Math 211, Section 01, Fall 2017
Final Exam, Friday, December 22, 2017
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. (11 points) Find an equation for the plane that contains the line ~r(t) = h2 + t, 3t, 1 − ti
and the point (2, 3, −1).
2. (12 points) Let f (x, y) = 2 + ln(3x + y 2 ). Write the equation of the tangent plane to
the surface z = f (x, y) at the point (−1, 2, 2), and then use it to estimate f (−1.2, 2.1).
3
3
2x + 4xy − y
if (x, y) 6= (0, 0),
3. (17 points) Let f (x, y) =
3x2 + y 2
0
if (x, y) = (0, 0).
3a. Prove that f is not continuous at (0, 0).
3b. Compute fx (0, 0) and fy (0, 0).
4. (20 points) Find and classify (as local minimum, local maximum, or saddle point) every
critical point of the function f (x, y) = 6x2 + 3y 2 − xy 2 + 9.
5. (15 points) Find the maximum and minimum values of the function
f (x, y) = x2 + 8y
on the ellipse x2 + 2y 2 = 18.
6. (20 points) Let E be the solid bounded below by the surface z = y 2 , bounded above
by the plane z = 1, and bounded in the back and front by the planes y = x and x = 2,
respectively. Suppose that the density of E is given by ρ(x, y, z) = 10z. Compute the mass
of E.
7. (15 points) Let E be the solid lying
• inside the sphere x2 + y 2 + z 2 = p
1,
• above (i.e., inside) the cone z = x2 + y 2 , and
• inside
Z Zthe
Z first octant.
Compute
8z dV .
E
8. (15 points) Let C be the quarter-circle
Z arc of radius 2 in the first quadrant of the
xy-plane, as shown in the figure. Compute
x3 ds.
C
2
C
2 9. (15 points) Let F~ (x, y) = 6x − 3x2 y, 6y 2 − x3 .
9a. Show that F~ is conservative by finding a potential function for F~ .
9b. Let C be the quarter-circle path running from (2, 1) to (1, 2), counterclockwise
along
Z
F~ · d~r.
the arc of the circle (x − 1)2 + (y − 1)2 = 1, as shown below. Compute
C
C
2
1
1
2
10. (20 points) Let C be the boundary of the triangle with vertices (0, 0), (2, 1), and (0, 1),
~
oriented counterclockwise,
as shown in the figure. Let G(x,
y) = hxy 2 , 3x2 y + cos8 yi.
Z
~ · d~r.
Compute
G
C
1
1
2
11. (20 points) Let S be the closed surface consisting of
• the portion of the paraboloid z = 2 − 2x2 − 2y 2 above the xy-plane, and
• the disk x2 + y 2 ≤ 1 in the xy-plane,
~
oriented Zoutward.
Let G(x,
y, z) = h2x3 , y 2 z, −yz 2 i. Use the Divergence Theorem to compute
Z
~ · dS
~ of G
~ through S.
the flux
G
S
12. (20 points) Let S be the portion of the surface z = x2 + y 2 that lies in the first octant,
and below
the plane z = 4, oriented downward. Let F~ (x, y, z) = hy, −x, xzi. Compute the
ZZ
~ of F~ through S.
F~ · dS
flux
S
OPTIONAL BONUS A. (2 points) Let CZ be the portion of the graph of y = sin x from
(9x2 y 2 + y) dx + (6x3 y − sin y) dy.
the point (0, 0) to the point (π, 0). Compute
C
OPTIONAL BONUS B. (2 points) Find a vector field F~ (x, y, z) such that
curl(F~ ) = 3yz 2 − xy, 4xyz − 3x3 , yz − 2xz 2 .
OPTIONAL BONUS C. (1 point) In the past week, it has been reported that the US
Department of Health and Human Services is prohibiting officials at the Centers for Disease
Control and Prevention from using seven terms in official documents. List three of those
seven terms.