .................................................................... Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trigonometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Ciphering & Mathbowl Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Various Forgettable Topics from Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Approximation (Differentials) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average Value vs. Average Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logarithmic Differentiation/Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Obscure Topics from Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagated Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marginal Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Epsilon-Delta Definition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page 2 Page 3 Page 4 Page 4 Page 8 Page 9 Page 9 Page 9 Page 10 Page ? Page ? Page ? Page ? Page ? Page 13 Page 14 Page ? Page ? Page ? Page ? Page ? Page ? Page 15 Page ? Page ? Page ? Page ? Introduction Here’s a bunch of tournament techniques and knowledge you can use. It’s a work in progress as of now. I’ll try to update it regularly. Also, feel free to skim around, since you might know a lot of it already. Enjoy! Sweep! Page 3 Derivative Tricks 1 xn Differentiating xn Know these three facts: • The nth derivative of xn is n! • The (n − 1)th derivative of xn is n! · x • The (n + 1)th derivative of xn is 0 Memorizing this is extremely useful for ciphering/mathbowl and will save time on individual tests. Example: Example 1 What is the 69th derivative of x70 ? Answer: 70! · x Example 2 What is the 70th derivative of x70 ? Answer: 70! Example 3 What is the 71th derivative of x70 ? Answer: 0 Example 4 What is the 72th derivative of x70 ? Answer: 0 Example 5 What is the 89th derivative of x70 ? Answer: 0 Page 4 Notice that for any n greater than n+1, the derivative of xn will also be 0. Keep that in mind. Practice Problem 1 What is the 30th derivative of x31 ? Practice Problem 2 What is the 10th derivative of x11 ? Practice Problem 3 What is the 6th derivative of x5 ? Practice Problem 4 What is the 77th derivative of x77 ? What about something like 39th derivative of x70 There is an easy trick to this too! Page 5 Differentiating xn less than (n − 1) times Suppose α is a number less than (n − 1): The αth derivative of xn is n · (n − 1) · (n − 2) · . . . · (n − α + 1) · x(n−α) This may look intimidating at first, but looking at a few examples will make it obvious how simple it is: Example 1 What is the 50th derivative of x70 ? 70 · 69 · 68 · . . . · (70 − 50 + 1) · x70−50 = 70 · 69 · 68 · . . . · 21 · x20 Example 2 What is the 20th derivative of x50 ? 50 · 49 · 48 · . . . · 31 · x30 Example 3 What is the 24th derivative of x78 ? 78 · 77 · 76 · . . . · 55 · x54 Example 4 What is the 19th derivative of x44 ? 44 · 43 · 42 · . . . · 26 · x25 Example 5 What is the 5th derivative of x10 ? 10 · 9 · 8 · 7 · 6 · x5 Page 6 Try some out yourself. Finding the pattern is key here. Practice with smaller derivatives if you are having trouble with this. Practice Problem 1 What is the 30th derivative of x42 ? Practice Problem 2 What is the 10th derivative of x57 ? Practice Problem 3 What is the 6th derivative of x12 ? Practice Problem 4 What is the 77th derivative of x99 ? Page 7 2 sin & cos cycles Differentiating sin x n times Divide n by 4, and find the remainder, r. If: r = 0, f n = sin x r = 1, f n = cos x r = 2, f n = − sin x r = 3, f n = − cos x Example 1 What is the 64th derivative of sin x? 64/4 leaves remainder 0, so it is sin x Example 2 What is the 33th derivative of sin x? 33/4 leaves remainder 1, so it is cos x Example 3 What is the 74th derivative of sin x? 74/4 leaves remainder 2, so it is − sin x Example 4 What is the 19th derivative of sin x? 19/4 leaves remainder 3, so it is − cos x Example 5 What is the 25th derivative of sin x? 25/4 leaves remainder 1, so it is cos x For cos x, it’s the same as sin x, but one forward. So, what I usually do is, if asked to do 6th derivative of cos x, I just do 7th derivative of sin x instead. General case: The nth derivative of cos x is the (n + 1)th derivative of sin x. If you would rather memorize the ones for cos x, go ahead and derive them yourself! Page 8 Everything About Limits Here are some limit techniques: 1 Limits by Direct Substitution Plug in the x-value. Example: lim (23x) = 69 x→3 2 Limits with Fractions Plug x-value, if indeterminate ( 00 , ∞ , ∞ − ∞, 00 , 1∞ , ∞0 ), use one of following: ∞ L’Hopital’s Rule: Differentiate top and bottom. (No quotient rule). Example: cos x sin x ⇒ lim ⇒1 lim x→0 x→0 x 1 Factoring: Find the common factor. Example: ! x x (e − 1) 1 e −1 ⇒ lim ⇒ lim x x x→0 x→0 e2x − 1 (e− 1)(e + 1) 2 Rationalization: You multiply by the conjugate. Example: ! ! √ ! ! √ x+1 x+5+2 ( x + 1) · ( x + 5 + 2) x+1 √ ⇒ lim √ ⇒ lim ⇒4 lim √ x→−1 x→−1 x→−1 x + 1 x+5−2 x+5−2 x+5+2 Page 9 3 Limits Involving Number e Limit Definition of Derivative lim 1 + n→∞ 1 n n =e 1 lim (1 + n) n = e n→0 r x tx 1 2x 3x lim x→∞ 1+ = ert Example 1 lim x→∞ 4 1+ = e3/2 Limits at Infinity √ Use dominance (Note: x2 = −x if approaching −∞) : Example 1 x6 x6 + 18x3 ⇒ lim =0 n→∞ 48x8 n→∞ 2x2 + 48x8 lim Example 2 lim x − n→−∞ lim x − n→−∞ q √ x2 + 4 ⇒ lim x − n→−∞ √ x2 + 4x + 4 ⇒ (x + 2)2 ⇒ lim x − (−(x + 2)) ⇒ lim 2x − 2 ⇒ DN E n→−∞ n→−∞ Practice Problem 1 s 4 lim x2 − x + 2 + x x→−∞ 3 Answer: 8/3 Page 10 5 Trigonometric Limits Know that sin x and cos x oscillate between 1 and -1. This is important when taking dominance into account. If a limit ends in oscillation, it is DNE. If the trig fraction results in 0/0 L’hopital’s is possible. sin x/x sin x =1 n→0 x lim 1 - cos x/x 1 − cos x =0 n→0 x lim Example 1 lim n→0 1 sin x = 2x 2 Example 2 sin 5x =5 n→0 x lim Example 3 sin 2x 2 7 = n→0 x 7 lim Example 4 sin 2x =0 n→∞ x lim Example 5 x = DN E n→∞ sin x lim Example 6 lim n→0 sin 2x 2 = sin 5x 5 Page 11 Extra 1 Evaluate: lim tanx = 0 x→0 Extra 2 lim arctan(x) = x→∞ 6 π 2 Limit Definition of a Derivative (and the Alternate) Limit Definition of Derivative f (x + h) − f (x) = f 0 (x) h→0 h f (x) − f (α) Alternate ⇒ lim = f 0 (α) x→α x−α lim f (x) means that lim+ f (x) and lim− f (x) Regular ⇒ lim x→α x→α x→α Example 1 f 0 (x + h) − f 0 (x) = f 00 (x) h→0 h lim Example 2 lim x→α sin(x) − sin(α) = cos α x−α Example 3 Evaluate limx→5 Solution: x+5 . x−5 DNE, since lim+ x→5 x+5 x+5 = ∞ and lim− = −∞ are not equal x→5 x − 5 x−5 Page 12 Example 4 (Really Good Problem) Find α such that limx→0 Solution: tan(αx+π/4)−1 x =4 tan(αx + π/4) − 1 tan(αx + π/4) − 1 ⇒ lim α· ⇒ αf 0 (π/4) ⇒ α sec2 (π/4) ⇒ α·2 x→0 x→0 x αx lim Therefore, α · 2 = 4, so α = 2 Example 5 (Another Really Good Problem) Find limx→2 Solution: f (x)−f (2) √ √ x− 2 √ √ √ √ √ f (x) − f (2) x+ 2 f (x) − f (2) √ √ ·√ √ ⇒ lim lim √ · ( x + 2) ⇒ f 0 (2) · ( 2 + 2) x→2 x→2 x−2 x− 2 x+ 2 √ = 2 2f 0 (2) Note: for limit definition of integral, go to Various Forgettable Topics section 7 Change of Limit If all nothing else works, this is your last technique to rely on. The trigger to use this is if you see a fraction, typically x1 . Know that: 1 lim = lim If u = , x→∞ u→0 x Example 1 1 sin u ) ⇒ lim =1 lim x sin( x→∞ u→0 u x Page 13 Ciphering & Mathbowl Tricks Gotta go fast. Use these tricks when applicable and if you’re comfortable with them. Here: Solving answer while saying it - ex: f (x) = x cos x, find f 0 (x) you can raise your hand immediately and solve it without writing it down. (mathbowl trick) Skipping derivatives - ex: f (x) = 3x2 + 2x + 1, find f 00 (x) don’t bother with first derivative, go straight to second (use the xn derivative trick) you can also use this for inflection point questions R1 √ Circle - ex:√Evaluate −1 1 − x2 dx R 2 2 If you see k − x2 , it is √the same as finding the area of a circle (remember circle is x +y = 2 2 2 r and rewritten is y = r − x ). find radius and plug into area formula (make sure to look at bounds, it could be a half/quarter of the circle) dy . save like 1-2 secs. maybe even make Shorthand notation - write less, use y 0 instead of dx your own for some (like maybe t(x) instead of tan(x)) Page 14 Various Forgettable Topics Linear Approximation (Differentials) Practice: http://tutorial.math.lamar.edu/Problems/CalcI/Differentials.aspx Ex: Compute the differential for y = x2 as x changes from 1 to 1.01 dy Answer: dx = 2x ⇒ dy = 2xdx ⇒ dy = 2(1)(.01) ⇒ dy = 0.02 You just compute derivative, but multiply by dx at the end. Plug in “integer” value of x (i.e. x = 1) for x, and ∆x for dx. Inverse Functions (f −1 (x)) Practice: http://tutorial.math.lamar.edu/Problems/CalcI/InverseFunctions.aspx Find the inverse function by swapping x and y. If it is not easy to differentiate, do the following: Let g(x) = f −1 (x) and g(f (x)) = x. Differentiate and solve. Average Value vs. Average Rate of Change Practice (Value): http://tutorial.math.lamar.edu/Problems/CalcI/AvgFcnValue.aspx Practice (ROC): https://www.khanacademy.org/math/algebra/algebra-functions/average-rate-of-change-word-problems/ a/average-rate-of-change-review Rb Average Value is a f (x)dx . b−a Average ROC is f (b)−f (a) . b−a Cross-sections Practice: https://www.khanacademy.org/math/ap- Rcalculus- ab/ab- applications-of- integration- new/ab- 8- 7/e/volumes- with- square- and- rectangle- cross- sections Volume with cross-sections: ab A(x)dx, where A(x) is the area of cross-section Inverse Trigonometric Functions Practice (Diff.): https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/invtrigderivdirectory/InvTrigDeriv.html Practice (Int.): https://www.intmath.com/methods-integration/6-integration-inverse-trigonometric-forms.php You can derive with triangle or implicit differentiation, or memorize them. Logarithmic/Exponential Differentiation Practice (Logs): https://www.intmath.com/differentiation-transcendental/5-derivative-logarithm.php Practice (Expo): https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/logdiffdirectory/LogDiff.html 1 All you really have to know is derivative of logα x is x·lnα and derivative of αx is αx · ln α lnx This can be easily derived from knowing logα x = lnα and αx = eln α·x Quick Examples: Differentiating logsin x x is the same as differentiating lnlnsinx x , which you know how to do. Differentiating xsin x is the same as differentiating eln x·sin x , which you know how to do. Page 15 Limit Definition of Integral Practice (Probs #9-13): https://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/defintdirectory/DefInt.html Basically, n→∞ lim ∞ X i=1 Quick Example: f (ci )∆xi = Z b a f (x)dx, where ∆xi = b−a n and ci = a + b−a n ·i ∞ X 4i 2 4 Z 6 2 x dx lim (2 + ) · = n→∞ n n 2 i=1 Hard Example: i2 n2 n X n X i lim = lim 2 2 n→∞ n→∞ i=1 i + n i=1 i2 n2 + n2 n2 = i n lim i 2 n→∞ i=1 ( n ) + n X 1 Z1 x · = dx 1 n 0 x2 + 1 Shell Method Practice: https://www.khanacademy.org/math/old-ap-calculus-ab/ab-applications-definite-integrals/ab-shell-method/ a/stacy-scaling-3 Difficult to explain through text alone. I recommend looking at Khan Academy’s video on shell method. This is a good alternative to ring/washer method sometimes when working with difficult solids of revolution. Page 16 Obscure Topics from Calculus The chances these topics will show up on locals is little to none. This is just for if you’re extra. Euler’s Method Practice: https://www.khanacademy.org/math/ap-calculus-bc/bc-differential-equations-new/bc-7-5/e/euler-s-method You make a chart with x, y, and dy/dx on the top. It’s difficult to explain with just text. Watching the Khan Academy video and doing some problems is the easiest way to learn it. Propagated Error Practice (lol): https://www.youtube.com/watch?v=R7t9Iv_lTc8 The same thing as differentials, except you plug in the error value into dx (or whatever variable you are with respect to). Example: There is a square with side s = 12 inches, with a possible error of 1/64 inches. Calculate the propagated error in computing the area of the square. A = s2 (area of square) dA = 2s ds dA = 2sds dA = 2 · 12 · ±1/64 ⇒ ± 83 in.2 Marginal Cost Practice: http://tutorial.math.lamar.edu/Problems/CalcI/BusinessApps.aspx Marginal cost is a joke. It’s basically find the derivative at x = some constant. Example: A company’s production cost when producing x toys is represented by C(x) = x2 . What is the marginal cost when x=2? Answer: Evaluate C 0 (2) = 4 Epsilon-Delta Definition of Limit Practice: https://brilliant.org/wiki/epsilon-delta-definition-of-a-limit/ This one is easy, but hard to explain. Solving some problems is usually the best way to learn this. ACTM (from the resources file) has quite a few of these. Page 17
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