Problem Points Score
M161, Midterm 2, Fall 2010
1
25
Name:
2
25
Section:
3
18
Instructor:
4
22
5
10
X
100
Time: 75 minutes. You may not use calculators or other
electronic devices on this exam
d
d
d
sin(x) = cos(x),
cos(x) = − sin(x),
tan(x) = sec2 (x),
dx
dx
dx
d
d
d
1
1
1
,
,
,
asin(x) = √
acos(x) = − √
atan(x) =
2
2
dx
dx
dx
1 + x2
1−x
1−x
d
1
1
d
d
,
,
acsc(x) = − √
asec(x) = √
sec(x) = sec(x) tan(x),
2
2
dx
x x − 1 Zdx
x x −1
Zdx
sin(2x) = 2 sin(x) cos(x)
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. Short answer questions. Put your answer in the box. No work outside the box will be
graded.
(a) Find the limit of the sequence: { 42 , 49 , 16
, 25 , 36 , 49 , ...}.
8 16 32 64
Z
∞
(b) Does
1
(c) Does
R∞
1
√
x8 − x2 + 1
dx converge?
2x4 − 3x + 1
2+sin(x)
x
dx converge?
(d) For what values of p does the function 4x grow at a faster rate than
A) For all values of p.
B) For p < 0.
C) For p < 2.
D) For p < 4.
E) For no values of p.
√
xp ?
Z
∞
π
− arctan(x)) dx. Which of the following is true?
2
0
A) The integral diverges by the Divergence Test.
B) The integral converges by the Divergence Test.
C) The integral diverges by a Limit Comparison Test with x1 .
D) The integral converges by a Limit Comparison Test with x12 .
E) None of the above.
(e) Consider the integral
(
2. Evaluate the following integrals.
Z
(a) x5 ln(x) dx.
Problem 2 continued: Evaluate the following integrals.
Z
5x − 7
(b)
dx.
2
x − 3x + 2
Z
(c)
√
x2
dx.
4 − x2
3. Your calculus instructor made an infinitely long funnel by rotating the graph of y = 1/x
from x = 1 to x = ∞ around the x-axis.
(a) It turns out that the volume of the solid region inside the funnel is:
Z ∞
π
dx.
x2
1
Is the volume finite? Explain how you found your answer.
(b) It turns out that the surface area of the solid is:
Z ∞ r
2π
1
1 + 4 dx.
x
x
1
Is the surface area finite? Explain how you found your answer.
(c) True/False: If the funnel were a paint can, it could hold enough paint to color its
surface. Explain your answer.
4. A bacterial culture starts with population y(0) = 50 and grows at a rate proportional
to its size. After three hours, there are 200 bacteria.
(a) What type of formula for dy/dt expresses the condition ’grows at a rate proportional to its size’ ?
(b) Find a formula for y(t), the number of bacteria after t hours.
(c) How many bacteria are there after 6 hours? Why is this a whole number?
(d) How many bacteria are there after 10 hours?
5. Write an N and proof that limn→∞
n2 + 1
= 1.
n2
Extra Credit: Let f (x) : R → R be a positive continuous
decreasing function with
Z ∞
lim f (x) = 0. Let f −1 (x) denote its inverse function. If
f (x) dx diverges, which of the
x→∞
following
R 1 must be true?
A) 0 f −1 (x)dx must converge.
R1
B) 0 f −1 (x)dx must diverge.
C) lim f −1 (x) = 0
x→∞
D) lim f −1 (x) = ∞
x→∞
E) None of the above.
0