Math 227-104 (CRN 24680)
Carter
Test 2 Spring 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.
Egg, bean, and cheese makes a tasty breakfast burrito.
1. (10 points) Sketch the level curves z = −4, −1, 0, 1, and z = 4 for the function
z = f (x, y) = x2 − y 2 .
2. (20 points) Compute the gradient of z = f (x, y) = x2 − y 2 . Use your computation to
compute the equation of the plane that is tangent to the surface at the point (8, 1, 63).
3. (20 points) Find optimal (Maximal and minimal) values of the function
f (x, y) = x2 − 2x − y 2 + 4y − 8
over the triangular region bounded by y = 0, x = 3 and y = −3x.
4. (20 points) Sketch the region, reverse the order of integration, and evaluate the integral.
Z
0
π
Z
x
π
sin (y)
dy dx
y
5. (20 points) Express the volume of a tetrahedron that has vertices (0, 0, 0), (A, 0, 0),
(0, B, 0), and (0, 0, C) as a triple integral. You don’t need to compute the integral!
6. (20 points) Find the volume of the ice-cream cone bounded by the sphere x2 +y 2 +z 2 =
25, and the cone described by φ = π3 .
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