Problem Points Score
M161, Test 3, Fall 2007
1
20
Name:
2
20
3ab
20
3c
10
4
30
∑
100
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
asin(x) = √
,
dx
1 − x2
d
1
acsc(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
acos(x) = − √
,
dx
1 − x2
1
d
asec(x) = √
,
x x2 − 1
Zdx
ln xdx = x ln x − x +C
1 + cos(2x)
cos2 (x) =
2
d
tan(x) = sec2 (x),
dx
d
cot(x) = − csc2 (x),
dx
1
d
atan(x) =
,
dx
1 + x2
1
d
acot(x) = −
1 + x2
Zdx
sec(x)dx = ln | sec(x) + tan(x)| +C
1 − cos(2x)
sin2 (x) =
2
Taylor series of the function f (x) about x = a:
∞
f 0 (a)
f 00 (a)
f (n) (a)
f (a) +
(x − a) +
(x − a)2 + · · · = ∑
(x − a)n
1!
2!
n!
n=0
Theorem (The Derivative Rule for Inverses) If f has an interval I as domain and f 0 (x) exists and
is never zero on I, then f −1 is differentiable at every point in its domain. The value of ( f −1 )0 at a
1
point b = f (a) in the domain of f −1 is given by ( f −1 )0 (b) = 0 .
f (a)
1)
a) Determine the value of the following infinite sum:
8 + 2 + 1/2 + 1/8 + 1/32 + · · · .
∞
b) Consider the series
∞
form
n+2
. We re-index this series to start at n = 0, i.e. convert to the
n
n=3 2
∑ an. Determine an.
n=0
∑
2)
a) Calculate the Taylor polynomial for ln(cos(x)) of degree 2 about a = 0.
b) Calculate the Taylor polynomial for x4 + x2 − 2x + 1 of degree 3 about a = 1.
3) Determine, for example by appropriate manipulation of known power series or by calculating
the Taylor series about a = 0, power series for the following functions. Your solution should make
clear what steps were performed with series of what functions.
a) sin(x3 )
b)
x2
1 − x3
c)
1
(Hint: Find a series for the antiderivative first.)
(1 − x)2
4) Solve the following initial value problem using power series. (A solution in form of a power
series is sufficient, you do not need to recognize the function.)
(1 − x)y0 − y = 0,
y(0) = 1