White
M161, Test 1, Fall 2011
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
.
′
−1
f ( f (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) Let f (x) = (x − 2)3 + x for any real number x.
a) Show that f is one-to-one on the set of real numbers. (If you refer to a criterion or test, indicate
why the test applies! If you refer to the graph of f you must explain why your sketch of the graph is
good enough for such an argument.).
b) Determine a such that f (a) = 12
c) Let g(x) = f −1 (x) the inverse of f . Determine g ′ (12).
2) Evaluate the following integrals. Show your work.
1
√
dx
a)
25 − 9x 2
∫
b)
∫ sin(x) ⋅ cos(x)dx
3
c)
∫
(ln x)5
dx
x
3) The table below gives three differential equations. For each equation, indicate which of the four
possible general solutions I-IV (i.e. one is not used) is a general solution for this equation. (You do
not need to solve the equations, just to verify which of the functions is a solution!) Furthermore, we
list for each equation one initial value setting. Find the correct value for C in the respective general
solution that will solve this given initial value problem.
This problem will be graded correct answer only.
dy
+ 3y = e x
dx
dy
+ ex = x2
dx
dy
+ 3y = e −x
dx
y(0) = 0
y(0) = 2
y(0) = 3/2
General solution
Initial Value Problem
Value of C to solve the IVP
Possible General Solutions:
I
II
III
IV
y(x) = (1/2 e 2 x + C) e −3 x
y (x) = 1/4 e x + Ce −3 x
y (x) = −1/2 x 2 + e x + C
y (x) = −e x + 1/3 x 3 + C
Hint: The correct values for C are to be chosen from the set {±1/4, 0, ±1, ±2, ±3}, no two of the
initial value problems have the same C-value.
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.)
If f (x) = ln(x + 4 + e −3x ), then f ′ (0) is
2
1
1
2
a
b nonexistent
c
d −
e
5
4
5
5
B)
If x = 3 tan θ, what is sec θ?
A)
a
3
b
√
9 + x2
x
√
9 + x2
c
3x
d
√
9 + x2
x2
is:
C)
An antiderivative of 2
x +1
a 1 + arctan(x)
b x + arctan(x + 1)
d arctan(x + 1)
e x − arctan(x)
c
9 + x2
√
9 + x2
3
e
arctan(x 2 + 1)
ex
decreasing?
x
a only (−∞, 0) and (0, 1)
b only (1, ∞)
c only (−∞, 0)
d it is never decreasing
e only (0, 1)
D)
On what interval(s) is the function
E)
Which condition guarantees that f (x) =
a
d
ax+b
cx+d
has an inverse?
ab − cd > 0 b ad − bc > 0 c ab + cd > 0
We cannot determine without knowing the values.
e
ac − bd < 0
F)
The number of bacteria in a culture is growing according to the function
f (t) = 3000e 2t/5 . At a certain time t = t1 the number of bacteria present was 7,500.
Find the number present at t = t1 + 5
a
7500e 2
b
15000/7e 7
c
3000e 2
d
1200e 2
e
7500e 5
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?