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MATH237: Calculus

创建日期：2024-04-22 15:49:12

所属分类：数学mathematics

标签： MATH237

Calculus

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MATH237: Calculus

Written Assignment 4

Instructions
There are five submission slots on Crowdmark: Q1, Q2, Q3 and Q4. Please upload your
solutions into the appropriate slots.
Don’t forget about the LATEX Bonus and the opportunity to earn up to +2% on your final
course grade. See LEARN→ Activities and Assignments→Written Assignments and scroll
down to ‘LaTeX Bonus’ for more info.
For Q1, your sketches may either be hand-drawn or graphed with software. You may submit
them as separate image files on Crowdmark, or you may insert them alongside the rest of
your write-up. If you’d like to do the latter, a code snippet is provided in the .tex template.
[If you submit as separate image files, you will still be eligible for the LaTeX bonus.]
Problems
Q1. Let f : R2 → R be given by f(x, y) = 2x3 − 3x2 + y2 − 2y.
(a) Find the critical points of f and classify them into local maxima/minima and saddle
points.
(b) Let D be the region in the xy-plane that lies inside the quadrilateral with vertices at
(0, 0), (2, 0), (2, 2) and (1, 2), together with its boundary edges. Sketch D.
(c) Find the global maximum and minimum values of f on D and the points at which they
occur. Mark these points on your sketch of D.
Q2. Consider the lines L1 and L2 in R3 given by the vector equations
L1 : ~v = (2, 4, 4) + t(4, 1, 5) and L2 : ~v = (1,−3, 2) + t(−2, 3, 1).
(a) Find an expression for the distance between two points on L1 and L2. Your expression
should be a function f : R2 → R.
(b) By finding the extreme values of f , find the shortest distance between L1 and L2. Be
sure to fully justify your answer.
[Hint: The expression for the distance-squared is often simpler than the expression for
the distance. So you may find it easier to work with f2 instead of f . If you do so, be
sure to clearly explain how the process of optimizing f2 is related to optimizing f .]
Q3. The Cauchy–Schwarz inequality says that if ~a = (a1, . . . , an) and ~b = (b1, . . . , bn) are two
vectors in Rn, then
|~a ·~b| ≤ ‖~a‖‖~b‖.
In this exercise you will give a proof of this inequality using multivariable calculus.
(a) Assume that the inequality is true for all ~b ∈ Rn with ‖~b‖ = 1. Deduce from this that
the inequality must then be true for all ~b ∈ Rn.
Page 1 of 2
MATH237: Calculus 3
Written Assignment 4

(b) Find the maximum and minimum values of the function f(x1, . . . , xn) =
∑n
i=1 aixi
subject to the constraint ‖~x‖ = c where c ∈ R>0 is a fixed positive real number.
[Hint: The Lagrange multipliers algorithm applies in the same way to a function of n
variables as it does to functions of 2 or 3 variables. You may use, without proof, the fact
that the set S = {~x ∈ Rn : ‖x‖ = c} is closed and bounded and has no “edge points.”]
(c) Using your findings in parts (a) and (b), give a proof of the Cauchy–Schwarz inequality.
Q4. Let ~u = (u1, u2) be a unit vector in R2 and let f : R2 → R be defined by
f(x, y) =

x2y
x2 + y2
if (x, y) 6= (0, 0),
0 if (x, y) = (0, 0).
(a) Find D~uf(0, 0).
(b) Using your solution to (a), find ∇f(0, 0).
(c) Use the Lagrange multipliers algorithm to find the maximum and minimum directional
derivatives at (0, 0). [Hint: What are you trying to optimize? What is the constraint?]
(d) If you’ve solved (b) and (c) correctly, you will have found that the maximum and mini-
mum directional derivatives are not equal to ‖∇f(0, 0)‖ and −‖∇f(0, 0)‖. This appears
to contradict the Greatest Rate of Change Theorem given in Unit 7.2. What went wrong?
Explain.

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