M161, Test 1, Spring 05
NAME:
SECTION:
Problem
1
Points
12
2
15
3ab
10
3cde
15
4
19
5
12
6
8
7
9
Total
100
Score
INSTRUCTOR:
You may not use calculators on this exam.
1 + cos 2θ
2
cos θ =
2
1 − cos 2θ
2
sin θ =
2
1. (a) Use the propertiesof logarithms
to
1
2
.
simplfy ln(3x − 9x) + ln
3x
x
(b) Evaluate the expression sin arctan √
x2 + 1
(c) Solve for y in terms of x. ln(y 2 − 1) −
ln(y − 1) = ln(sin x)
2. Calculate the following derivatives (you
do not have
s to simplify).
d
ln
(a)
dx
(x + 1)5
(x + 2)20
d 4√x+x2
(b) e
dx
d
(c)
tan−1(ln x)
dx
3. Evaluate the following integrals. You
must
your work.
Z show
1
1
q
(a)
dx
2
0
1 − x4
(b)
Z
1
dx
3x − 2
ex + e−x
dx
cosh x
(c)
Z
(d)
Z ln 3
ex dx (Simplify your answer.)
ln 2
(e)
Z
cosh(2x) dx
p
4. Consider the function f (x) = ln(x) + 1.
(a) Find the domain and range of f (x).
(b) Plot f on the axis below.
4
3
2
1
0
−1
−2
−3
−4
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
(c) Give a short explanation why you know
that f will have an inverse.
(d) Find f −1(x).
(e) Find the domain and range of f −1(x).
5. Calculate the following limits.
(a) lim x cot x
x→0+
1 − cos x
(b) lim
x→0 x + x2
ln x
(c) lim √
x→∞ 2 x
6. Which of the following functions grows
faster or slower than x2 (or do they grow at
the same speed) as x approaches infinity?
Explain.
p
(a) x4 + x3
(b) x2e−x
7. Find the solution to the differential equady 2xy + 2x
tion
= 2
. Write your solution as
dx
x −1
a function of x.