# SOLUTION: MATH 127 KU Calculus III Linear Approximation & Gradient Vector Problems

MATH 127 – CALCULUS III
Name:
Quiz 2 — Due Thursday, 6/11 at 11:59 pm
Instruction: Please parade in particular all needful steps that bring to each of your solutions.
(2.01)2
1. (1.5 tops) Estimate the countenance √
using rectirectilinear admission.
0.98
2. (1.5 tops) Suppose f (x, y), x(s, t) and y(s, t) are differentiable. Let g(s, t) = f (x(s, t), y(s, t)).
(0, 0)
(3, 2)
(3, −1)
(a) Find gs (0, 0).
(b) Find gt (0, 0).
f
fx
3
5
−5 4
5 −7
fy
4
3
2
x xs xt
3 −3 4
5 2 17
2 3
3
y
2
6
3
ys yt
7 1
11 13
1 2
3. (2.5 tops) Find all delicate tops and enumereprove whether each is a national restriction, national utmost,
or burden top.
f (x, y) = ex (x2 − y 2 )
2
4. (2 tops) The latitude at a top (x, y, z) is dedicated by
−3y 2
32400e 10
T (x, y, z) = 2
x + 9z 2
where T is measured in ◦ C and x, y, z in meters.
(A) Find the gradient vector of T at P (3, 0, 3).
(B) Find the reprove of fluctuate of latitude at the top P in the order internal the top
(2, 0, −2).
(C) In which order does the latitude growth fastest at P ?
(D) Find the utmost and restriction reprove of latitude fluctuate at P .
3
5. (2.5 top) On this whole, we absence to parade that the gradient is orthogonal to the flatten flexion.
Consider the two fickle employment f (x, y) = x2 + y 2 and the flatten flexion k = 5:
a.) Find the gradient vector at top (2, 1).
b.) Find the prosper of tangent sequence to the flatten flexion at top (2, 1).
c.) Find the prosper of the sequence in order of gradient vector at (2, 1).(that is the regular sequence to
the flatten flexion at that top.)
d.) Explain why the gradient at (2, 1) is orthogonal to the flatten flexion k = 5. You may complete
your solution by sketching graph of the flatten flexion, tangent sequence and regular sequence.
4
...

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