Problem Points Score
M161, Midterm 3, Fall 2010
1
18
Name:
2
22
Section:
3
18
Instructor:
4
22
5
20
X
100
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
1
1
d
d
,
,
acos(x) = − √
atan(x) =
dx
dx
1 + x2
1 − x2
d
1
d
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
Taylor series of f (x) about x = a:
∞
X f (n) (a)
f 0 (a)
f 00 (a)
f (a) +
(x − a) +
(x − a)2 + · · · =
(x − a)n .
1!
2!
n!
n=0
1. Short answer questions. Put your answer in the box. No work outside
the box will be graded.
(a) Express the number 0.34 = 0.343434 . . . as a fraction in lowest
terms.
(b) What are the center and radius of the interval of convergence of
the power series
∞
X
(3x − 2)n
?
n
n=1
Center:
Radius of convergence:
(c) 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?
2. For each of the following series, state whether it converges or diverges
and give a short justification.
(a)
∞
X
n4
n=0
(b)
∞
X
4n
:
cos(πn):
n=0
(c)
∞
X
ln(n)
n=1
n2
:
3. Match the following functions to their corresponding power series:
e2x − 1
8x
4−2x
2xe2x
10 10
x
2x 1−2
1−2x
arctan(2x) + 12 ln(1 + 4x2 )
sin(2x) + cos(2x) − 1
∞
X
xn
A.
2n−2
n=1
B.
10
X
2n xn
n=1
C.
∞
X
n=1
D.
∞
X
2n
n=1
E.
2n
xn
(n − 1)!
∞
X
n!
xn
(−1)
n2 +n+2
2
(−1)
n2 +n+2
2
n=1
F.
∞
X
n=1
2n n
x
n!
2n n
x
n
4. The Taylor series for
1
1−x
is
P∞
n=0
xn = 1 + x + x2 + x3 + x4 + · · ·.
(a) The partial sum sn (x) = 1 + x + · · · + xn equals
1 + tn (x)
for some
1−x
function tn (x). What is tn (x)?
(b) For what values of x does limn→∞ sn (x) exist?
(c) Complete this sentence: If the sequence of partial sums of a series
has a finite limit, then the series
.
(d) Use the Taylor series for
1
1−x
to find the Taylor series for
1
.
(1−x)2
5. Let f (x) = cos(2x) and let a = π/8.
(a) Compute the degree 3 Taylor approximation T3 (x) for f (x) at
x = a. (This is the one that ends with a constant times x3 ).
(b) What is the coefficient of x2010 in the Taylor series for f (x) centered
at x = a?
(c) Complete the sentence: The polynomial T3 (x) is the best
for f (x) near x = a.