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May 5, 2003 MATH 208 Final Exam Spring Semester, 2003
Name: Section: Instructor:
Instructions:
(1) You must show supporting work to receive credits.
(2) Use exact values whenever possible, e.g. π/4 instead of 0. 785398 ...
(3) Calculators are allowed.
(4) Make sure you have all the 7 pages of questions.
Problem 1 2 3 4 5 6 7 8 9 10 11 12 13 Total
Point 14 16 16 16 16 18 14 18 12 16 14 14 16 200
Credit
- (14 points) Let f (x, y) =
4 x + 5y.
(a) (10 points) Find the quadratic approximation to f (x, y) at the point (4, −3).
(b) (4 points) Use the quadratic approximation of (a) to approximate f (4. 1 , − 2 .9).
- (16 points) Let S be the surface given by
6 − z
2
x
2
2
(a) (4 points) Verify that point P = (1, 1 , 2) is on the surface.
(b) (6 points) Find a normal vector of S at the point P.
(c) (6 points) Write down an equation of the tangent plane to S at P.
- (16 points) Use polar coordinates to evaluate the following integral:
2
0
4 −y
2
0
e
x
2 +y
2
dx dy.
- (18 points) Let f (x, y, z) = 2x
2 ln(y) + xe
2 z , let P = (− 2 , 1 , 0) and Q = (2, 5 , −2).
(a) (6 points) Find and simplify 5 f (− 2 , 1 , 0).
(b) (6 points) Find and simplify the rate of change of f at P in the direction from P to Q.
(c) (6 points) Find and simplify the unit vector in the direction in which f decreases most rapidly at P.
- (14 points) Consider
1
0
√
1 −x
2
0
1 −x
2 −y
2
0
(x + z)dzdydx. Change the integral to an iterated triple
integral in the spherical coordinates. (DO NOT EVALUATE the triple integral.)
- (18 points) Let
F (x, y) = (x
2
i + (x
2
j.
(a) (6 points) Use a derivative test to verify that the vector field
F (x, y) is a conservative vector field.
(b) (8 points) Find a potential function f for
F (x, y).
(c) (4 points) Find the
C
F (x, y) · d~r, where C is a smooth path from (1, 0) to (0, 1).
- (12 points) Consider the integral
2
0
8 x
x
4
f (x, y)dydx.
(a) (4 points) Sketch or describe the region of integration.
(b) (8 points) Switch the order of integration.
- (14 points) Suppose
F is a smooth vector field defined everywhere such that div
F = 10. Find the flux of
F out of a closed cylinder (with cover and base) of height 5 and radius 2, centered on the z-axis with base
in the xy-plane.
- (16 points) Let
F = (y + 2z)
i + 4x~j + yz
k.
(a) (6 points) Find the curl of
F : 5 ×
F.
(b) (10 points) Use your result from part (a) to find the line integral around the circle of radius 1 in the
xy-plane, centered at the origin, oriented counterclockwise when viewed from above.
(... End )
