MATH 172.504
Examination 3
April 19, 2012
NAME
SIGNATURE
“An Aggie does not lie, cheat, or steal or tolerate those who do.”
This exam consists of 10 problems, numbered 1–10. For partial credit you must present
your work clearly and understandably and justify your answers.
The use of calculators is not permitted on this exam.
The point value for each question is shown next to each question.
CHECK THIS EXAMINATION BOOKLET BEFORE
YOU START. THERE SHOULD BE 10 PROBLEMS
ON 7 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
1–5
6
7
8
9
10
Total
Points
Possible Credit
30
10
10
10
18
18
96
NAME
MATH 172
Examination 3
Page 2
Multiple Choice: [6 points each] In each of Problems 1–5, circle the best answer.
1.
Find the formula for the n-th term of the sequence {an } that starts
1
1
3
9
27
81
a0 = − , a 1 = , a 2 = − , a 3 = , a 4 = − , a 5 = .
3
4
7
10
13
16
(A)
(−1)n · (3n − 3n−1 )
3n − 1
(B)
(−1)n · 3n
n+3
(−1)n+1 · 3n−1
(C)
3n + 1
2.
(D)
(−1)n+1 · 3n2
3n + 1
(E)
(−1)n+1 · 3n+1
3n + 9
3 cos(2πn)
.
1
n→∞
−2
n
Evaluate lim
(A) 0
(B) 3
(C) −
1
2
(D) −
3
2
(E) The limit does not exist.
April 19, 2012
NAME
3.
The series
MATH 172
Examination 3
Page 3
∞
X
32n + (−2)n+1
10n
n=0
(A) converges to
25
3
(B) converges to
17
2
(C) converges to 8
(D) converges to 9
(E) diverges.
4.
∞
X
1
Consider the series S =
(−1)n+1 n . By the Alternating Series Estimation The11
n=1
orem, what is the smallest number of terms m of this series that we would need to
take so that the partial sum
m
X
1
(−1)n+1 n
11
n=1
is within 0.001 of the value of the sum S?
(A) 1
(B) 2
(C) 3
(D) 4
(E) The series diverges.
April 19, 2012
NAME
5.
MATH 172
Examination 3
Page 4
Which of the following integrals represents the surface area of the surface obtained
by rotating the graph of y = 2x2 over the interval [1, 3] about the x-axis?
Z
3
√
x 1 + 16x2 dx
3
√
x2 1 + 16x2 dx
3
√
x2 1 − 16x2 dx
3
√
x 1 − 4x dx
3
√
x2 1 + 4x4 dx
(A) 2π
1
Z
(B) 4π
1
Z
(C) 4π
1
Z
(D) 2π
1
Z
(E) 4π
1
6.
[10 points] Find the arclength of the graph of the curve defined by y = 2x3/2 from
x = 0 to x = 5/3.
April 19, 2012
NAME
7.
MATH 172
Examination 3
Page 5
[10 points] Find the limits of the following sequences.
2n3 − 3
(a) an = √
9n6 + 20n
(b) bn =
8.
n−1
n
n
[10 points] Consider the series
∞ X
n=1
1
1
−
2n + 1 2n + 3
.
(a) Find a formula for the partial sum Sm of the series.
(b) Find the sum of the series or show that it diverges.
April 19, 2012
NAME
9.
MATH 172
Examination 3
Page 6
[18 points] Determine whether each series below converges or diverges. For each
problem use one or more of the tests we have studied: absolute convergence test,
alternating series test, direct comparison test, geometric series test, integral test,
limit comparison test, p-series test, ratio test, or test for divergence (n-th term test).
Be sure to specify which test(s) you are using and show how they apply.
(a)
∞
X
cos(n)
n=1
(b)
∞ 1/n
X
1
n=1
(c)
n2
2
∞
X
(3n)!
n=1
(n!)3
April 19, 2012
NAME
10.
MATH 172
Examination 3
Page 7
[18 points] Consider the following function:
∞
X
(x + 1)n
√
f (x) =
.
n · 3n
n=1
(a) What is the open interval of convergence for f (x)?
(b) Does the series converge at the endpoints of the interval in (a)? Why or why not?
(c) Write down a power series expression for g(x) = f 00 (x).
April 19, 2012