MATH261 EXAM II FALL 2014
NAME:
SI:
Problem Points
1
18
2
10
3
18
4
18
5
20
6
16
Total
100
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit.
GOOD LUCK !!!
Score
1. Consider the function
f (x, y, z) = xy 2 sin(πz) + cos(πx2 ) + tan−1 z, and suppose that
√
x = u2 + w, y = u w and z = cos(π(u + 2w)).
(a) (6pts) Write down the general multivariable Chain Rule that computes ∂f /∂u in
this case. DO NOT actually compute the partial derivatives involved until part
(b).
(b) (12pts) Use your formula in (a) to compute ∂f /∂u when u = −1, w = 1.
p
2. Consider the function f (x, y, z) = sin(yz) + x2 + y.
(a) (7pts) Find the rate of change of f (x, y, z) in the direction of i + k at the point
(2, 2, π).
(b) (3pts) Give a vector (it need not be a unit vector) in whose direction the MAXIMUM rate of change of f (x, y, z) occurs at the point (1, 1, 1).
3. Consider the surfaces F (x, y, z) = 4x2 + 9y 2 − z 2 = 16 and
G(x, y, z) = x2 yz + π1 cos(πyz) = 2. Complete the following.
(a) (12pts) At the coordinate point P = (2, 1, 3), find a vector perpendicular to
surface F and another vector perpendicular to surface G.
(b) (6pts) In vector form, give an equation of the line that is tangent to both surfaces
at P .
4. Consider the function f (x, y) = x2 y − xy 2 + xy.
(a) (12pts) One of the critical points is (−1, 0). Classify it as a local minimum, a
local maximum, or a saddle point.
(b) (6pts) Find all remaining critical points. DO NOT CLASSIFY.
5. (20pts) Using Lagrange multipliers, find the point on the line 2x + y = −26 that is
closest to (1, −3). Circle all the equations that must be solved. The solution should
be neat enough to read.
6. (16pts) Evaluate the integral below. Your answer should include a graph of the region
with points of intersection clearly included.
Z 4Z 2 p
x + y 2 dy dx
√
0
x