Final Review
Calculus I, Fall 2006

Given a percent growth rate of 10%, what is
the value of a in the equation f(x)=Pax
A. 0.01
B. 0.001
C. 1.10
D. 1.01
E. None of the Above

Find the indefinite integral of the function
graphed below:
A. ln|sec(s)tan(x)|+C
B. sec2(x)+C
C. ln|sec(x)|+C
D. tan(x)+C
E. None of the Above

Given logbx=A, what is the equivalent
exponential expression?
A. x=Ab
B. lnA=x
C. bA=x
D. b=Ax
E. None of the Above

Given the graph of f(x)=x(x2-3) below, what is
the area between the curve and the x-axis from
0 to 1?
A. 1/4 - 3/2
B. 3/2 – 1/4
C. -2
D. 2
E. None of the Above

Exponential functions have a constant rate of
change
A. TRUE
B. FALSE

What is the equation describing the following graph?
5
3
1
-6
-4
-2
2
4
6
 x 
A. 2 sin 
3
 4 
 1

B. 5 cos
(x   )  3
 4

 x
C. 5 sin    3
2
 x
D. 2 sin    3
2
E. None of the above

Given the graph of f(x)=x(x2-3) below, what is the total
change in the anti-derivative function F(t) from -1 to 1?
1
A.

B.
C.
D.
E.
4
4
0
None of the above
1
x( x 2  3) dx

Given y=16sin(2x)+5, what is the period?
A. 2
B. 4
C. 2
D. 
E. None of the Above

What is the equation describing the following graph?
1
-3/2
-
/2
-/2
-1

3/2


A. sin  x  
2



B.  sin  x  
2

C. cos( x   )
D. All of the above
E. None of the above

logb(C+D)=logbC•logbD
A. TRUE
B. FALSE

Which of the following is the graph of
log 3 x
B
A
D
C
E None of these

Given the following graph of g(x), order:
a) g’(2)
b) average value of g(x) over the interval 0<x<8
c) Integral of g(x) from 1 to 3
from least to greatest.
A. a<c<b
B. a<b<c
2
4
6
8
C. c<a<b
D. c<b<a
E. None of
the Above

Given F(x)=f’(x), what does the shaded region
represent?
F(x)
A.
B.


4
1
4
1
f ( x)dx
F ( x)dx
C . f ( 4)
D. B & C
E. None of the above

Given the graph of f(x), which of the following is the
graph of f(x+k)?
A
B
C
D None of these

The following graph shows the velocity of a
rocket taking off from earth versus time.
When did the rocket hit the ground?
2
4
6
A. t=2
B. t=4
C. t=6
D. t=8
E. None of the Above
8

Given the following graph of g(x), determine which of
the following statements are true about the antiderivative.
g(x)
t
G(t )   g ( x)dx
1
i.
2
G(t) is positive for all 0<t<2
ii. Over the interval 1<t<2, G(t) is increasing
iii. G(t) is concave up at t = 1/2
4
6
8
A. i only
B. ii only
C. iii only
D. i and ii only
E. ii and iii only
F. None of the above

Given sin2()-1=x, x is equal to
A. csc 
B. cos 
2
C. cos 
D. csc 2 
E. None of the above

Given the following table of values, find a formula
for f (x)
x
0
1
2
3
4
f(x)
16
24
36
54
81
16
A.
x  16
2
B. x 2  3.43 x  16
C. 16(1.5)
x
D. None of the above

Evaluate the following integral:
 7 cos(7 x)dx
A. 7 sin( x)  C
B.  sin( 7 x)  C
C.  7 sin( x)  C
D. sin( 7 x)  C
E. None of the above

A person runs around a circular track. Their
horizontal distance from the starting line is a
periodic function. If the track is 400 meters long,
and the runner takes 1 minute to run completely
around the track once, which of the following
equations could model the distance from the starting
line as a function of time given the runner starts
running at time t=0?
A. 400 sin( x)
B. 400 cos( 2x)
C. 200 sin( 2x)
D. 200 cos( x)
E. None of the above

Evaluate the following integral:
x
 x 2  4 dx
A.
B.
C.
1 2
( x  4)  C
2
1 2
x 4C
2
2( x 2  4)  C
D. ln( x 2  4)  C
E. None of the above

The total number of undergraduate students at the
Willamette University is given in the following
table. What was the average rate of change in the
number of students from 1990 to 2005?
year
1985
1990
1995
2000
2005
enrollment
1531
1507
1525
1659
1842
A.
B.
C.
D.
22.33
15.55
1674.5
None of the above

Given a runner’s horizontal distance from the
starting line on a circular track is given by the
equation
d = 200sin(2x)
Which of the following formulas gives the total
horizontal distance run in 4 minutes?
A. 200 sin( 2 4)

4
C.

4
D.
200 sin( 2 4)
B.
0
0
200 cos( 2x)dx
400 cos( 2x) dx
E. None of the above

 
d xe 2 x
e

dx
A. e
xe 2 x
2 x
xe
B. (1  2 x)e
C. e
2 x
 2 x xe 2 x
e
2 x
 xe
xe 2 x
D. e
E. None of the above

Find the critical points of
f ( x)  x  9 x  48 x  52
3
2
A. 8
B. 2
C.  2
D. A and B
E. None of the above



d
3 ln( 4e x ) 
dx
A. 12 ln( 4e x )
3
B.
4e x
12
C.
4e x
D. 3
E. None of the above

d
tan( x) 
dx
A.  sec ( x)
2
B. cot( x )
2
C. csc ( x)
2
D. 1 / cos ( x)
E. None of the above



d
12 x sin( 12 x) 
dx
A. ln( 12)
x
cos(
12
x
)(
12
)
x
B. ln( 12)(12 )( sin( 12 x) ) 
2 sin( 12 x)
x
cos(
12
x
)(
6
)(
12
)
x
C. ln( 12)(12 )( sin( 12 x) ) 
sin( 12 x)
ln( 12)(12 x )( sin( 12 x) )  cos(12 x)(12)(12 x )
D.
sin( 12 x)
E. None of the above

Given F’(x)=f(x), the shaded area on the graph of f(x)
represents
1
b
a
-1
A. F ' (b)
B.

b
a
F ( x)dx
C. F (b)  F (a)
D. None of the above

What is the derivative of g (t )  t ln( t )
1
2
A. ln( t )
t
B. 2t ln( t )
C. ln( t )  1
D. 1
E. None of the above

Given F’(x)=f(x), the marked quantity on the graph of
f(x) represents:
a
b
A1=4
A. F ' (a )
B. F (b)  F (a)
F (b)  F (a)
C.
ba
D. The average rate of change of F ( x) between a and b
E. The instantane ous rate of change of F ( x) at x  a

Given F’(x)=f(x), which of the following represents
1
f ( x ) dx
ba 
on the graph of F(x)?1
b
a
B
A
-3/2
-
/2
-/2
a
b
a
-3/2

b
-
3/2
/2
-/2
D. None of the above
a
a
b
-1
-1
C
D. None of the above
-3/2
-
a
/2
b
-1

Given F’(x)=f(x), which of the following represents
f(a) on the graph of F(x)
1
A
B
a
-3/2
-
a
/2
-/2
-1
C
a

3/2
D. None of the above

d  sin( x) 



dx  cos( x) 
A. tan( x)
B. tan( x) sec( x)
2
C. sec ( x)
D. None of the above



d ln(x )
e

dx
1
x
B. 1
A.
C.
1
e ln(x )
D. None of the above

d
dx
 4x  2x 
3
A. (6 x 2  1)( 4 x 3  2 x) 1/ 2
1
B.
(12 x 2  2) 1/ 2
2
C.
12 x 2  2
2
D.
(4 x 3  2 x) 3 / 2
3
E. None of the above



d 3 4x
xe 
dx
A. 3 xe  4 x e
4x
3 4x
B. 12 xe  16 x e
x
3 x
C. 4 x e  3 x e
3 4x
2 4x
D. None of the above

d
ln( x) 
dx
A. x 1
1
B.
x
C.
1
e ln(x )
D. All of the above
E. None of the above

 
d x 2
2

dx
A. 2
x2
B. 2 x 1
C. 4 ln( 2)2
x
D. None of the above

d
log 6 x  
dx
1
A.
6x
1 1
B.
ln( 6) x
C. 6 log x
D. None of the above

f ( x)  x  6 g ( x)  2 x  x
4
3
2
d
 f ( x)  g ( x)
Find
dx
A. 4 x 3  6 x  6 x 2  2 x
B. 4 x 3  6 x 2  2 x
1 5
1 4 1 3
C.
x  6x  x  x
5
2
3
D. None of the above

2
x
y
x4
dy
Find
dx
A. 2 x
2x  x2
B.
( x  4) 2
x 2  8x
C.
( x  4) 2
D. None of the above

y  3  5e
x
x
dy
Find
dx
A. ln( 3)3 x  5e x
B. 3 x  5 ln( e)
1
1
C. x  x
3 5e
D. None of the above

2
t
 1  t 3 dt 
A. ln 1  t 3  C
1
B.
ln 1  t 3  C
3
C. 3 ln 1  t 3  C
D. None of the above

1
 (1  t ) 2 dt 
A. 2 ln 1  t  C
B. ln (1  t ) 2  C
2
C.
C
3
(1  t )
1
D.
C
(1  t )
E. None of the above

What is the amplitude of the function
3 
f ( x)  12 tan  x   3
 
A. 12
B.
3

C. 6
2 2
D.

3
E. None of the above

Fill in the blanks
1.
F(x)
F’(x)
Increasing
positive
2.
decreasing
negative
3.
critical point
0
4.
Concave up
increasing
5.
Concave down
decreasing
6.
Inflection point
a. Concave down
c. zero
e. Concave up
g. Max of Min
Max/Min
b. decreasing
d. inflection point
f. increasing
h. critical point

Which is the best method to use to integrate

A.
B.
C.
D.
(e x  3)dx
Substitution
Algebra
Can' t be done
None of the above

d
sec( x) tan( x) 
dx
A. sec( x)
B. sec 2 ( x) tan( x)
C. sec 3 ( x)  tan 2 ( x) sec( x)
D. None of the above

g (t )  3  cos(t ) is the derivative of
A. 3  sin( t )  C
B. 3t  sin( t )  C
C. 3t  sin( t )  C
D. sin( 3t )  C
E. None of the above

Which of the following equations describes the graph
of the sine function shifted down by C units, left by D
units, stretched horizontally by A units, and vertically
by B units?
 2

A. y  B sin 
( x  D)   C
 A

1

B. y  B sin  ( x  D)   C
A

 2

C. y  B sin 
( x  D)   C
 A

1

D. y  B sin  ( x  D)   C
A

E. None of the above

csc( x)

d
csc( x)
dx
A.  1
B.  tan( x)
C.  ln csc( x)
D. None of the above

What is the equation describing the following graph?
5
-6
-4
-2
2
4
6
5
  5
A. y   cos x  
2
2  2
5   5
B. y  sin  x  
2 2  2
1  5
C. y  5 cos x  
2  2
1  5
D. y  5 sin  x  
2  2
E. None of the above

Given f(t) gives the velocity of a ball in feet per
second as a function of the time in seconds since the
ball was thrown into the air, what are the units of

3
1
f (t )dt
A. seconds
B. ft/sec
C. feet
D. ft/sec2
E. None of the above

Given F(5)=6, use the graph of F’(x) below to find
F(-4).
A3=5
A1=4
A2=1
A. F’(-4)=-2
B. F’(-4)=14
C. F’(-4)=16
D. None of the above

Which of the following is a possible formula for the
function graphed below?
 
 
A. y  sin  2 x     2
2 
 
 
 
B. y  sin  2 x     2
2 
 
 
 
C. y  cos 2 x     2
4 
 
D. All of the above
E. None of the above

Given 3log(x)=12, which of the following
must be true?
A. x=1036
B. 3x=1012
C. x3=1012
D. x=104
E. None of the Above

Given 19t+5=25, what is t?
A.
B.
C.
D.
E.
log 25
t
5
log 19
25
t
5
19
log 25
t
log 19  5
log 19
t
log 25
None of the above

Which of the following formulas is/are correct?
 A  ln( A)
A. ln   
 B  ln( B)
ln( A)
B. ln( A  B) 
ln( B )
 A
C. ln    ln( A)  ln( B)
B
D. ln( A  B)  ln( A)  ln( B)
E. None of the above

Given f(x)=ax+b, when a is greater than one,
f(x) is everywhere ______________.
A. positive
B. negative
C. increasing
D. decreasing
E. None of the Above

Given the following graph of F(x), which are
inflection points of F(x)?
K
J
A B C
I
D
L
H
E
F
G
A. A, C, F, K, M
B. E, G, M
C. E, G
D. B, D, H, I, J, L, N
E. None of the Above
M
N

Suppose p is a critical point of the function
f(x). p is a local minimum if _____________.
A. f’(p) is positive
B. f’(p) is negative
C. f’(x) switches from positive to negative at p
D. f’(x) switches from negative to positive at p
E. None of the Above

Given Aln(x)=C, what is x?
eC
A. x  A
e
B. x  e
C.
D.
E.
F.
ln C
ln A
1
ln( A )
xC
B and C only
All of the above
None of the above

Find the second derivative of f(x)=-13x5
A.
f ' ' ( x)  65 x 4
B.
f ' ' ( x)  260 x
C.
f ' ' ( x)  260 x
D.
f ' ' ( x)  65 x
4
3
3
E. None of the above

Given the following graph of the derivative
F’(x), which are inflection points of F(x)?
K
J
A B C
I
D
L
H
E
F
G
A. A, C, F, K, M
B. E, G, M
C. E, G
D. B, D, H, I, J, L, N
E. None of the Above
M
N

Given the following graph of the derivative
F’(x), which are local minima or maxima of
F(x)?
K
J
A B C
I
D
L
H
E
F
G
A. A, C, F, K, M
B. E, G, M
C. E, G
D. B, D, H, I, J, L, N
E. None of the Above
M
N

How many critical points does f(x)=2x3 have?
A. 0
B. 1
C. 2
D. 3
E. None of the Above

What is the global maximum of f(x)=-(x-2)2+4
A. (-2,4)
B. (4,-2)
C. (4,2)
D. (2,4)
E. None of the Above

Where is f(x)=x3 concave up?
A. (-, )
B. (- ,0]
C. [0, )
D. (0, )
E. None of the Above

Given F’(x)=f(x), the marked quantity on the graph of
f(x) represents:
a
b
A1=4
A. F ' (a )
B. F (b)  F (a)
F (b)  F (a)
C.
ba
D. The average rate of change of F ( x) between a and b
E. The instantane ous rate of change of F ( x) at x  a

The following is a graph of the velocity of two cars A
and B. Given this graph, which of the following
statements are true?
A. Car A travels further
Car A
B. Car A went faster
Car B
C. Car A stops first
D. Car B travels further
E. Car B went faster
F. A, B, and C
G. B, C, and D
H. A, C, and E
I. None of these

For which graph is the statement “has a local and
global minimum at x=3, but no local or global
maximum” true?
A
C
B
D
E. None of these

Which of the following is the graph of F(x), which is
the graph of F’(x), and which is the graph of F’’(x)?
F’(x)
F(x)
F’’(x)

Between x=-1 and x=2, the graph of the derivative
f’(x) is __________.
A. increasing
f(x)
B. decreasing
C. both
D. neither

Between x=-1 and x=2, the graph of the derivative
f’(x) is __________.
A. positive
f(x)
B. negative
C. both
D. neither

Which of the following statements can be true of a
continuous function F(x) for - ≤ x ≤ 
1. F(x)>0, F’(x)<0, F’’(x)>0
A. 1
2. F(x)<0, F’(x)<0, F’’(x)<0
B. 2
3. F(x)<0, F’(x)>0, F’’(x)>0
C. 3
D. 1 and 2
E. 1, 2, and 3
F. None of these