M161, Test 1, Spring 2010
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
12
2
30
3
22
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
d
2
y (t) = e −t − 2 y (t) t
dt
2
a) Show that the family of functions y (t) = (t + C) e −t is a solution to this equation.
b) Determine the constant C that fulfills the initial value condition y(2) = 0.
2) Evaluate the following integrals. Show your work.
a)
∫ sin(x) ⋅ cos(x) dx
b)
∫ √−x
3
4
1
2
− 8x − 15
dx
c)
1
∫ x ln(3x ) dx
2
3) Let f (x) = ln(3x) − ln(5x). Does f have an inverse? Explain! (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!)
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 this page and do
not need to hand it 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
B)
a
C)
a
D)
What is the domain of acos(x)?
[− π2 , π2 ]
b
[−1, 1]
c
[0, π]
d
[0, π2 ]
e
R
The set of all points (e t , t) where t is a real number is the graph of y =
1/e x
b
e 1/x
c
xe 1/x
d
1/ln(x)
e
ln(x)
If f (x) = (x 2 + 1)2−3x then f ′ (1) =
−1/2 ln(8e)
− ln(8e)
b
An antiderivative of √
c
−3/2 ln(2)
1
(x − 4)(6 − x)
d
1/8
e
−1/2
is:
arccos (x − 5) b arcsin (x − 5) c arctan (x − 5)
√
√
d ln (5 − x + (x − 4)(x − 6))
e ln (x − 5 + (x − 4)(x − 6))
a
E)
a
F)
a
Evaluate
−1
+C
2e x 2
∫e
x
b
x2
dx
−1
+C
ex2
Differentiate f (x) =
−2e x
(e x − 1)2
b
1
+C
2e x 2
c
1
+C
ex2
d
e
x2
+C
2e x 3
ex + 1
ex − 1
2e 2x
(e x − 1)2
c
2e x
(e x − 1)2
d
2
(e x − 1)2
e
ex
2(e x − 1)2
If you are done and have time left, check your answers on all the problems. Is in each problem
clear, what your answer is? Did you tick the correct boxes on the multiple choice questions?