MATH 152
SPRING 2016
SAMPLE EXAM II
1. Find the limit of the sequence an = (−1)n e
1
e
b) diverges
a)
c) 0
d) 1
e) -1
2.
32n
=
n
n=1 (−10)
∞
X
a)
b)
c)
d)
e)
10
19
9
19
1
8
−10
−9
19
3. The series
∞
X
n
n=1 3n + 1
1
3
b) Converges to 0
a) Converges to
c) Converges to 1
1
d) Converges to
4
e) diverges
ln n
n
4. If the nth partial sum of the series
∞
X
an is sn =
n=1
a) (i) an =
∞
X
1
an = 1
and (ii)
(n + 1)(n + 2)
n=0
b) (i) an =
∞
X
2
an = 1
and (ii)
(n + 1)(n + 2)
n=0
∞
X
n+1
an .
, find (i) an and (ii)
n+2
n=1
∞
X
1
c) (i) an =
an = 1
and (ii)
(n + 1)
n=0
d) (i) an =
∞
X
n
an = 1
and (ii)
(n + 1)(n + 2)
n=0
e) None of these, the series diverges by the Test for Divergence.
5. The integral
Z
∞
1
sin2 x
dx
x2
a) converges to 0
b) diverges to ∞
c) diverges by oscillation
1
dx
2
Z ∞1 x
1
e) diverges by comparison to
dx
x
1
d) converges by comparison to
6.
Z
∞
xe−x
2 /2
dx =
0
a) divergent
1
b)
2
c) 0
1
d)
4
e) 1
Z
∞
7. Find the length of the curve y = 4x3/2 from (0, 0) to (2, 4).
√
1
(73 73 − 1)
54
√
1
(73 73 − 1)
b)
27
√
1
(37 37 − 1)
c)
54
√
1
(37 37 − 1)
d)
27
e) None of these.
a)
3
. Given that the terms
an
of the sequence are increasing and bounded, find the limit of the sequence.
8. Consider the sequence whose terms are defined as a1 = 2, an+1 = 4 −
a) L = 1
b) L = 3
c) L = 2.5
d) L = 2.8
e) The sequence diverges.
9. Which of the following integrals gives the area of the surface obtained by rotating the curve
y = e2x , 0 ≤ x ≤ 1 about the y-axis?
a)
1
Z
√
2π 1 + 4e4x dx
0
s
1
2π 1 + e4x dx
4
0
Z 1
√
c)
2πe2x 1 + 2e2x dx
b)
Z
d)
Z
e)
Z
1
0
1
0
1
0
√
2πe2x 1 + 4e4x dx
√
2πx 1 + 4e4x dx
10. Which of the following sequences is both bounded and decreasing?
a) an = e−n
b) an = cos n
c) an =
−1
2
n
d) an = ln n
1
n2
e) an = 1 −
11. Which of the following integrals gives the area of the surface obtained by rotating the curve
y = ln(3x + 1), 0 ≤ x ≤ 1 about the x-axis?
a)
s
Z
1
Z
ln 4
Z
ln 4
Z
1
Z
ln 4
2πx 1 +
0
b)
2πy 1 +
0
c)
0
d)
0
e2y
dy
9
s
e2y
ey − 1
dy
1+
2π
3
9
s
2πy 1 +
0
e)
s
9
dx
(3x + 1)2
e4y
dy
9
s
2
ey
e −1
dy
1+
2π
3
9
y
Part II - Work Out Problems
12. Find the surface area obtained by rotating the curve parametrized by
π
x(t) = cos2 t, y(t) = sin2 t, 0 ≤ t ≤ about the y axis.
2
13.
Z
dx
√
=
x2 1 − x2
14. Find the sum of the series
(i) The series whose nth partial sum is given by sn = ln(n) − ln(2n + 1)
(ii)
(iii)
22n+3
n
n=0 5
∞
X
∞
X
n=1
n2
2
+ 2n
15. Integrate:
16.
Z
−1
−2
√
x2
Z
x4 + 1
dx
x2 − 4
dx
=
+ 4x + 5
17. The integral
18. Integrate:
Z
Z
3
0
dx
4x − 1
2x2 − x + 4
dx
x3 + 4x
1
x4
+ 2 , 1 ≤ x ≤ 3, about the x
19. Find the surface area obtained by rotating the curve y =
4
8x
axis.