Name
Sec
MATH 251
Final Exam
Section 200/511
1-12
/48
14
/16
13
/16
15
/25
Total
/105
Spring 2011
Solutions
P. Yasskin
Multiple Choice: (4 points each. No part credit.)
1. Find the projection of the vector u
12 , 3 , 4
13 13 13
12 , 3 , 4
13 13 13
26 , 13 , 26
9
9
9
26 , 13 , 26
9 9
9
108 , 9 , 36
13 13 13
a.
b.
c.
d.
e.
u v
24
proj v u
2. If |u |
3 8
u vv
|v | 2
along the vector v
2, 1, 2 .
Correct Choice
13
|v |
13 2, 1, 2
9
4 and |v |
12, 3, 4
4
1 4
3
26 , 13 , 26
9 9
9
3 and the angle between u and v is
3
then |u
v|
Correct Choice
a. 6 3
b. 6
c. 3 3
d. 3
e. 12 3
|u
v|
|u ||v | sin
4 3
3
2
6 3
3. Find the point where the line through the origin perpendicular to the plane 3x
intersects that plane. At this point x
a. 2
y
3y
4z
17
z
Correct Choice
b. 3
c. 5
d. 17
e. 1
2
The normal to the plane is:
N
3, 3, 4 .
The line through the origin O in the direction N is X O tN or x, y, z
3t, 3t, 4t .
Plug the line into the plane and solve for t:
2x 3y 4z 3 3t 3 3t 4 4t
34t 17.
1.
3 , 3 ,2
Plug into the line:
x, y, z
So
x y z 2
So
t
2
2 2
1
t 2 , 2t, ln t
4. Find the arc length of the curve r t
a. e 2
e 2 , 2e, 1 .
and
1, 2, 0
Correct Choice
b. e 2
1
2
1
c. e
between
d. 2 e 3
3
e. 2 e 3
3
e
5
3
e
t 2 , 2t, ln t
rt
e 2 ,e,1
L
2t, 2, 1
t
v
e
ds
e
1,2,0
1
t
2t
|v | dt
1
1
4t 2
|v |
t2
dt
1
t2
4
ln t
e
1
1
t
2t
e2
1
2
2t
1
t
e2
1
5. The pressure, P, density, , and temperature, T, of a certain ideal gas are related by P
At the point 1, 2, 3 , the density and its gradient are
4 and
2, 1, 3 .
while the temperature and its gradient are T 300 and T
Hence the pressure is P 3 4 300 3600 and its gradient is P
a.
1802. 4, 2704. 8, 2701. 2
b.
600. 8, 901. 900. 4
c.
68, 132, 42
d.
204, 372, 126
e.
1. 2, 1. 2, 0. 9
P
x
P
P
y
P
P
z
P
3 T.
0. 2, 0. 4, 0. 1 ,
Correct Choice
x
P
T
T
x
3T
y
P
T
T
y
3T
z
P
T
T
z
3T
x
y
z
3
T
x
3 300 0. 2
3 4 2
204
3
T
y
3 300 0. 4
3 4 1
372
3
T
z
3 300 0. 1
3 4 3
126
6. Find the equation of the plane tangent to the graph of x 2 yz 2
2y 2 z 3
10 at the point
3, 2, 1 .
The z-intercept is
a.
b.
c.
d.
e.
6
25
5
25
6
60
5
3
F x, y, z
N
N X
Correct Choice
x 2 yz 2
F 3, 2, 1
N P
2y 2 z 3
F
23 2 , 3
12x
y
12z
2
2xyz 2 , x 2 z 2
4 2 ,2 3
12 3
2
2
4yz 3 , 2x 2 yz
2
62
12 1
6y 2 z 2
2
12, 1, 12
50
z-intercept
50
12
25
6
2
7. Han Deut is flying the Millennium Eagle through a dangerous zenithon field whose density is
xyz.
If his current position is x, y, z
1, 1, 2 , in what unit vector direction should he travel to
decrease the density as fast as possible?
2, 2, 1
2, 2, 1
3 3 3
2, 2, 1
2, 2, 1
3 3 3
2, 2, 1
3 3 3
a.
b.
c.
d.
e.
Correct Choice
yz, xz, xy
2, 2, 1
1, 1,2
4
He should travel in the direction v
8. Compute
a.
b.
2
c.
4
d.
e
9
e
9
1
4
1
e.
e
x2 y2
3
2, 2, 1 .
3 3 3
or the unit direction v
2, 2, 1
dA over the quarter circle x 2
1
y2
9 in the first quadrant.
9
e
2
e
4
x2 y2
1
9
e
Correct Choice
9
e
/2
3
dA
e
0
9. Compute
8
2
0
x 1/3
r2
e
r dr d
2
0
r2
3
2
0
4
1
e
9
cos y 4 dy dx
HINT: Reverse the order of integration.
a. 1 sin 4
1
4
4
1
1
b.
sin 64
4
4
1
c.
sin 64
4
1
d. 1 sin 16
4
4
1
e.
sin 16
Correct Choice
4
Plot the region. Reverse the order.
x 1/3
Compute new limits: y
8
0
2
x
cos y 4 dy dx
1/3
2
0
y 3 cos y 4 dy
2
0
y3
x
cos y 4 dx dy
0
y
y3
2
cos y 4
0
sin y 4
4
2
y o
1 sin 16
4
x
y3
x 0
dy
2
1
0
0
2
4
6
8
x
3
10.
z dV over the solid sphere x 2
Compute
given in spherical coordinates by
/2
2
2 cos
2
cos
0
0
/2
2
/2
sin d d d
from
0
to
/2
cos 6
6
8
F ds for F
2, 4, 6
2
0
4 cos 5 sin d
1
2
1
0. What does this say about ?
0
11. Compute
z
2 cos .
HINT: The whole sphere has z
a. 2
3
4
b.
Correct Choice
3
c. 4
6
d.
3
e.
12
2
z
cos
dV
sin d d d
z dV
y2
2x, 2y, 2z
2 cos
4
cos sin d
4
0
4
3
0
2, 4, 6
t t t
along the curve r t
1, 2, 3 .
HINT: Find a scalar potential.
a.
70
b.
42
Correct Choice
c. 0
d. 42
e. 70
F
f
F ds
12. Compute
xf
2x
f ds
yf
2y
f 1, 2, 3
F ds for F
zf
2z
12
f 2, 4, 6
sec x 3
x2
f x, y, z
22
5y, cos y 5
32
3x
22
y2
42
z2
62
42
counterclockwise around the triangle with
vertices 0, 0 , 8, 0 and 0, 4 .
Hint: Use Green’s Theorem.
a. 12
b. 16
c. 32
d. 64
e. 128
P
sec x 3
F ds
Correct Choice
5y
Q
P dx
Q dy
cos y 5
xQ
3x
xQ
y P dx dy
yP
3
8 dx dy
5
8Area
8
8
1
2
8 4
128
4
Work Out: (Points indicated. Part credit possible. Show all work.)
13. (16 points) Use Lagrange multipliers to find 4 numbers, a, b, c, and d, whose product is
and for which a
2b
3c
4d is a minimum.
Minimize f
2b
3c
4d subject to the constraint g
a
2
3
2.
3
abcd
f
1, 2, 3, 4
g
bcd, acd, abd, abc
f
g:
1
bcd
2
acd
3
abd
4
abc
Make the right sides all be abcd and equate:
abcd a 2b 3c 4d
a
a
a
So
b
c
d
Substitute into the constraint:
2
3
4
4
a
2
2 24 16
2,d
a4
a 2, b 1, c
abcd a a a a
2 3 4
24
3
3
3
1
2
14. (16 points) Find the mass and the y-component of the center of mass of the quarter of the cylinder
x2
y2
xyz
4 with 0
0
y
15.
0
3
y dV
0
M xz
M
48 1
5 9
x
0, y
if the mass density is
0
F dV
for the vector field F
2
2xy , 2yx , z x
2 x2
y2
2
4
3
0
1 9
2 2
32 1
5
3
9
9
2
48
5
F dS
V
V
2
xyz.
dV
2
(25 points) Verify Gauss’ Theorem
above the cone z
0, z
r dr d dz
/2
r 4 2 sin 2
z2 3
r 3 sin cos z dr d dz
4 0
2
2 0
0
0
/2
2
/2 2
3
5
sin
r
z2
r 4 sin 2 cos z dr d dz
5
3
2
0
0
0
0
16
15
/2
dV
M xz
r 2 sin cos z
r cos r sin z
3
M
3 in the first octant
z
y
2
and the solid
below the plane z
2.
Be careful with orientations. Use the following steps:
First the Left Hand Side:
a. Compute the divergence:
F 2y 2 2x 2 x 2 y 2 3x 2 3y 2
b. Express the divergence and the volume element in the appropriate coordinate system:
F 3r 2
dV r dr d dz
c. Find the limits of integration:
0
2
2 x2 y2
z 2 becomes 2r z 2
2r 2 when r 1
d. Compute the left hand side:
2
1
2
F dV
0
V
1
12
0
r3
r 4 dr
0
1
3r 2 r dz dr d
2
2r
12
3r 3 z
0
r4
4
r5
5
1
12
0
1
4
2
1
dr
2r
6
1
5
3
5
r3 2
2r dr
0
5
Second the Right Hand Side:
The boundary surface consists of a cone C and a disk D with appropriate orientations.
e. Complete the parametrization of the cone C:
R r,
r cos , r sin ,
2r
f. Compute the tangent vectors:
er
cos , sin , 2
e
r sin , r cos , 0
g. Compute the normal vector:
2r sin
k r cos 2
r sin 2
N î 2r cos
This is up and in. Reverse N
2r cos , 2r sin , r
2
2
2
2
h. Evaluate F
2xy , 2yx , z x y
on the cone:
2
3
3
F
2r cos sin , 2r sin cos 2 , 2r 3
2r cos , 2r sin , r
R r,
i. Compute the dot product:
F N 4r 4 cos 2 sin 2
4r 4 sin 2 cos 2
j. Compute the flux through C:
2
1
F dS
0
C
2r 4 4 sin 2 cos 2
2r 4
1 dr d
0
2r 4 4 sin 2 cos 2
5
2 r
5
2 2 cos 2 2 d
2 2 1 cos 4
5 0
5 0
2
k. Complete the parametrization of the disk D:
R r,
r cos , r sin ,
d
1
2
0
0
1
5
1
sin 2 2
sin 4
4
1 d
2
0
2
5
2
l. Compute the tangent vectors:
er
cos , sin , 0
e
r sin , r cos , 0
m. Compute the normal vector:
N î0
0 k r cos 2
r sin 2
0, 0, r
This is correctly up.
n. Evaluate F
2xy 2 , 2yx 2 , z x 2 y 2
on the disk:
2
3
3
F
2r cos sin , 2r sin cos 2 , 2r 2
R r,
o. Compute the dot product:
F N 2r 3
p. Compute the flux through D:
2
1
F dS
0
D
2r 3 dr d
2
0
1
r4
2
0
q. Compute the right hand side:
F dS
V
F dS
C
F dS
D
2
5
3
5
which agrees with (d).
6