MATH 152 Honors
FINAL EXAM
Sections 201-202
Solutions
Spring 2012
P. Yasskin
Multiple Choice: (15 problems, 4 points each)
1.
e
Compute
9x 2 ln x dx
1
a.
2e 3
1
b.
2e
3
2
c.
2e 3
d.
3e 3
3e 2
e.
3e 3
3e 2
u
3
ln x dv
1 dx
v
x
du
2.
CORRECT
2
Compute
x
1
a.
b.
c.
d.
e.
2
1
3.
9x 2 dx
3x
3
1
2
4/3
e
3x 2 ln x dx
3x 3 ln x
b.
c.
d.
e.
1
2e 3
1
dx
3
CORRECT
1
x 2
dx
4/3
61
54
16
9
11
9
1
9
1
54
1
0
L
e
x3
3
1
L
u
3x 3 ln x
1
3x
2
1/3
2
1
lim x
2
Find the arclength of the parametric curve
a.
3x 2 dx
16
1
36
x
3
2
t4
x
3
1 2
1/3
y
1/3
1 t6
2
for 0
t
1.
CORRECT
dx
dt
9t 4
2
du
25
u du
16
dy
dt
2
1
dt
2
3t 5
2
1
dt
0
36t 3 dt
1
36
4t 3
2 u 3/2
3
1 du
36
25
16
16t 6
9t 10 dt
0
1
t 3 16
9t 4 dt
0
t 3 dt
1 25 3/2
54
16 3/2
1 125
54
64
61
54
1
4.
A 2 meter bar has linear density
Find the average density of the bar.
a.
2 kg/m
b.
3 kg/m
c.
4. 5 kg/m
d.
5 kg/m
e.
6 kg/m
ave
5.
CORRECT
2
1
2
0
5 m
7
5 m
6
6 m
5
7 m
5
42 m
5
b.
c.
d.
e.
2
1
M
2
x 3 dx
2
M1
x
0
42
5 6
x4
4
1
2
3
4
5
y dy
x
3
x 3 kg/m where x is measured from one end.
1
2
4
6
0
x5
5
2
0
42
5
7
5
dy
dx
If y x satisfies the differential equation
condition y 0
3, find y 4 .
a.
b.
c.
d.
e.
4
2
x2
2
x 3 dx
x1
M1
M
x
1 2
2
0
CORRECT
0
6.
x4
4
1 x
2
x 3 dx
1
A 2 meter bar has linear density
Find the center of mass of the bar.
a.
x 3 kg/m where x is measured from one end.
1
x
y
and the initial
CORRECT
y2
2
x dx
0 & y
3
x2
2
9
2
C
0
2
C
x2
y
C
9
2
2C
y
x2
9
y4
5
2
7.
8.
dy
dx
Find an integrating factor for the differential equation
a.
b.
c.
d.
e.
e cos x
e sin x
e cos x
2
ex
2
e x
dy
dx
2xy
2xy
sin x.
CORRECT
sin x
P
2x
I
A sequence is defined recursively by:
e
P dx
e
x2
4 and a n
a1
10a n
1
16 . Find nlim a n .
2
4
6
8
CORRECT
Diverges
a.
b.
c.
d.
e.
If L
lim a n exists, then L
n
10L
16 .
2
So L 10L 16 0 or L 2 L 8
0 or L 2 or 8.
a1 4
a2
10 4 16
24
4 a1.
So a n is increasing from 4 and the limit is probably 8.
To see this rigorously, let P n be the statement a n 1 a n and 4 a n 8.
Since a 2 a 1 and 4 a 1 4 8, we have P 1 is true.
Assume P k is true, i.e. a k 1 a k and 4 a k 8.
Then 10a k 1 10a k and 40 10a k 80 and 10a k 1 16 10a k 16 and 24 10a k 16
Since 10a k 16 24 0, we can take square roots without modifying inequalities.
So 10a k 1 16
10a k 16 and 24
10a k 16
64
Or a k 2 a k 1 and 4
24
ak 1
64
8 which is P k 1 .
So a n is increasing and bounded above by 8. So the limit exists and must be 8.
64
3n
9.
n 2
2 2n
1
a.
2
b.
9
14
9
CORRECT
2
4
Diverges
c.
d.
e.
a
32
24 1
9
8
r
3
4
|r|
9
8
3n
1
n 2
2 2n
1
1
9
3
4
8
6
9
2
3
10. Find the radius of convergence of the series
n 1
a.
b.
c.
d.
e.
L
0
1
3
1
2
2
3
an 1
lim
an
n
11. lim sin x
x 0
b.
c.
d.
x
2
3 n.
CORRECT
3 n1 n 1 2
2|x 3|
2n x 3 n
n 2 2
1 . So R
1
3| 1 or |x 3|
2
2
2n
lim
n
Convergent if L
a.
2n
n 1
2|x
1
x
x cos x
x3
1
6
1
3
1
2
2
3
CORRECT
e.
sin x
x3
3!
x
lim sin x
x 0
x
x cos x
x3
lim
x3
6
3
x x
6
cos x
x
x3
2
x cos x
lim
x3
x 0
12. Suppose the series
x2
2
1
ne
x 0
x3
6
x
x3
2
1
2
x3
9
n2
is approximated by its 9 th partial sum
n 1
x3
2
ne
n2
1
6
1
3
.
n 1
Use an integral to bound the error in this approximation.
a.
b.
c.
d.
e.
1e
2
1e
2
1e
2
1e
2
1e
2
64
81
CORRECT
100
121
144
The error is E
ne
n2
.
n 10
So E
ne
9
n2
dn
1e
2
n2
9
0
1e
2
92
1e
2
81
4
13. Find the angle between the vectors u
0°
30°
45°
60°
90°
a.
b.
c.
d.
e.
u v
1
1, 1, 1
and v
1, 2, 1 .
CORRECT
2
1
0
u v
|u ||v |
cos
0
90°
14. If u points South-West and v points Up, which way does u
a.
South-East
b.
North-East
c.
North-West
d.
45° Up from North-West
e.
45° Down from North-West
v point?
CORRECT
Hold your right fingers South-West with the palm facing Up. Then your thumb points
North-West.
15. Find a unit vector perpendicular to both a
a.
2, 4, 2
b.
2, 4, 2
and b
1, 0, 1 .
1, 2, 1
c.
d.
1 , 2 , 1
6
6
6
e.
1 ,
6
a
3, 2, 1
b
CORRECT
2 , 1
6
6
ı
k
3
2 1
1
0
16
a
b
4
a
a
b
b
1
2 6
ı 2
0
3
1
k0
2
2, 4, 2
1
4
24
2, 4, 2
2 6
1 ,
6
2 ,
6
1
6
5
Work Out (4 questions, 12 points each)
Show all you work.
4
16. Compute
2
x
2 sec
x
2 @
4
2
x
3
8
x2
x
8
x2
3
dx
dx
4
2 sec tan d
1 or
sec
/3
dx
0
4
/3
1
0
4 @
/3
8 2 sec tan d
8 sec 3 4 sec 2
4
cos 2 d
2
0
x
0
sin 2
2
1
2
2 or
sec
sec d
sec 3
/3
1
2
0
3
/3
3
cos 2 d
0
1 sin 2
2
3
6
3
8
x 2 is rotated about the y-axis to form a bowl. If the bowl contains 8 cm 3 of
water, what is the height of the water in the bowl?
17. The curve y
r2.
The silce at height y is a circle. So its area is A
The radius is
So the area is
A
and the volume of the slice of thickness dy is
dV
r
x
y
y
y dy.
So the volume up to height h is
y2
2
h
V
dV
y dy
0
We equate the volume to 8
2
h2
8
h2
16
h
0
2
h2
and solve for h:
h
4 cm
6
18. A leaking sandbag is lifted 20 ft at 2 ft/sec. The sandbag starts out weighing 50 lb but is leaking
sand at 3 lb/sec. How much work is done to lift the sandbag?
HINT: What is the weight of the bag when it is y ft above the ground?
In t sec the bag is lifted y
ft
sec
2
t sec
2t ft.
During this time it loses F 3 lb/sec t sec 3t lb of sand.
y
y
F 50 3 . So the work done is
So F 3 and the weight is F 50
2
2
20
20
2
y
3y 2 20
W
Fdy
50 3
dy
50y
50 20 3 20
700 ft-lb
2
4
4
0
0
0
1 n2n
n!
19. Determine if the series
n 0
diverges.
If it converges, find the sum.
If it diverges, does it diverge to
2 n . Since e x
n!
The related absolute series is
n 0
Alternatively, apply the ratio test:
an 1
lim
an
n
lim
n
2 n 1 n!
n 1 ! 2n
converges absolutely, converges but not absolutely or
lim
n
n
2
n!
an
2
n
0
1
an
n 0
,
x n , we conclude
n!
or neither?
n 0
2n
n!
e2.
n 1
2
n 1 !
1
1
Absolute series converges and original series converges absolutely.
Similarly, since e x
n 0
x n , we conclude
n!
n 0
1 n2n
n!
e 2.
2
Original series converges to e .
Circle One:
Converges Absolutely
Fill in the Blank:
Converges Conditionally
Converges to
Or Circle One:
Diverges to
e
Diverges
2
Neither
7