Math 227-103 (CRN 10968)
Carter
Test 2 Fall 2015
General Instructions. Write your name on only the outside of your blue books. Do not
write on this test sheet, do all of your work inside your blue books. Write neat complete
solutions to each of the problems in the blue book. Please put your test sheet inside the blue
book as you leave. There are 110 points.
The microwave setting for baked potatoes is pretty good. Scrub the potato thoroughly
before cooking it. Then stab it several times with a fork. Put in the microwave at the potato
setting. You can stop midway and turn it over so the cooking is more even. Fix with your
favorite toppings (e.g. butter, sour cream, or chives).
1. Consider the function
z = f (x, y) = 2x2 − 4x + y 2 − 2y + 1
restricted to the triangle bounded by the lines x = 0, y = 0, x + y = 5. The following
values of the function are helpful: f (0, 0) = 1, f (1, 0) = −1, f (2, 3) = 4, f (0, 5) = 16,
and f (0, 1) = 0.
(a) (5 points) What are the optimal values of f along the line x = 0?
(b) (5 points) What are the optimal values of f along the line y = 0?
(c) (5 points) What are the optimal values of f along the line y = 5 − x?
~ = ~0?
(d) (5 points) For which (x, y) is ∇f
(e) (5 points) What are the absolute Maxima and minima of f on this region?
2. (20 points) A closed can is to hold 16π cubic centimeters of whoop-em. What are the
dimensions (height h and radius r) of the can that uses the least area?
3. (10 points) Interchange the order of integration and compute the integral:
Z 1Z 1
x2 exy dx dy.
0
y
4. (15 points) Compute the volume of the bullet shaped region that is bounded above by
z = 16 − x2 − y 2 and below by the (x, y)-plane z = 0. Hint: You may want to use polar
coordinates.
5. (15 points) Use polar coordinates to compute the double integral
Z Z √
1−y 2
1
−1
−
√
ln (x2 + y 2 + 1) dx dy.
1−y 2
1
6. (10 points) Express the volume of a sphere by means of a triple integral using spherical coordinates. You do not need to compute the integral, but the volume form and
the limits of integration should be correct!
7. (15 points) Show that the differential form that is expressed in the integral is exact (so
M dx + N dy + P dz = dF for some real-valued function w = F (x, y, z)). Find such a
function. And compute the integral:
Z
(3,5,0)
yz dx + xz dy + xy dz.
(,1,1,2)
2