Name
MATH 152H
1-14
/56
Exam 2 Spring 2016
15
/15
P. Yasskin
16
/15
17
/18
Total
/104
Sections 201/202 (circle one)
Multiple Choice: (4 points each. No part credit.)
1. Find the general partial fraction expansion of f x
a. A
x
Dx E
x2 4 2
Bx C
x2 4
b. A
x
d. A
x
Bx
x2
Bx
x2
e. A
x
Bx
x2 4
c. A
x
C
4
C
4
x
3
x2 4
4x x 4
16
.
Dx E
x2 4 2
Dx E
x2 4
Dx E
x2 4 2
Dx
2
x2 4
2. In the partial fraction expansion
x3
4x
x4
32
16
Ax
x2
B
4
C
x
2
x
D ,
2
which coefficient is INCORRECT?
a. A
1
b. B
4
c. C
1
d. D
1
e. All of the above are correct.
1
3. Compute
4
1
1 dx
x2
a. 5
4
b.
5
4
c. diverges to
d. diverges to
e. diverges but not to
4. The integral
3
sin x
dx is
x
a. divergent by comparison with
2 dx.
x
b. divergent by comparison with
3 dx.
x
c. divergent by comparison with
4 dx.
x
d. convergent by comparison with
2 dx.
x
e. convergent by comparison with
4 dx.
x
2
5. Compute
4/3
0
a. 16
1
9x 2
16
dx
9
1
b.
9
c.
d.
e.
12
24
48
6. Which of the following integrals results after performing an appropriate trig substitution for
4/3
1
a.
b.
c.
d.
e.
1
20
1
20
5
64
5
64
1
16
1
dx
25x 2 16
arcsec 5/4
csc d
arcsec 5/3
arcsec 5/3
csc d
arcsec 5/4
arcsec 5/4
sec d
arcsec 5/3
arcsec 5/3
sec d
arcsec 5/4
arcsec 5/3
cot 2 d
arcsec 5/4
3
7. Compute
3/5
0
a.
b.
c.
d.
e.
1
dx
25x 2 16
1
20
1
20
1 ln 7
40
1
1 ln 7
20
40
1 ln 7
1
40
20
8. Find the length of the curve x
2t 2 , and y
t 3 for 0
t
1.
a. 732
b. 108
c. 7
d. 61
27
e. 1
3
4
9. The curve y
sin x for 0
x
is rotated about the x-axis.
Which integral gives the area of the resulting surface?
2 x 1
a.
cos 2 x dx
0
b.
2 sin x 1
cos 2 x dx
2 cos x 1
sin 2 x dx
0
c.
0
sin 2 x dx
cos x 1
d.
0
sin x dx
e.
0
10. The curve y
sin x for 0
x
is rotated about the y-axis.
Which integral gives the area of the resulting surface?
2 x 1
a.
cos 2 x dx
0
b.
2 sin x 1
cos 2 x dx
2 cos x 1
sin 2 x dx
0
c.
0
cos x 1
d.
sin 2 x dx
0
sin x dx
e.
0
5
11. Consider the sequence a n
arctan
3n 6
3n 6
. Compute lim
an.
n
a.
b.
c.
d.
e.
2
3
4
6
2n
12.
n 1
32n
3
12
a.
b. 12
9
c.
d. 9
e. diverges
n
13.
n 1
3 2n
3
a. 3
b. 6
c. 9
d. 12
e. diverges
6
ln
14.
n 3
n2
n2
1
a. ln 3
b.
c.
d.
e.
4
ln 3
2
ln 3
ln 1
4
ln 1
2
Work Out: (Points indicated. Part credit possible. Show all work.)
15.
15 points
Compute
3x
x4
3 2
dx. Simplify to a single ln.
81
7
16.
15 points
Compute
4/3
1
1
dx. Simplify to a single ln.
25x 2 16
8
17.
18 points
Consider the sequence defined recurrsively by
a1
a.
6 pts
i.e. a n
1
ii. Assume a k
Therefore, a n
6 pts
1
1
ak
an
an
1
5
4
an
0:
ak
2
1
0:
0 for all n.
Use mathematical induction to show a n is bounded above by 8.
ii. Assume a k
Therefore, a n
6 pts
a1
0 and prove a k
i. To get started, show a 1
c.
and
Use mathematical induction to show a n is positive and increasing for all n,
a n 0.
i. To get started, show a 2
b.
2
8 and a 2
8 and prove a k
1
8:
8:
8 for all n.
What do you conclude about lim
an?
n
Find lim
an.
n
9