MATH 152, Spring 2014
COMMON EXAM II - VERSION A
FIRST NAME:
LAST NAME:
INSTRUCTOR:
SECTION NUMBER:
UIN:
DIRECTIONS:
1. The use of a calculator, laptop or computer is prohibited.
2. In Part 1 (Problems 1-15), mark the correct choice on your ScanTron using a No. 2 pencil. For your own records,
also record your choices on your exam!
3. In Part 2 (Problems 16-20), present your solutions in the space provided. Show all your work neatly and concisely
and clearly indicate your final answer. You will be graded not merely on the final answer, but also on the quality
and correctness of the work leading up to it.
4. Be sure to write your name, section number and version letter of the exam on the ScanTron form.
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Question
Points Awarded
Points
1-15
60
16
9
17
10
18
7
19
6
20
8
100
1
PART I: Multiple Choice: 4 points each
1. Evaluate
Z
4
0
4x2
dx.
x+2
(a) 64 + 16 ln(3)
(b) 16 ln(12)
(c) 64 + 16 ln(12)
(d) 16 ln(3)
(e) Integral diverges
2. If the nth partial sum of the series
∞
P
an is given by sn =
n=1
(a) a4 =
(b) a4 =
(c) a4 =
(d) a4 =
(e) a4 =
1
2
1
63
55
63
1
99
89
99
n
, what is a4 ?
2n + 1
3. Which of the following series diverges by the Test for Divergence?
∞
P
1
(a)
cos
n
n=1
∞ (−1)n
P
(b)
4
n=1 n + 2
∞
P
n
(c)
3
n=1 n + 1
∞
P
1
(d)
sin
n
n=1
∞
P
4n
(e)
n+2
n=1 9
2
4. After making the appropriate trigonometric substutition, the integral
(a)
Z
π/4
16 tan2 θ sec4 θ dθ
0
(b)
Z
π/2
8 sin2 θ cos θ dθ
0
(c)
Z
(d)
Z
π/2
8 sin2 θ cos2 θ dθ
0
π/2
16 sin2 θ cos2 θ dθ
0
(e)
Z
π/4
8 tan2 θ sec2 θ dθ
0
5. The sequence an = ln(n) − ln(4n + 2)
(a) converges to 0
(b) converges to ln(4)
1
(c) converges to ln
4
(d) diverges to ∞
(e) diverges to −∞
6. Find the length of the curve y = x3/2 , 0 ≤ x ≤ 4.
(a)
(b)
(c)
(d)
(e)
√
3
(10 10 − 1)
2
64
27
√
2
(10 10 − 1)
3
8 √
10 10
27
√
8
(10 10 − 1)
27
3
Z
2
x2
0
p
4 − x2 dx becomes:
7.
Z
tan4 x sec4 x dx =
(a)
(b)
(c)
(d)
(e)
8.
∞
P
n=1
1
1
tan7 x + tan5 x + C
7
5
1
1
tan6 x + tan4 x + C
6
4
1
1
7
− tan x + tan5 x + C
7
5
1
1
7
tan x − tan5 x + C
7
5
1
1
tan5 x + sec4 x + C
5
5
3n
4n+1
=
(a) 1
(b) 0
3
(c)
28
(d) 4
3
(e)
4
9. Which of the following integrals gives the surface area obtained by revolving the curve y = arctan(x), 0 ≤ x ≤ 1,
about the x-axis?
s
2
Z 1
1
(a)
2πx 1 +
dx
x2 + 1
0
s
2
Z 1
1
dx
(b)
2π arctan(x) 1 +
x2 − 1
0
s
2
Z 1
1
dx
(c)
2π arctan(x) 1 +
x2 + 1
0
s
2
Z 1
1
dx
(d)
2πx 1 +
x2 − 1
0
s
2
Z 1
1
2π arctan(x) 1 + √
(e)
dx
x2 + 1
0
4
10. The integral
Z
2
−1
dx
x2
(a) diverges to −∞
(b) diverges to ∞
(c) diverges to 2
1
(d) diverges to
2
(e) diverges to ln 4
11. Which of the following sequences is both bounded and decreasing?
(a) an = e−n
(b) an = cos n
(c) an = ln n
n
−1
(d) an =
2
1
(e) an = 1 − 2
n
12. Find the length of the curve x =
(a)
(b)
(c)
(d)
(e)
t3
t2
, y = , 0 ≤ t ≤ 1.
2
3
5
6
2 √
(2 2 − 1)
3
3 √
(2 2 − 1)
4
11
6
1 √
(2 2 − 1)
3
5
13.
Z
sin3 x cos2 x dx =
(a)
(b)
(c)
(d)
(e)
14.
Z
1
1
cos3 x + cos5 x + C
3
5
1
1
− cos3 x − cos4 x + C
3
4
1
1
3
− cos x − cos5 x + C
3
5
1
1
3
− cos x + cos5 x + C
3
5
1
1
4
3
sin x
cos x + C
4
3
x+2
dx =
− 1)
x2 (x
(a) −3 ln |x| +
(b) 3 ln |x| +
2
+ 3 ln |x − 1| + C
x
2
+ 3 ln |x − 1| + C
x
2
+ 3 ln |x − 1| + C
x
2
(d) −3 ln |x| − + 3 ln |x − 1| + C
x
2
(e)
+ 3 ln |x − 1| + C
x
(c) −
15. Find the surface area obtained by rotating the curve y = x2 , 0 ≤ x ≤
(a)
(b)
(c)
(d)
(e)
π √
(5 5 − 1)
6
26π
3
13π
3
9π
2
π √
(5 5 − 1)
8
6
√
2, about the y-axis.
PART II WORK OUT
Directions: Present your solutions in the space provided. Show all your work neatly and concisely and Box your
final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work
leading up to it.
16. (9 pts) Integrate
Z
4
dx.
x3 + 4x
7
1
1
cos
17. (a) (6 pts) Find sn , the nth partial sum, for the series
− cos
.
n
n+1
n=1
∞
P
1
1
cos
− cos
.
(b) (4 pts) Using the nth partial sum found in part (a), find the sum of the series
n
n+1
n=1
∞
P
18. (7 pts) Using a comparison theorem, prove that the integral
8
Z
∞
1
dx
converges.
x + e4x
19. (6 pts) Find
Z
sin2 x cos2 x dx
20. (8 pts) Find
Z
dx
√
4x2 + 9
9