M161, Test 1, Fall 2009
Problem Points Score
Name:
Section:
Instructor:
Time: 75 minutes. You may not use calculators on this exam
d
d
sin(x) = cos(x),
cos(x) = − sin(x),
dx
dx
d
1
1
d
asin(x) = √
acos(x) = − √
,
,
dx
dx
1 − x2
1 − x2
d
1
1
d
acsc(x) = − √
asec(x) = √
,
,
dx
x x 2 − 1 dx
x x2 − 1
sin(2x) = 2 sin(x) cos(x)
ln xdx = x ln x − x + C
1 + cos(2x)
tan2 (x) + 1 = sec2 (x)
cos2 (x) =
2
∫
1
16
2
30
3
18
4
36
∑
100
d
tan(x) = sec2 (x),
dx
1
d
atan(x) =
,
dx
1 + x2
d
sec(x) = sec(x) tan(x),
dx
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 ′ (x) exists and
is never zero on I, then f −1 is differentiable at every point in its domain. The value of ( f −1 )′ at a
point b in the domain of f −1 is the reciprocal of the value of f ′ at the point a = f −1 (b): ( f −1 )′ (b) =
1
.
f ′ ( f −1 (b))
Multiple Choice Answer Block
A a
b
c
d
e
D a
b
c
d
e
B a
b
c
d
e
E a
b
c
d
e
C a
b
c
d
e
F a
b
c
d
e
1)
Consider the differential equation
y ′ + 2y = e −x
a) Show that the family of functions y = Ce −2x + e −x is a solution to this equation.
b) Determine the constant C that fulfills the initial value condition y(2) = 2.
2) Evaluate the following integrals. Show your work.
log(x 2 )
dx
a)
x
∫
b)
∫x
2
1
dx
+ 10x + 34
1
Let f (x) = ln ( ) for 0 < x < ∞.
x
a) What is the range of f ?
3)
b) Explain why f has an inverse. (If you refer to a criterion or test, indicate why the test applies. If
you refer to the graph of f you must show that your sketch of the graph is good enough!)
c) Determine ( f −1 )′ (5)
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.
It is strongly recommended that you work out the problems until the correct answer is uniquely
determined and don’t just try to solve them by “intuition” or “guessing” – doing so is likely to result
in a wrong pick.
Each correct anwer is worth 6 points, each incorrect answer is counted as 0 points. (Unanswered
questions are 1 point, questions in which more than one answer is ticked are considered to have been
answered wrongly.)
A)
a
e
B)
a
ax + b
(for c =/ 0 and x > −d/c) have an inverse?
cx + d
ab − cd ≠ 0 b ab − cd > 0 c ac − bd ≠ 0 d ad − bc ≠ 0
We cannot determine without knowing the values.
When does f (x) =
Simplify e t
t2
3 ln ( )
4
b
2
(ln 3)/4
3
t 2 /4
c
3e
t 2 /4
d
3t 2
( )
4
e
t2 3
( )
4
C)
a
D)
a
If f ′ (x) = 3 f (x) and f (0) = 3, then f (3) =?
3e
3e 3
b
c
9e 3
d
3e 9
e
9e 9
If f (x) = e 1/x , then f ′ (x) =
1/x
− ex 2
b
−e 1/x
c
e 1/x
x
d
e 1/x
x2
e
1 (1/x)−1
xe
E)
a
If sin x = e y , 0 < x < π, what is d y/dx in terms of x?
− tan x
b
− cot x
c
cot x
d
tan x
e
csc x
F)
A radioactive substance decays according to an exponential law. Suppose
that you have initially 27g of this substance and after 2 years there are 18g left.
How much will be left after 6 years?
a
1g
b
3g
c
8g
d
9g
e
12g