Problem Points Score
M161 Midterm 2, Spring 11
1
12
2
18
3
20
4
20
5
15
6
15
X
100
Name:
Section:
Instructor:
Exam Time: 75 minutes. You may not use calculators or
other electronic devices.
d
sin(x) = cos(x),
dx
d
1
,
asin(x) = √
dx
1 − x2
d
1
,
acsc(x) = − √
dx
x x2 − 1
sin(2x) = 2 sin(x) cos(x)
d
d
cos(x) = − sin(x),
tan(x) = sec2 (x),
dx
dx
d
d
1
1
,
,
acos(x) = − √
atan(x) =
2
dx
dx
1 + x2
1−x
d
d
1
,
asec(x) = √
sec(x) = sec(x) tan(x),
x x2 − 1
Zdx
Zdx
ln xdx = x ln x − x + C
sec(x)dx = ln | sec(x) + tan(x)| + C
tan2 (x) + 1 = sec2 (x)
cos2 (x) =
1 + cos(2x)
2
sin2 (x) =
1 − cos(2x)
2
1. Multiple Choice Questions: Select the correct response and write the letter of your
answer in the box. No work outside the box will be graded.
√
Z
A.)
6
0
a
The integral diverges to ∞
Z
B.)
1
dx?
6 − x2
π
b 0
c
2
√
What is the value of
The improper integral
2
∞
d
π
√ −1
2 6
e
π
√
2 6
t4 − 1
dt:
t6 − t2 − 1
R∞
diverges as seen by using the Direct Comparison Test with R2 1t dt
∞
b converges as seen by using the Limit Comparison Test with 2 t16 dt
R∞
c converges as seen by using the Limit Comparison Test with 2 t12 dt
R∞
d diverges as seen by using the Limit Comparison Test with 2 1t dt
R∞
e converges as seen by using the Direct Comparison Test with 2 t12 dt
a
2. The following list contains six limits of functions. The limits of these functions are:
0, −6, −1/6, ln(2), −ln(2), and undefined.
Match these values with the limits and put the answer in each box.
limx→0
(1/2)x − 1
x
limx→0
sin(x) − x
x3
limx→∞
x − 12x2
2x2 − 6x
limx→∞ x3 e−x
limx→∞ (ln(2x) − ln(x + 3))
limx→∞ sin(
πx
)
2
3. Evaluate the following integrals:
(a)
Z
2x − 3
dx.
x2 − 4
(b)
Z
(4x2
8
dx.
+ 1)2
4. (a) Using a suitable comparison test, determine whether the following integral converges or diverges. You do not need to evaluate the integral. State clearly the
comparison test you are using and the function you are comparing with.
Z ∞
2 + cos x
dx.
x
π
(b) Evaluate the following improper integral.
Z ∞
1
dx.
x(ln(x))2
2
5. (a) Suppose f (x) and g(x) are nonnegative functions and limx→∞
Which of these statements is true?
(A) There is a value beyond which f (x) is smaller than g(x).
(B) There is a value beyond which f (x) is larger than g(x).
(C) There is a value beyond which f (x) is 0.
(D) No conclusions can be drawn.
(b) Which of these functions decays faster?
1
(A)
.
(x + 1) ln(x)
1
x(ln(x))2
(c) Justify your answer to part (b).
(B)
f (x)
g(x)
= 0.
6. (a) The following definition is wrong.
Definition: the sequence {an } converges to the number L if for every > 0, there
corresponds a number N , such that |an − L| < for some integers n > N .
Changing just one word will make the definition correct. Circle that word.
(b) To make the definition correct, what word should replace your circled word?
(c) Use the definition to prove that the sequence { n12 } converges to 0.