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University of Waterloo
MECHANICAL AND MECHATRONICS ENGINEERING
DESIGN PROJECT: EXTREMA OF FUNCTIONS
MTE 203 – ADVANCED CALCULUS
Due Date: Monday, November 12, 2018
Project Description:
On topographic maps of mountainous regions, the curves that represent constant
elevation (height above sea level) are level curves (usually referred as contours) of
complicated functions that represent the terrain. Modeling the variation in temperature
at the surface requires the solution of constrained extreme value problems. Extreme
value problems with algebraic constraints on the variables can be solved using the
method of Lagrange Multipliers which usually results on a system of non-linear
equations. The use of mathematical software such as MATLAB® makes it possible to
graph complex functions and to also solve non-linear equations with only a few
keystrokes which is helpful for the analysis of complex multivariable problems.
In the following project you will use MATLAB® to help you solve a constrained extreme
value problem.
Project Submission:
Reports should reflect your individual work!!!
Present your analysis and results in a clear and concise report. The report and
calculation process can be typed or handwritten. However, presentation should be neat
and professional.
Please, save your report and all associated *.m files as a *.zip file and post it in the
dropbox created in LEARN for that purpose. If all your project files are not zipped in a
single *.zip file, 5% of your mark will be deducted.
The report should include the following sections:
•
Cover Page: The cover page should include the name and course number, the
date of submission, your name and your ID. Please use the template posted in
UW-Learn (MTE203 – Project 1 Cover Page Template.docx)
•
Section 1: Summary of Analysis
o Detailed explanation of all the techniques used
1 MTE 203 - Fall 2018
Instructor: Patricia Nieva
University of Waterloo
MECHANICAL AND MECHATRONICS ENGINEERING
o Explanation of all your results
o Required plots clearly including axis names, labels and legends
o Problems encountered during the analysis of the project
•
Section 2: Summary of Results
o A summary of the results need to be tabulated appropriately. Use as
many tables as you consider relevant and make sure to show all the
required answers. See an example of a table that can be used to report
all critical points at the end of this project description.
o If you decide to type your project, you can attach all hand calculations as
an appendix. If you do so, when reporting, clearly indicate the section of
the appendix you are referring to.
•
Section 3: Appendices
o Reproduce your Matlab codes as appendices
o Attach all hand calculations as appendices with appropriate numbering
for referencing
Notes on Appendices:
•
Indicate clearly the section to which each appendix belongs.
•
Appendices can be typed or handwritten. If handwritten, they should be neat
and clear. If we do not understand your hand writing, that specific section will
not be graded
2 MTE 203 - Fall 2018
Instructor: Patricia Nieva
University of Waterloo
MECHANICAL AND MECHATRONICS ENGINEERING
Project Details:
Functions representing terrains are usually very complex but can easily provide
important information such as the volume of the surface and the degree and orientation
of the slopes. They are also a very flexible way to lay the groundwork for other site
modeling enterprises. The terrain studied here can be modeled by the following
function
𝑧 = 𝑓(𝑥, 𝑦) = ln(𝑥 4 + 1) (4𝑥 4 + (2𝑦)2 ) 𝑒 (−0.5𝑥
2−(𝑦−0.8)2−3)
(𝑠𝑖𝑛(2𝑥 + 0.05𝑦 4 ) + 2 cos(0.75𝑦))
Where 𝑥, 𝑦 and 𝑧 are in kilometers and the domain of the function is defined by the
open region1,
𝑅: {−5 < 𝑥 < 5, −2 < 𝑦 < 4}
In your analysis, consider that the 𝑥𝑦-plane represents the sea level and that the east
and north directions are represented by the direction of the positive x and positive y
axes respectively (see Figure 1).
(N)
NW
(W)
SW
(E)
Figure 1. Directions of the Terrain
(S)
1
The region does not include any points on the edge of the rectangle. In other words, we aren’t allowing
the region to include its boundary and so it’s open.
3 MTE 203 - Fall 2018
Instructor: Patricia Nieva
University of Waterloo
MECHANICAL AND MECHATRONICS ENGINEERING
The temperature distribution within the limits of the terrain is a function of the height
and it is given by
𝑇(𝑥, 𝑦, 𝑧) = −2 ∗ z2 − exp(−0.1 ∗ ((0.1 ∗ x − 2) − (0.05 ∗ y − 3)2 − (z − 1)2 )) + 10
[℃]
Where 𝑥, 𝑦 and 𝑧 are in kilometers and T is in degrees Celsius.
Part I:
In this part of the problem you will be required to plot the terrain and its contour map
(level curves at heights above sea level), and find the location of highest and lowest
elevation of the terrain.
a. Use MATLAB® to plot the terrain and its contours, showing a reasonable number
(25-30) of different elevations.
b. By observation of the contour lines, determine where in the terrain the slope is the
steepest. Justify your answer and verify your finding mathematically
c. Calculate the highest and lowest elevation of the terrain using the first and second
partial derivative tests. List and classify all the critical points you have found in a
table (see template for the table at the end of this document)
Hint: You might want to check for a non-obvious critical point around the southwest and south-east vicinities of the terrain
Part II:
In this part of the project you are required to calculate the temperature at different
locations of the terrain.
a. Calculate the temperatures at the highest and lowest elevation of the terrain
b. Suppose that a hiker is standing at the point (2.2, 0.5) on the terrain.
i.
Calculate the temperature that the hiker feels at this point
ii.
Use MATLAB® to draw a reasonable number of isotherms at the hiker's
elevation
4 MTE 203 - Fall 2018
Instructor: Patricia Nieva
University of Waterloo
MECHANICAL AND MECHATRONICS ENGINEERING
c. Suppose that the hiker decides to walk in the north direction (see note 1 below
and Figure 1)
i.
Using partial differentiation determine if the hiker would be ascending or
descending
ii.
Calculate the rate of change in temperature that the hiker would
experiences as she walks in this direction.
d. Suppose now that the hiker decides to walk in the southwest direction (see note
2 below)
i.
Using partial differentiation determine if the hiker would be ascending or
descending
ii.
Calculate the rate of change in temperature that the hiker experiences as
he walks in this direction.
e. Using MATLAB®, plot again the terrain 𝑓(𝑥, 𝑦) but this time, use the temperature
function as the color map for the terrain. Explain how this plot is useful to the
hiker.
f.
Note that 𝑇(𝑥, 𝑦, 𝑧) can be written entirely in terms of 𝑥 and 𝑦. On a separate
graph, plot 𝑇(𝑥, 𝑦) simply as a function of 𝑥 and 𝑦. This means that 𝑇 is shown on
the vertical axis. Explain how this plot is useful to the hiker.
g. Using Lagrange Multipliers, determine the location of the lowest temperature on
the surface of the island.
Note:
1. To determine the direction of the hiker as he/she ascends/descends in the
northwest/southwest direction, use a vector that defines such direction in the
𝑥𝑦-plane (i.e. a 2D vector)
5 MTE 203 - Fall 2018
Instructor: Patricia Nieva
University of Waterloo
MECHANICAL AND MECHATRONICS ENGINEERING
2. To calculate the rate of change of temperature that the hiker experiences in the
northwest/southwest direction, use a vector that defines such direction in the
tangent plane to the surface at the hiker’s initial position (2.2,0.5).
Example of a table listing critical points:
List of critical point(s) of the terrain
Point #
(x,y)
f(x,y)
⋮
⋮
⋮
A
B
C
⋮
⋮
⋮
D = B2-AC
Type of point2
⋮
⋮
Note:
1. A, B, and C are the Hessian parameters and they are given by:
2 f
2 f
2 f
A = 2 ;B =
;C = 2
xy
x
y
2. List all the points you have found and clearly distinguish which points are
acceptable and which are not. Explain your reasoning.
Indicate clearly whether the critical point corresponds to a relative maximum, minimum, saddle or fails
the second derivative test
2
6 MTE 203 - Fall 2018
Instructor: Patricia Nieva
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