M161, Test 1, Fall 2008
Problem Points Score
Please circle where you took M160 (or equivalent):
I CSU
AP
Community College
0
1
1
19
2
20
3
20
4
40
∑
100
other College
You get 1 point for this indication.
Name:
Section:
Instructor:
You may not use calculators on this exam
cos2 θ =
1+cos 2θ
2
sin2 θ =
1−cos 2θ
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 value of ( f −1 )0
at a point b in the domain of f −1 is the reciprocal of the value of f 0 at the point a = f −1 (b):
1
.
( f −1 )0 (b) = 0 −1
f ( f (b))
Multiple Choice Answer Block
A a
b
c
d
e
E a
b
c
d
e
B a
b
c
d
e
F a
b
c
d
e
C a
b
c
d
e
G a
b
c
d
e
D a
b
c
d
e
H a
b
c
d
e
1) a) Rewrite the expression log(5) + 3 log(x4 − 2) + 2 log(x + 1) as a single logarithm log(A).
Determine A.
b) Determine
d
arctan(2x )
dx
2) Z Evaluate the following integrals. Show your work.
1
dx
a) √
49 − 4x2
Z
b)
1
√
dx
x log x
3)
Solve the separable differential equation
dy
√
y = x−2
dx
for the initial condition u(1) =
1
.
36
4) The following multiple choice problems will be graded correct answer only. (You do not need
to show work, and no partial credit will be given. Record your answer in the answer block on the
front page. Answers given on these pages will not be scored. You also may tear off these pages
and do not need to hand them in.) Each correct anwer is worth 5 points, each incorrect answer is
counted as −2 points. (Unanswered questions are 0 points.) You cannot get less than 0 points in
this part.
A)
a
B)
a
d
C)
a
If x = 3 tan θ, what is sec θ::
9 + x2
b
3x
√
9 + x2
c
3
√
9 + x2
d
√
9 + x2
3
1
is::
x2 − 4 x + 13
1
1
x−3
(ln (x − 5) − ln (x + 1)) b
arctan
6
2
2
x−2
x+2
1
1
arctan
e
arctan
3
3
3
3
e
x
√
9 + x2
An antiderivative of
c
1
x+3
arctan
2
2
Indicate all intervals, on which the function x − 4 ln(x + 1) is one-to-one:
(0, 3)
b
(0, ∞)
c
(0, e − 1)
d
(3, ∞)
e
(e/4 − 1, ∞)
x
2 dx:
ex
−1
b
2 +C
ex
Z
Evaluate
D)
a
−1
2 +C
2ex
1
2 +C
2ex
c
d
1
2 +C
ex
e
x2
3 +C
2ex
A bacteria culture grows at a rate proportional to its size. Suppose that the
initial size is 8000 bacteria, and after 2 hours there are 12000 bacteria. How
many bacteria are there after 6 hours?:
E)
a
20000
24000
c
27000
d
36000
e
48000
Suppose that A, B and C are numbers satisfying
A
B
ln
= 3.5
and
ln
= 2.1
C
C
F)
Find
a
b
A
:
B
3.5 − 2.1 = 1.4
b
3.5
2.1
c
e3.5−2.1 = e1.4
d
e3.5 − e2.1
3.5
e
e 2.1
Which of the following functions is a general solution to the differential
equation y0 + 2y = e−x ?:
G)
a
d
y(t) = 2e−x +C
b y(t) = Ce−2x + e−x
c
−2x
−x
−2x
−Cx
y(t) = e +Ce
e y(t) = e
+e
Z 4
Evaluate
H)
a
1
b
2
x
√
dx:
0
25 − x2
π
c 5
d
2
e
1
−π
2
y(t) = Ce−x