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241 Exam Final
Show all work. No calculators allowed.
Lawrence Sze
Name
June 10 2011
No Calculators allowed. You may assume the following formulae:
If P x, y, z
P r, , z , then
y
sin
x r cos
r
x2 y2
2
x y2
y
If P x, y, z
P , ,
sin cos
x2
y
sin sin
y
tan −1 x
z
cos
cos −1
P , ,
sin
z
cos
D
D
x
2
y2
y2
z2
y
sin
x2
x
cos
z
y2
x2
x
2
y2
x2
sin
z2
y2
x2
y2
y2
z2
1
m
D
then
r
x, y dA
x
cos
, then
x
If P r, , z
m
y
tan −1 x
r sin
r2
z
cos
D a, b
x
z2
r
2
z
2
sin
f xx a, b f yy a, b − f xy a, b
1
m
D
x x, y dA
r
r
2
z2
2
y
y x, y dA
1. (5 points) Describe in words the surface given by
∂z
∂z
and ∂y
if
2. (10 points) Find ∂x
ln x
yz
3. (15 points) Let F 〈2xz y 2 , 2xy, x 2 3z 2 .
a. Find a function f such that F ∇f.
b. Evaluate F dr along the curve C : x
C
1
2
sin 2 cos 2
cos 2
4
xy 2 z 3
t 241 , y
1
t 241 , z
1
t 241 − t 241 0 ≤ t ≤ 1.
4. (10 points) Find the limit if it exists, or show that the limit does not exist.
x3 y3
lim
x,y → 0,0 x 2
y2
5. (15 points) Find the equation of the the tangent plane to the surface z
1, 0, 0
6. (9 points) Match the graph with it’s equation
A______________
1
xe y cos z at
B___________________ C_________________
a. x 2 4y 2 9z 2 1
b. y 2x 2 z 2
c. x 2 − y 2 z 2 1
d. −x 2 y 2 − z 2 1
e. x 2 2z 2 1.
7. (20 points) First, compute y 3 dx − x 3 dy directly, and then by applying Green’s Theorem
C
where C is the positively oriented circle x 2 y 2 4.
8. (15 points) Why is the line integral 2x sin ydx x 2 cos y − 3y 2 dy independent of path?
C
Also, evaluate the integral where C is any path from −1, 0 to 5, 1 .
9. (20 points) Find the mass and the center of mass of E where E is the solid that lies
between the spheres x 2 y 2 z 2 1 and x 2 y 2 z 2 4 in the first octant. Assume the
solid is constructed of a material with uniform density 1.
10. (15 points) Find the maximum and minimum values of the function
f x, y, z
2x 6y 10z subject to the constraint x 2 y 2 z 2 35
11. (15 points) Evaluate the integral used to evaluate
xydA, where R is the region
R
bounded by xy 1, xy 3, y x, and y 12x obtained by transforming the region R to
a rectangular region using an appropriate change of variables.
12. Evaluate
5, 0 ,
x
R
5 5
,
2 2
1
13. Evaluate
0
ydA where R is the trapezoidal region with vertices given by 0, 0 ,
and
1
5
2
1
x y3 1
, − 52 .
dy dx