M161, Test 2, Fall 2007
Name:
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
1
d
asin(x) = √
,
dx
1 − x2
d
1
acsc(x) = − √ 2
,
dx
x x −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) = √ 2
,
x x −1
Zdx
ln xdx = x ln x − x + C
1 + cos(2x)
cos2 (x) =
2
Problem Points Score
1
15
2ab
18
2cd
18
2ef
18
3
16
4
15
X
100
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
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
1
value of (f −1 )0 at a point b = f (a) in the domain of f −1 is given by (f −1 )0 (b) = 0 .
f (a)
1) Set up the partial fraction decomposition with constants A, B, C and so on for the rax3 + 2x + 7
. (You do not need to solve for the constants
tional function
(x − 1)(x + 5)2 (x2 + 1)2 (x2 + 2)
or to find an antiderivative!)
2) Z Evaluate the following integrals. Show your work.
1
a)
dx
(x − 5)(x − 6)
b)
Z
6x cos(2x)dx
c)
d)
Z
sin5 (x) cos2 (x)dx
Z
4xex dx
2
e)
f)
Z
x+3
dx
x2 + 1
Z
√
x3
dx
9 − x2
3) For each of the following improper integrals determine, with explanation, whether the
integral converges or diverges. (You do not need to calculate the values of convergent integrals.)Z
∞
1
dx
a)
3
x +1
1
b)
Z ∞
1
2 + sin(x)
dx
x
4) a) Complete the definition for limit of a sequence:
The sequence {an } converges to the number L if for every positive ε there exists an integer
N such that
n4 + 1
b) Now consider the sequence an =
and let L = 1. For a given ε > 0 calculate N
n4
such that the condition in a) is fulfilled.
c) Write a a proof that lim an = 1.
n→∞