Midterm 2 practice 2
UCLA: Math 31B, Spring 2017
Instructor: Noah White
Date:
Version: practice
• This exam has 4 questions, for a total of 40 points.
• Please print your working and answers neatly.
• Write your solutions in the space provided showing working.
• All final answers should be exact values. Decimal approximations will not be given credit.
• Indicate your final answer clearly.
• Full points will only be awarded for solutions with correct working.
• You may write on the reverse of a page or on the blank pages found at the back of the booklet however
these will not be graded unless very clearly indicated.
• Non programmable and non graphing calculators are allowed.
Name:
ID number:
Discussion section:
Question
Points
1
8
2
10
3
15
4
7
Total:
40
Score
UCLA: Math 31B
Midterm 2 (practice 2) (solutions)
Spring, 2017
1. Calculate the following limits using any technique you like.
(a) (4 points)
lim x(ln x)2
x→0+
(b) (4 points)
lim xsin x .
x→0+
Solution:
(a) We turn the 0 · ∞ indeterminate into an
∞
∞
2
lim x(ln x) = lim
x→0+
indeterminate and apply L’Hôpital’s rule twice:
(ln x)2
= lim
1
x
x→0+
= lim
x→0+
2 ln x
2(ln x) ·
−1
x
= lim
x→0+
−1
x2
x→0+
1
x
2
x
1
x2
= lim 2x = 0.
x→0+
sin x
(b) Let y(x) = ln x
. Then y(x) = (sin x)(ln x). Thus,
lim y(x) = lim (sin x)(ln x) = lim
x→0+
x→0+
x→0+
= − lim
x→0+
ln x
1
sin x
= lim
x→0+
sin x
sin x
sin x sin x
·
= − lim
· lim
= −1 · 0 = 0.
x→0+
x→0+ cos x
x
cos x
x
We conclude that limx→0+ xsin x = e0 = 1.
1
x
− cos x
sin2 x
UCLA: Math 31B
Midterm 2 (practice 2) (solutions)
Spring, 2017
2. For each of the following improper integrals say whether it converges or diverges. If the integral converges,
you should say what value it converges to.
R ∞ 2x
(a) (5 points) 0 (1+x
2 )2 dx.
R1
(b) (5 points) 0 ln x dx.
Solution:
(a) It converges. We use the u-sub, u = 1 + x2 , so that du = 2xdx.
Z
0
∞
2x
dx = lim
S→∞
(1 + x2 )2
Z
S
0
2x
dx
(1 + x2 )2
1+S 2
−1
1
du = lim
= lim
S→∞
S→∞ 1
u2
u 1
−1
+ 1 = 1.
= lim
S→∞ 1 + S 2
Z
1+S 2
(b) It converges. We do integration by parts. We also use the fact that limS→0+ S ln S = 0 (use
for example L’Hopital’s rule),
Z
1
Z
1
ln x dx = lim
0
S→0+
= lim
S→0+
ln x dx
S
1
x ln x − x
S
= lim [−1 − S ln S + S]
S→0+
= −1.
UCLA: Math 31B
Midterm 2 (practice 2) (solutions)
Spring, 2017
3. For each of the following series say whether it converges or diverges. You do NOT need to justify your
answer.
Grading scheme: 0 points for wrong, 1 point for no response, 3 points for correct.
(a) (3 points)
P∞
(b) (3 points)
P∞
(c) (3 points)
(d) (3 points)
n 2n
n=1 (−1) 1+n2 .
√ n
n=2 n5 −4 .
P∞
2n
n=0 (1+n2 )2 (Hint: once of
P∞
n=1 an where the sequence
the previous questions in the midterm might help).
of partial sums (sN )∞
N =1 is described by
sN = N − sin N
(e) (3 points)
P∞
n=1
−1
.
an where the sequence of partial sums (sN )∞
N =1 is described by
sN =
N
X
−1
n− sin n .
n=1
Solution:
(a) Converges (use alternating series test).
√
(b) Converges (use limit comparison test with bn = 1/ n3 ).
(c) Converges (use integral test and question 2b).
(d) Converges (to 1, see question 1b).
(e) Diverges (since an → 1 6= 0).
UCLA: Math 31B
Midterm 2 (practice 2) (solutions)
Spring, 2017
P∞ √
4. (7 points) Does the series n=1 n n − 1 converge or diverge? Full points will only be given to solutions
that justify the answer clearly.
√
Solution: Let an = n n − 1 = n1/n − 1. First we check that an → 0. We can do this using
L’Hopital’s rule. Note that
1
ln (n)
lim ln n1/n = lim
n→∞ n
n→∞
1
= lim
= 0.
n→∞ n
So limn→∞ n1/n = 1 and limn→∞ an = 0. Thus the series has a chance to converge.
At this point we might be a bit stuck. It is not obvious how to integrate this function and it also
may not be obvious how to use the comparison
like this it is worth trying to
P 1 test. In situations
1
.
Let
b
=
.
compare the series to a “standard” one, say
n
n
n
an
= lim n(n1/n − 1) = ∞.
n→∞
bn
P∞ √
diverges, then by the limit comparison test n=1 n n − 1 does as well.
lim
n→∞
Since
P
1
n
UCLA: Math 31B
Midterm 2 (practice 2) (solutions)
Spring, 2017
This page has been left intentionally blank. You may use it as scratch paper. It will not be graded unless
indicated very clearly here and next to the relevant question.
UCLA: Math 31B
Midterm 2 (practice 2) (solutions)
Spring, 2017
This page has been left intentionally blank. You may use it as scratch paper. It will not be graded unless
indicated very clearly here and next to the relevant question.