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Name:
Math 206A: Fall 2011
Final Exam
Put your name on your exam and turn it in with your exam book. Write all of your answers in the exam
book. Label problems clearly and circle final answers.
Correct answers accompanied by incorrect or incomplete work will not receive full credit.
Good Luck!
- (15 points) For each of the following equations, identify the name of the surface, match it with a sketch,
and match it with a set of level curves. (It is not necessary to show work for this question. But if you
get the question wrong, some work might be worth partial credit.)
(a) 2 x
2 − 3 y
2 − z = 0
(b) 4 x
2
2 − z
2 = 0
(c) 3 x
2
2
2 = 1
(i) (ii) (iii)
(iv) (v) (vi)
(U)
z=1, z=-
z=
(V)
z=
z=
(W)
z=
z=
z=
z=-1 z=-
(X)
z=.25,
z=-.
z=
(Y)
z=
z=
z=-
z=-
z=
(Z)
z=.25, z=-.
z=
- (14 points) 2 x − 3 y + z = 12 is the equation of a plane.
(a) Find a point on the plane.
(b) Find a vector perpendicular to the plane.
(c) Find a vector parallel to the plane.
(d) Find a vector that is perpendicular to your both of your vectors in (b) and (c).
- (5 points) Suppose u is a function of x and y, which in turn are functions of s and t. Suppose that at
the moment of interest, u x
= 2, u y
= 3, x s
= 4, x t
= 5, y s
= 6, y t
= 7. What is u s
- (8 points) Let
f : R
2 → R
2 be defined by
f (x, y) = (2xy, 3 x − y
2
f (− 1 , 1)?
- (12 points) Let f (x, y) = 3xy − x
3 − y
3 .
(a) Find all critical points of f.
(b) Use the second derivative test to classify each critical point you found in (b) as a local maximum,
local minimum, or saddle point. If you cannot use the second derivative test to describe the
critical points state that and explain why.
- (9 points) Let C be the upper half of the circle with radius 2 centered at (1, −2).
(a) Write a parameterization of C.
(b) A “fence” is built over C with height h(x, y) = x
2
- y + 20. Set up (but do not evaluate)
the integral that represents the area of one side of the fence. Simplify so that there is no vector
notation in your answer.
- (6 points) For the following integral, sketch the region of integration and reverse the order of integration.
Do NOT evaluate the integral.
2
0
√
2 y
−
√
2 y
(3x + 2y) dxdy
- (9 points) Let
F = (3x
2 y
3
4 , 3 x
3 y
2
4
3 ).
(a) Show that
F is path independent.
(b) Use the fundamental theorem of path integrals to evaluate
C
F · d~x, where C is the line segment
from (1, 2) to (− 1 , 3).
- (18 points) Let R be the region in the first quadrant which is bounded by x = 0 and x = −y
2
Let
F (x, y) = (x + y, xy). Illustrate Green’s theorem for
F on R. In other words, evaluate the two
integrals involved with Green’s Theorem and show that they are are equal as the theorem claims. You
may assume that all hypotheses of Green’s Theorem are satisfied. (Note that the path corresponding
to x = −y
2
f (t) = (−t
2
- (4 points) What is your favorite color?
