Math 211, Section 04, Fall 2016
Final Exam, Tuesday, December 20, 2016
Instructions: Do all twelve numbered problems. If you wish, you may also attempt the three
optional bonus questions. Show all work, including scratch work. Little or no credit may
be awarded, even when your answer is correct, if you fail to follow instructions for a
problem or fail to justify your answer. If your answer for a given problem is a sum of
fractions with different denominators, you may leave it that way. Otherwise, simplify your
answers whenever possible. If you need more space, use the back of any page. If you have
time, check your answers.
WRITE LEGIBLY.
NO CALCULATORS.
1. (10 points) Find an equation for the line of intersection of the planes 2x + 5y − z = 3
and x − y = 4.
2x3 + 3xy + 3y 2
if (x, y) 6= (0, 0),
x2 + 2y 2
2. (15 points) Let f (x, y) =
0
if (x, y) = (0, 0).
2a. Prove that f is not continuous at (0, 0).
2b. Compute the directional derivative D~u f (0, 0), where ~u is the unit vector in the
direction of h1, −1i.
3. (15 points) Find the maximum and minimum values of the function
f (x, y) = 4x2 y
on the ellipse 4x2 + y 2 = 36.
4. (15 points) Find and classify (as local minimum, local maximum, or saddle point) every
critical point of the function f (x, y) = x2 y − 2x2 − 6y 2 − 12y.
5. (15 points) Let C be the curve in R3 parametrized by ~r(t) = hcos t, sin t, t3 i, for 0 ≤ t ≤ 1.
Z
5a. Compute
z ds.
C
Z
5b. Compute
F~ · d~r, where F~ (x, y, z) = hyz, −xz, 2zi.
C
6. (20 points) Find the volume of the solid bounded by the surfaces z = 2y, y = 0, z = 2,
x = 0, and x = z 2 .
7. (18 points) Let E be the solid lying inside the sphere
x2 + y 2 + z 2 = 1, above the cone
Z
Z
Z
p
10xy dV .
z = x2 + y 2 , and inside the first octant. Compute
E
p
8. (17 points) Let E be the solid enclosed by the cone z = x2 + y 2 and the plane z = 2.
Suppose that the density of E at the point (x, y, z) is x2 . Compute the mass of E. 9. (15 points) Let F~ (x, y) = hx − cos(2y), y 3 + 2x sin(2y)i.
9a. Show that F~ is conservative by finding a potential function for F~ .
9b. Let C be Z
the (spiral) curve parametrized by ~r(t) = ht cos πt, t sin πti for 1 ≤ t ≤ 4.
Compute
F~ · d~r.
C
10. (15 points) Let C be the path in the xy-plane that begins at (2, 0), runs counterclockwise along the arc of the circle of radius 2 centered at the originZ to the point (−2, 0), and
6x2 y dx + (2x3 − xy) dy.
then goes straight right along the x-axis back to (2, 0). Compute
C
-2
2
11. (20 points) Let S be the closed surface consisting of the portion of the paraboloid
z = 1 − x2 − y 2 above the xy-plane, together with the diskZxZ2 + y 2 ≤ 1 in the xy-plane,
~
~ · dS.
~
oriented outward. Let G(x,
y, z) = hxz, 3yz, x2 y 4 i. Compute
G
S
2
2
12. (25 points) Let S be the portion of the sphere x + y + z 2 = 9 that lies in the first
octant, and let C be the boundary of S, oriented counterclockwise when viewed from above.
That is, C consists of three quarter-circle arcs: from (3, 0, 0) to (0, 3, 0) in the xy-plane; then
from (0, 3,Z0) to (0, 0, 3) in the yz-plane; and finally from (0, 0, 3) to (3, 0, 0) in the xz-plane.
z 2 dx + cos(y 5 ) dy + xz dz.
Compute
C
OPTIONAL BONUS A. (2 points) Recall that on the homework, you verified that
D −y
x E
∂Q
∂P
,
has
−
= 0, but you computed
the vector field F~ = P, Q =
2
2
2
2
x +y x +y
∂x
∂y
Z
F~ · d~r and worked out that it was not zero, where C1 is the circle of radius 1 centered at
C1
the origin,Zoriented counterclockwise.
Compute
F~ · d~r, where C2 is the curve x6 + y 6 = 64, oriented counterclockwise.
C2
OPTIONAL BONUS B. (2 points) Find a vector field F~ (x, y, z) such that
curl(F~ ) = x − yz, y − xz, xy − 2z .
OPTIONAL BONUS C. (1 point) In the past two weeks, the prime minister of Italy
resigned, and the president of South Korea was impeached and (at least temporarily) removed
from office. Name these two recently-former heads of state.