M161, Test 2, Spring 2008
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
d
1
asin(x) = √
,
dx
1 − x2
1
d
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
d
1
acos(x) = − √
,
dx
1 − x2
d
1
asec(x) = √ 2
,
x x −1
Zdx
ln xdx = x ln x − x + C
1 + cos(2x)
cos2 (x) =
2
Problem Points Score
1
12
2ab
28
2cd
28
3
20
4
12
X
100
d
tan(x) = sec2 (x),
dx
d
cot(x) = − csc2 (x),
dx
d
1
atan(x) =
,
dx
1 + x2
d
1
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) Perform a partial fraction decomposion (you do not need to integrate) for the function
4x3 + 5x2 + x − 4
x2 − 1
2) Z Evaluate the following integrals. Show your work.
1
a)
dx
(x2 + 1)3/2
b)
Z
1 · ln(x2 + x)dx
c)
Z
1
dx
x2 − 5x + 6
√
d)
Z
x2 − 9
dx
x3
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)
x ln(x)
2
b)
Z ∞
1
1
dx
x
sin
4) Give examples of sequences with the specified properties. Give a brief (one sentence)
justification:
a) A sequence that is increasing but does not converge.
b) A sequence that converges with limit π.
c) A sequence that is increasing and bounded from above.