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Math 212 Spring 2006 Final Exam
Name: Section:
Instructions: You have 3 hours to complete this exam. You should work alone, without access to the textbook or class notes. You may not use a calculator. Do not discuss this exam with anyone except your instructor.
This exam consists of 9 questions. Except for the first problem (multiple choice), you must show your work to receive full credit. Be sure to indicate your final answer clearly for each question.
Note: If you use a major theorem (such as Green’s, Stokes’, or Gauss’ Divergence theorem), you must indicate it. Points will be deducted for failure to indicate the use of a major theorem. You must clearly indicate each time you use such a theorem.
Before turning in the exam, be sure to:
- Staple your exam with this cover sheet on top,
- Pledge your exam,
- Write your name and section number above.
The exam is due by Wednesday, May 10, 4 p.m. Good luck!
Pledge:
Problem Value Score
1 10
2 10
3 10
4 10
5 10
6 10
7 10
8 10
9 10
Total 90
- In the following problem, let S be a unit disk in the plane z = 5, centered at (0, 0 , 5), and oriented upward. Let C be a straight path from (2, 2 , 2) to (0, 0 , 0). Also, let
f (x, y, z) = xy 2 z 3 , F(x, y, z) = −yi + xj, G(x, y, z) = 3i + j + 2k.
Now, for each of the given quantities, decide if it is positive, negative, or zero. (You do not need to justify your answers. No partial credit will be given.)
(a)
S
∇ × F · dS.
(b)
C
∇f · ds.
(c)
S
F · dS.
(d)
S
G · dS.
(e)
∂S
∇f · ds.
- (a) Find all critical points of the function g(x, y) = 2y 3 − 6 y + x 2 in R 2 and classify them.
(b) Let B be the unit ball in R 3 , i.e. the set of points (x, y, z) satisfying x 2 + y 2 + z 2 ≤ 1. Let f (x, y, z) = 2x + 4y + 6z. Find the minimum and maximum values of f restricted to B.
- (a) Let f (x, y) be a C 1 function whose domain is the unit disk in the xy–plane, such that f (x, y) ≥ 0 everywhere. Suppose that the level set f (x, y) = 0 is exactly the unit circle. Let S be the graph of f (x, y), oriented upward, and let
F(x, y, z) = curl(x 2 − z, ez^ + 2x, π).
Determine
S
F · dS.
(b) Let g(x, y) be a C 1 function whose domain is the unit disk in the xy–plane, such that g(x, y) ≤ 0 everywhere. Suppose that the level set g(x, y) = 0 is exactly the unit circle. Let T be the graph of f (x, y), oriented downward.
Determine
T
F · dS.
- Evaluate
∫ √^73
0
∫ √^73
√ (^3) x x(
1 + y 7 ) dy dx. Hint: Consider the region of integration.
