Problem Points Score
M161, Midterm 3, Spring
2010
1
12
2-3
15
4-5
15
6
18
7
20
8
20
X
100
Name:
Section:
Instructor:
Time: 75 minutes. You may not use calculators or other
electronic devices on this exam
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
1
d
1
,
acos(x) = − √
,
atan(x) =
2
dx
dx
1 + x2
1−x
d
1
d
,
asec(x) = √
sec(x) = sec(x) tan(x),
2
x x −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
Taylor series of f (x) about x = a:
∞
X f (n) (a)
f 00 (a)
f 0 (a)
(x − a) +
(x − a)2 + · · · =
(x − a)n .
f (a) +
1!
2!
n!
n=0
Show all your work. Partial credit will be given on all questions except
#2 and #6. Put your final answer in the boxes or lines provided.
1. A kid is given 5 mgs of medicine every hour. At the end of the hour,
half of the medicine is left in the body.
(a) The concentration of medicine in the body after the 5th dose is:
d=
4
X
5
.
n
2
n=0
What is d?
(b) The long-range concentration of medicine in the kid’s body is
about:
∞
X
5
.
D=
2n
n=0
What is D?
2. Correctly complete the following statements using one of the words:
CONVERGES, DIVERGES, or INCONCLUSIVE. Suppose that an
and bn are positive numbers for all n ≥ 1.
P
P
an
(a) If lim
= ∞ and ∞
bn converges, then ∞
.
n=1
n=1 an
n→∞ bn
P
P
an
(b) If lim
= L and ∞
bn converges, then ∞
.
n=1
n=1 an
n→∞ bn
P
P
an
= 0 and ∞
bn converges, then ∞
(c) If lim
.
n=1
n=1 an
n→∞ bn
P∞
P
an
(d) If lim
= ∞ and ∞
b
diverges,
then
.
n
n=1
n=1 an
n→∞ bn
3. What are the center and radius of the interval of convergence of the
power series
∞
X
(2x − 6)n
?
n
n=1
Center:
Radius of convergence:
4. Does the following series converge absolutely, converge conditionally,
or diverge?
∞
X
(−1)n
√ .
n
n=1
5. The 4th order Taylor polynomial of f (x) centered at a = 0 is
1 + x − 3x2 + 4x3 − 5x4 .
Is f (x) increasing or decreasing near x = 0?
Is f (x) concave up or concave down near x = 0?
6. Match the following functions to their corresponding power series:
4
4+3x2
ln(1 + 3x2 /4)
arctan (3x/4)
sin (3x/4)
4
4−3x2
e−3x
A.
∞
X
(−1)n−1 3n x2n
4n · n
n=1
B.
∞
X
(−1)n 3n x2n
4n
n=0
C.
∞
X
3n x2n
n=0
D.
42n+1 (2n + 1)!
∞
X
(−1)n 3n x2n
n=0
F.
4n
∞
X
(−1)n 32n+1 x2n+1
n=0
E.
2 /4
4n · n!
∞
X
(−1)n 32n+1 x2n+1
n=0
42n+1 (2n + 1)
7. For each of the following series, state whether it converges or diverges
and give a justification. Be sure to specifically cite any tests you use.
∞
X
2n3
(a)
(2n)!
n=1
(b)
∞
X
sin(n) + cos(n)
n2
n=1
(c)
∞
X
ln(n)
n=1
n
8. Consider the function f (x) = ex and the value a = 1.
(a) Compute the Taylor series for f (x) centered at a.
(b) What is the interval of convergence of the series from part (a)?
(c) Compute the following infinite sum:
e + e(e − 1) +
e
e
(e − 1)2 + (e − 1)3 · · · .
2!
3!