Name
MATH 253H
1-12
/48
Fall 2014
10
/20
P. Yasskin
11
/10
12
/25
Total
/103
Final Exam
Sections 201-202
Solutions
Multiple Choice: (4 points each. No part credit.)
1. If z
c.
z
r
z
r
z
d.
z
e.
z
a.
b.
where x
f x, y
f
sin
x
f
cos
y
f
r sin
y
f
r cos
y
f
r cos
y
f
r sin
y
f
r cos
x
f
r sin
x
f
r sin
x
f
r cos
x
z
r
z
Solution:
and y
r cos
f x
x r
f x
x
then
r sin
Correct Choice
f y
y r
f y
y
f
cos
x
f
r sin
x
2. If a car is currently at the point P
f
sin
y
f
r cos
y
and has velocity v
3, 4
2, 1
at what rate is its distance
from the origin currently changing?
a. dD
b.
c.
d.
e.
dt
dD
dt
dD
dt
dD
dt
dD
dt
2
2
5
2
5
2
11
5
Solution: D
dD
dt
D dx
x dt
Correct Choice
x2
D
x
y2
D dy
y dt
3
5
x
x
2
2
y
4 1
5
2
y
D
y
3
5
6
4
5
x
2
y
2
4
5
2
1
3. Find the point where the line r t
R u, v
a.
b.
c.
d.
e.
3
u, 2
v, u
4t 1, 2t 1, 3t intersects the plane
v . At this point x, y, z what is x y z?
9
0
9
18
45
Correct Choice
Solution: r t
R u, v :
4t 1 3 u
2t 1 2
So
u 4t 2
v 3 2t
and
3t
4t 2
x, y, z
r1
5, 1, 3
x y z 9
v
3
3t
2t
u v
2t 1
t
1
t2, t3, 8 t2 .
At t 2, Ham releases a space torpedo which then glides with constant velocity equal to the
velocity of the Centenial Eagle at the instant of release. Where does the torpedo hit the surface of the
Death Star, which is at z 0?
4. Ham Duet is flying the Centenial Eagle above the Death Star along the curve r t
a.
b.
c.
d.
e.
12, 32, 4
8, 20, 0
Correct Choice
4, 8, 4
0, 4, 8
8, 8 6 2 , 0
Solution: r t
t2, t3, 8 t2 , r 2
4, 8, 4 , v t
2t, 3t 2 , 2t v 2
X P tv
4, 8, 4 t 4, 12, 4
4 4t, 8 12t, 4 4t
It hits the surface when z 4 4t 0. Or t 1. So X
8, 20, 0 .
4, 12, 4
5. Duke Skywater is flying the Centenial Eagle through a dangerous polaron field whose density is given
1
. If Duke is currently at the point P
1, 2, 2
x2 y2 z2
should he fly to REDUCE the polaron density as fast as possible?
by
a.
b.
c.
d.
e.
2, 1, 1
3 3 3
2, 1, 1
3 3 3
1, 2, 2
3 3 3
1, 2, 2
3 3 3
1, 2, 2
3 3 3
in what unit vector direction
Correct Choice
x
Solution:
x
2
2
2 3/2
y z
1 1 4 4
1
27
9
Fastest DECREASE along: u
,
y
x
2
y
2
z2
1
1,2,3
3/2
,
z
x2
9
y2
z2
3/2
1, 2 , 2
27 27 27
1, 2,2
1, 2 , 2
27 27 27
1, 2, 2
3 3 3
2
6. Queen Leah is flying the Centenial Eagle above Planet Tattoo along the curve r t
b.
c.
d.
e.
Correct Choice
3
5
3
10
9
3
2 3
t2, t3, 8
Solution: r t
t2
16t 36t 3
2 8t 2 9t 4
d |v |
dt
aT
t2 .
1?
3
What is her tangential acceleration, a T , at t
a. 10
t2, t3, 8
vt
aT 1
3
2t, 3t 2 , 2t |v |
16
36
3
27
8
1
2
9
9
7. Find the tangent plane to the graph of z
1
xy
at
4t 2
1
2
16
3
9t 4
4t 2
4
3
8t 2
9t 4
10
3
2, 3 . The z-intercept is
a. 0
b. 1
Correct Choice
2
c. 1
d. 1
6
e. 1
36
1 f y x, y
1
x2y
xy 2
1 f x 2, 3
1 f y 2, 3
1
f 2, 3
6
12
18
f tan x, y
f 2, 3 f x 2, 3 x 2 f y 2, 3 y 3
1
1 x 2
1 y 3
1
1 2
1
c
6
12
18
6
12
18
Solution:
z
z
f x, y
1
xy
f x x, y
8. Find the tangent plane to the graph of the ellipsoid x 2
3
3xy
1
2
2y 2
z2
21 at
2, 1, 3 . The
z-intercept is
a.
b.
c.
d.
e.
3
4
6
7
42
Correct Choice
Solution: F x 2 3xy 2y 2 z 2
P
N
F
7, 10, 6
N X N P
P
6c
42
c
2, 1, 3
7x 10y
F
2x 3y, 3x
6z 7 2 10 1
4y, 2z
63
42
7
3
9.
y
A wire has the shape of the spiral given in polar coordinates
by r
from
x2
0 to
and has linear density
-2
1
2
-1
4
6
x
-2
y 2 . Find its mass.
-3
Hint: Parametrize the curve.
a. 1 1
2 3/2
1
2 3/2
1
2 3/2
1
2 3/2
1
2 3/2
b.
c.
d.
e.
3
2
3
4
3
2
3
4
3
1
3
2
3
4
3
Solution: r
r
Correct Choice
cos , sin
cos
|v |
-4
sin
M
2
v
sin
cos
ds
10.
Compute
2 3/2
0
1 1
3
sin , sin
cos 2
|v | d
1
0
1 1
3
cos
2
2
2
cos
sin 2
sin 2
d
2x, 2y
y
counterclockwise
to
2
-2
2
4
6
x
-2
2 .
Hint: Use a theorem.
a.
b.
c.
d.
e.
2
1
1
3
2 3/2
F ds for F
from
cos 2
0
around the piece of the spiral given in polar coordinates
by r
2
-4
2
2
3
4
5
2
2
Correct Choice
2
2
Solution: To use the Fundamental Theorem of Calculus for Curves, we need a scalar potential and
the endpoints.
F
f
2x
2y
f x2 y2
xf
yf
r
is the point x, y
r cos , r sin
,0
r
2 is the point x, y
r cos , r sin
2 ,0
F ds
f ds
f 2 ,0
f
,0
2
2
2
3
2
4
e x y dA over the region D of all points
11. Compute
such that |x|
x, y
1.
|y|
D
Hints: Draw the region and use a change of coordinates with u
a. 2e
1
e
b. e
c. 2e
e. 1
e
y and v
y.
2
e
Correct Choice
y
Solution: Plot in each quadrant:
I:
x
y
1
II:
x
y
1
III:
x
y
1
IV: x
y
1
or
1
-1
I:
x
2
e
1
e
e
d. e
x
u
1
II: v
1
III: u
1
IV: v
1
1
x
-1
u
v
2x
u
v
v
D
1
12. Compute the integral
1
x2
1
2
ev
1
v
2
1
4
1
1
1
e
e
2xz 2 , yz 2 , z 3
F dS for F
between the paraboloid z
Hint: Use a theorem.
96
192
384
768
0
1
4
u
2
u
y
2
1
2
1
2
1
e v 1 du dv
2
1
1
e x y dA
a.
b.
c.
d.
e.
u
x
1
2
1
2
x, y
u, v
J
2y
over the complete surface of the solid
y 2 and the plane z
4, with outward normal.
Correct Choice
Solution: By Gauss’ Theorem,
F dS
F dV
V
V
We do the volume integral in cylindrical coordinates.
The bottom and top surfaces are z r 2 and z 4 which intersect when r 2
F 2z 2 z 2 3z 2 6z 2
dV r dr d dz
2
2
4
0
r2
F dV
V
0
2
4
64r
0
r 7 dr
2
6z 2 r dz dr d
2
0
4
32r 2
r8
8
2z 3
2
4
z
r2
r dr
4
43
r2
3
4
r
2
r dr
0
2
4 128
32
384
0
5
Work Out: (Points indicated. Part credit possible. Show all work.)
13. (20 points) For each integral, plot the region of integration and then compute the integral.
27
3
0
3y
a. I
x4
e
Solution: y
dx dy
30
x3
3
I
x4
e
0
e
0
3
20
3
dy dx
e
x4
y
0
x3
y 0
dx
x4 3
x dx
0
10
0
0
4 y2
2
b. J
e
2
Solution:
1
x2 y2
2
u
x4
I
1
4
3
4x 3 dx
du
81
1
4
e u du
0
x 3 dx
e
u 81
0
1 du
4
1 1
4
e
81
x
dx dy
|y|
y
/4
e
/4
0
1
-1
2
J
1
2
x
u
r2
J
1
2 2
4
r2
r dr d
0
du
4
2r dr
r dr
1 du
2
e u du
0
e
u 4
0
4
1
e
4
6
14.
(10 points) Compute
F ds for F
3y
2x 2 , 3y 2
2x
y
over the complete boundary of the shape at the right,
5
4
which is a square of side 2 under an isosceles triangle
3
with height 3.
2
If you use a theorem, name it.
1
-1
Solution: Green’s Theorem says
P dx
Q
x
Q dy
R
R
P
y
0
1
x
dx dy
Here R is the region inside the square and triangle and
F ds
2x 2 dx
3y
3y 2
2x dy
So P
3y
2x 2 and Q
3y 2
2x.
R
F ds
R
5
x
3y 2
1 dx dy
2x
5 Area
y
3y
2x 2
dx dy
2
3
dx dy
R
5 22
12 3
2
35
R
7
15.
(25 points) Verify Stokes’ Theorem
F dS
F ds
P
P
x 2 z, x
for the vector field F
x2
y
z 2 with y
z, z 2 x
and the paraboloid
9 oriented to the right
Use the following steps:
First the Left Hand Side:
Parametrize the paraboloid as:
r cos , r 2 , r sin
R r,
a. Compute the coordinate tangent vectors:
î
k
er
cos ,
2r,
sin
e
r sin ,
0,
r cos
b. Compute the normal vector:
î 2r 2 cos
N
r cos 2
0
r sin 2
2r 2 sin
k0
2r 2 cos , r, 2r 2 sin
2r 2 cos , r, 2r 2 sin
Reverse N
c. Compute the curl:
î
F
k
x
y
x2z x
î0
z
z
z2
1
x2
k1
1, z 2
0
x2, 1
z2x
d. Evaluate the curl on the surface:
F
1, r 2 , 1
R r,
e. Compute the left hand side:
2
LHS
2
3
0
0
F N dr d
0
P
Do the
3
F dS
0
2r 2 cos
r3
2r 2 sin dr d
integral first:
3
LHS
0
2r 2 sin
r3
2r 2 cos
3
2
0
dr
2
0
r 3 dr
2
r4
3
0
81
2
8
Second the Right Hand Side:
f. Parametrize the boundary circle:
r
3 cos , 9, 3 sin
g. Compute the tangent vector:
r0
3, 9, 0
r
Reverse v
0, 9, 3
2
x 2 z, x
z, z 2 x
on the circle:
3 sin , 27 sin 2 cos
27 cos 2 sin , 3 cos
r
which goes from x axis to z axis, or clockwise.
3 sin , 0, 3 cos
h. Evaluate F
F
We need to orient counterclockwise as seen from the positive y axis.
3 sin , 0, 3 cos
v
i. Compute the right hand side:
2
RHS
F ds
0
P
81
2
2
F vd
2
0
sin 2 2
81 sin 2 cos 2
0
d
81 1 2
2 2
81 sin 2 cos 2
2
d
162 sin 2 cos 2 d
0
81
2
which agrees with the left hand side.
9