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needs to be solved %100 correct please.
5 questions calculus III
It’s worth 10 points.
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MATH 127 – CALCULUS III
Name:
Quiz 2 — Due Thursday, 6/11 at 11:59 pm
Instruction: Please show in detail all necessary steps that lead to each of your answers.
(2.01)2
1. (1.5 points) Estimate the expression √
using linear approximation.
0.98
2. (1.5 points) 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 points) Find all critical points and determine whether each is a local minimum, local maximum,
or saddle point.
f (x, y) = ex (x2 − y 2 )
2
4. (2 points) The temperature at a point (x, y, z) is given 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 rate of change of temperature at the point P in the direction toward the point
(2, 0, −2).
(C) In which direction does the temperature increase fastest at P ?
(D) Find the maximum and minimum rate of temperature change at P .
3
5. (2.5 point) On this problem, we want to show that the gradient is orthogonal to the level curve.
Consider the two variable function f (x, y) = x2 + y 2 and the level curve k = 5:
a.) Find the gradient vector at point (2, 1).
b.) Find the slope of tangent line to the level curve at point (2, 1).
c.) Find the slope of the line in direction of gradient vector at (2, 1).(that is the normal line to
the level curve at that point.)
d.) Explain why the gradient at (2, 1) is orthogonal to the level curve k = 5. You may complete
your answer by sketching graph of the level curve, tangent line and normal line.
4
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