M161, Test 2, Spring 2010
Problem Points Score
Name:
Section:
Instructor:
Time: 75 minutes. You may not use calculators on this exam
d
sin(x) = cos(x),
dx
d
csc(x) = − csc(x) cot(x),
dx
d
1
arcsin(x) = √
,
dx
1 − x2
d
1
arccsc(x) = − √
,
dx
x x2 − 1
sin(2x) = 2 sin(x) cos(x)
tan2 (x) + 1 = sec2 (x)
d
cos(x) = − sin(x),
dx
d
sec(x) = sec(x) tan(x),
dx
1
d
arccos(x) = − √
,
dx
1 − x2
1
d
arcsec(x) = √
,
dx
x x2 − 1
ln xdx = x ln x − x + C
1 + cos(2x)
cos2 (x) =
2
∫
1ab
24
1cd
24
2
16
3
36
∑
100
d
tan(x) = sec2 (x),
dx
d
cot(x) = − csc2 (x),
dx
1
d
arctan(x) =
,
dx
1 + x2
1
d
arccot(x) = −
dx
1 + x2
sec(x)dx = ln ∣ sec(x) + tan(x)∣ + C
1 − cos(2x)
sin2 (x) =
2
∫
Multiple Choice Answer Block
A a
b
c
d
e
D a
b
c
d
e
B a
b
c
d
e
E a
b
c
d
e
C a
b
c
d
e
F a
b
c
d
e
1) Evaluate the following integrals. Show your work.
x +1
dx
a)
(x − 1)2
∫
b)
∫e
√
3
x
dx
(Hint: Start by substituting
√
3
x).
c)
d)
1
∫ x √x − 1 dx
∫
x3
dx
(1 + x 2 )3/2
2) For each of the following improper integrals determine, using a suitable comparison test, whether the integral converges or diverges. (You do not need to calculate the values of convergent integrals.) State clearly what
comparison you are using, what function you compare with, and show that f < g, respectively work out the
limit of f /g. No points will be given if no work is shown, or if only an argument of polynomial degrees without
explicit comparison
√ is done.
∞
x
dx
a)
2
x + 3x + 18
1
∫
b)
∫
1
∞
ln(x)
dx
x 3/2
3) The following multiple choice problems will be graded correct answer only. You do not need to show work,
and no partial credit will be given. Record your answer in the answer block on the front page. Answers given
on these pages will not be scored. You also may tear off these pages and do not need to hand them in.
It is strongly recommended that you work out the problems until the correct answer is uniquely determined
and don’t just try to solve them by “intuition” or “guessing” – doing so is likely to result in a wrong pick.
Each correct anwer is worth 6 points, each incorrect answer is counted as 0 points. (Unanswered questions
are 1 point, questions in which more than one answer is ticked are considered to have been answered wrongly.)
a
e 2x − 1
? (Use L’Hospital’s theorem)
x→0 tan x
0 c 1 d 2 e The limit does not exist
What is lim
A)
-1
b
B)
Integrating ∫ 2x arcsin(x)dx by parts, we obtain an expression of the form
A − ∫ Bdx. What is B?
√
x2
2x
arcsin(x)x 2
x2
1 − x2
a
b √
c
d √
e
1 − x2
2
x2
1 − x2
1 − x2
3n + 4
Consider the sequence a n =
, which has limit L = 3. Determine the smallest
n
1
for any n > N.
value of N, for which ∣a n − L∣ <
100
a 25
b 100
c 200
d 300
e 400
C)
Solve the integral
D)
2x
∫ (x + 2)(x + 1) dx
a
c
e
−4 ln ∣x + 2∣ + 2 ln ∣x + 1∣ + C b 2 ln ∣x∣ + 23 x + 21 x 2 + C
4 ln ∣x + 2∣ − 2 ln ∣x + 1∣ + C d ln ∣x + 2∣ + ln ∣x + 1∣ + C
ln ∣x + 2∣ + ln ∣x + 1∣ − 3x + C
E)
Let n be a number for which the improper integral
∫
e
∞
dx
x(log x)n
converges. Determine the value of the integral
log n
1
1
1
a
b
c
d
e
n+1
n
n−1
n+1
log n
n−1
Which of the following sums does not equal
F)
4
8
16
32
+
+
+
+⋯
3 ⋅ 4 5 ⋅ 6 7 ⋅ 8 9 ⋅ 10
2 ⋅ 2n (2n)!
b
∑
n=1 (2n + 2)!
∞
2n
∑
n=2 (2n − 1)(2n)
∞
a
e
2n+1
∑
n=1 (2n + 1)(2n + 2)
∞
?
2n (2n − 2)!
∑
(2n)!
n=2
∞
c
2n
∑
n=2 (2n)(2n + 1)
∞
d
If you are done and have time left, check your answers on all the problems. Is in each problem clear, what
your answer is? Did you tick the correct boxes on the multiple choice questions? Do your calculated antideriva-
tives differentiate correctly to the original function?