AP Calculus AB
Semester I Exam Review
2015-2016
Day
1/4
1/5
1/6
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1/11
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1/15
1/18
In Class Activity
Semester I Overview, Tabular Data
Free Response Practice
FTC Practice, MC Practice
Free Response Practice
FINAL EXAM PART I – FREE RESPONSE
Last minute MC questions
Exams 7, 8, 9
Exams 1, 2, 5
Exams 3, 4, 6
NO SCHOOL
NO SCHOOL
Name
Out of Class Practice
WS – Tabular Data
Complete FR Packet
Complete MC Packet 1
MC Packet 2
MC Packet 3
Study!!!
Relax
Relax some more because Q3 is difficult…
AP Calculus AB
Name
Notes – Reasoning From Tabular Data
Adapted from AP Calculus Curriculum Module
Example 1:
Let y (t ) represent the population of a town over a 20-year period, where y is a differentiable function of t. The table
below shows the population recorded at selected times.
a. Use data from the table to find an approximation for y '(12), and explain the meaning of y '(12) in terms of the
population of the town. Show the computations that lead to your answer.
b. Use data from the table and a trapezoidal approximation with four subintervals to approximate the average
population of the town over the 20-year period. Show the computations that lead to your answer.
Example 2:
The rate at which water flows into a tank, in gallons per hour, is given by a positive continuous function R of time t. The
table below shows the rate at selected values of t for a 12-hour period.
12
a. Use a midpoint Riemann sum with three subintervals to approximate:
 R  t  dt , and explain the meaning of the
0
definite integral in terms of the water flow, using correct units. Show the computations that lead to your answer.
1
750  24t  t 2  , where positive rate P is

60
measured in gallons per hour and time t is measured in hours. Use P (t ) to find the average rate of water flow during
b. A model for the rate of water flow is given by the function: P  t  
the 12-hour time period. Indicate units of measure.
Example 3.
Particle A moves along a horizontal line with a velocity vA  t  , where vA  t  is a positive continuous function of t. The
time t is measured in seconds, and the velocity is measured in cm/sec. The velocity vA  t  of the particle at selected
times is given in the table below.
a. Use data from the table to approximate the distance traveled by particle A over the interval 0  t  10 seconds by
using a right Riemann sum with four subintervals. Show the computations that lead to your answer, and indicate units
of measure.
b. Particle B moves along the same line with an acceleration of aB  t   2t  7 cm
sec 2
. At time t=1 second, the
velocity of particle B is 13 cm/sec. Which particle is traveling faster at time t=5 seconds? Explain your answer.
AP Calculus AB
Name____________________________
Worksheet on Tabular Data
Date__________________Period______
Use your graphing calculator, and give decimal answers correct to three decimal places.
1.
Let y (t ) represent the temperature of a pie that has been removed from a 450 F oven and left to cool
in a room with a temperature of 72 F, where y is a differentiable function of t . The table below
shows the temperature recorded every five minutes.
t (min)
0
5
10
15
20
25
30
450
388
338
292
257
226
200
y (t ) ( F )
a. Use data from the table to find an approximation for y '(18) , and explain the meaning of y '(18) in
terms of the temperature of the pie. Show the computations that lead to your answer, and indicate
units of measure.
b.
Use data from the table to find the value of  y '  t  dt , and explain the meaning of
25
10

25
10
y '  t  dt in
terms of the temperature of the pie. Indicate units of measure.
c.
A model for the temperature of the pie is given by the function: W (t )  72  380e0.036t , where t is
measured in minutes and W  t  is measured in degrees Fahrenheit
 F.
Use the model to find the
value of W '(18) . Indicate units of measure.
d.
Use the model given in part (c) to find the time at which the temperature of the pie is 300 F.
2. The rate at which water is being pumped into a tank is given by the continuous, increasing function R (t ) . A table of
selected values of R (t ) , for the time interval 0  t  20 minutes, is shown below.
t (min)
R(t) (gal/min)
0
25
4
28
9
33
17
42
a. Use a right Riemann sum with four subintervals to approximate the value of:
20
46

20
0
R(t ) dt
Is your approximation greater or less than the true value? Give a reason for your answer.
b.
A model for the rate at which water is being pumped into the tank is given by the function:
W (t )  25e0.03t , where t is measured in minutes and W (t ) is measured in gallons per minute. Use
the model to find the average rate at which water is being pumped into the tank from t  0 to t  20
minutes.
c.
The tank contained 100 gallons of water at time t  0 . Use the model given in part (b) to find the
amount of water in the tank at t  20 minutes.
3.
Car A has positive velocity VA  t  as it travels on a straight road, where V A is a differentiable function
of t . The velocity is recorded for selected values over the time interval 0  t  10 seconds, as shown in
the table below.
t (sec)
VA  t  (ft/sec)
0
0
2
9
5
36
7
61
10
115
a.
Use data from the table to approximate the acceleration of Car A at t  8 seconds. Indicate units of
measure.
b.
Use data from the table to approximate the distance traveled by Car A over the interval 0  t  10
seconds by using a trapezoidal sum with four subintervals. Show the computations that lead to your
answer, and indicate units of measure.
c.
Car B travels along the same road with an acceleration of aB  t   2t  2 ft/sec2 . At time t  3 seconds,
the velocity of Car B is 11 ft/sec. Which car is traveling faster at time t  7 seconds? Explain your
answer.
AP Calculus AB
Name____________________________
FR Review
Date__________________Period______
Directions: A graphing calculator is required to do some problems or parts of the problems. Show all your work and
justification for each problem. Round decimal approximations to 3 decimal places.
1.
Let v(t )  3t 2  12t be the velocity of a particle moving along the x-axis for time t, t  0 . When t  0 , the
particle is at x  6 .
(a)
(b)
(c)
(d)
2.
Determine the position of the particle at t  a .
Write an expression for the speed of the particle at any time t.
In which direction does the particle begin moving and when does it turn around?
When, if ever, is the particle at the origin?
The number of minutes of daylight per day, L ( d ) , at 40 North latitude is modeled by the function
 2
L(d )  167.5sin 
 d  80    731 , where d is the number of days after the beginning of 1996.
 366

(For Jan. 1, 1996, d  1 ; and for Dec. 31, 1996, d  366 since 1996 was a leap year.)
(a) Which day (d) has the most minutes of daylight? Justify your answer.
(b) What is the average (mean) number of minutes of daylight in 1996? Justify your answer.
(c) What is the total number of minutes of daylight in 1996? Justify your answer.
3.
f ( x)
Let f be a continuous function defined on the closed interval  2,3 . The graph of f consists of a semicircle and a
semi-ellipse, as shown above. Let G( x)  G(2) 

x
2
f (t ) dt ,
(a) On what intervals, if any, is G concave down? Justify your answer.
(b) If the equation of the line tangent to the graph of G ( x ) at the point where x  0 is y  mx  7 , what is the
value of m and the value of G (0) ? Justify your answer.
4.
This problem deals with functions defined by f ( x)  x3  3bx with b  0 .
(a) In a viewing window of  5,5 x  15,15 , graph the members of the family f ( x)  x3  3bx with b  1 ,
b  2 , and b  3 together on the same graph. Label each graph.
(b) Find the x- and y-coordinates of the relative maximum points of f in terms of b .
(c) Find the x- and y-coordinates of the relative minimum points of f in terms of b .
5.
Consider the relation defined by the equation tan y  x  y for x in the open interval 2  x  2 .
(a) Find
dy
in terms of y .
dx
(b) Find the x- and y-coordinate of each point where the tangent line to the graph is vertical.
(c) Find
6.
d2y
in terms of y .
dx 2
Let f be a function defined on the closed
interval  3,9 . The graph of f , consisting
of three line segments is shown to the right.
Let g ( x) 

x
0
f (t ) dt .
(a) Find g (4.5), g '(4.5), and g ''(4.5) .
(b) Find the average value of f on the interval
3,5 .
Show the work that leads to your
answer.
(c) Find the x-coordinate of any points of
inflection of g . Justify your answer.
(d) Find the coordinates of all maximum points of g .
7.
Air is being blown into a sphere at a rate of 6 cubic inches per minute. How fast is the radius changing when the
radius of the sphere is 2 inches?
8.
Water is flowing into an inverted right circular cone at a rate of 4 cubic inches per minute. The cone is 16 inches
tall and its base has a radius of 4 inches. At the moment the water has a depth of 5 inches, how fast is the radius
at the surface of the water increasing?
AP Calculus AB
Name
Notes – Applying the FTC
Date
Period
Learning Targets:


Use the Fundamental Theorem of Calculus to evaluate definite integrals
Use the Fundamental Theorem of Calculus to represent a particular antiderivative and the
analytic and graphical analysis of functions so defined.
Example 1:
Given
dy
 3x 2  4 x  5 with the initial condition y  2  1, find y  3 .
dx
Example 2.
 
Given f '  x   sin x 2 and f  2   5, find f 1 .
Example 3.
The graph of f’ on 2  x  6 consists of two line segments and a semicircle as show
at right. Given that f  2   5, find f  0 , f  2 , and f  6 .
Example 4
The graph of f’ is shown at right, with areas of regions enclosed by the graph
and the x-axis as indicated. Given that f  3  5, find f  0 , f  7  , and f  9 .
Sketch the graph of f on the blank axes.
Example 5
A pizza with temperature of 95 C is put into a 25 C room when t=0. The pizza’s temperature is decreasing at a rate of
r  t   6e0.1t C per minute. Estimate the pizza’s temperature when t=5 minutes.
Example 6


A particle moves along the x-axis so that at any time t > 0, its acceleration is given by a  t   ln 1  2t . If the velocity
of the particle is 2 at time t = 1, then the velocity of the particle at time t = 2 is
.
AP Calculus AB
Name____________________________
Sem I MC Review 1 – NO Calculator
Date__________________Period______

1.
4
2
a. 
x 2 dx
3
4
b. 
1
4
c.
1
4
d.
3
4
e.
5
8
If f ( x)   2 x  1 , then the 4th derivative of f ( x ) at x  0 is
4
2.
a. 0
b. 24
If y 
3.
a.
4.
4  x 
2 2
b.
3x
4  x 
2 2
c.
6x
d.
4  x 
2 2
e. 384
3
4  x 
dy
 cos  2 x  , then y 
dx
1
2
a.  cos  2 x   C
d.
d. 240
3
dy

, then
2
4 x
dx
6 x
If
c. 48
1 2
sin  2 x   C
2
1
2
1
e.  sin  2 x   C
2
b.  cos 2  2 x   C
c.
1
sin  2 x   C
2
2 2
e.
3
2x
8n3
n  5n3  2n 2  1
5.
lim
a. 8
b. 
8
5
c. 0
d.
1
5
e. nonexistent
If f ( x)  x , then f '(5) 
6.
a. 0
b.
1
5
c. 1
d. 5
e.
25
2
The position of a particle moving along a straight line at any time t is given by s(t )  t 2  4t  4 .
7.
What is the
acceleration of the particle when t  4 ?
a. 0
b. 2
If x 2  xy  y 3  0 , then, in terms of x and y,
8.
2x  y
a. 
x  3y2
x  3y2
b. 
2x  y
c. 4
d. 8
e. 12
dy

dx
c. 
2x
1 3y2
3
2
d. 
2x
x  3y2
e. 
2x  y
x  3y2 1
1
2
The velocity of a particle moving on a line at time t is v  5t  2t meters per second. How many meters did
the particle travel from t  0 to t  4 ?
9.
a.
32
3
b. 32
c. 64
d. 74
2
3
e. 84
10.
The function defined by f ( x)  2 x3  4 x 2 for all real numbers x has a relative maximum at x 
b. 
a. 2
11.
c. 4sin x cos x
b. 0
d. 2
e. no rel max
d. 2  cos x  sin x 
e. 2  cos x  sin x 
d  1 1
2
 3   x  at x  1 is
dx  x
x

a. 6
13.
c. 0
If y  cos 2 x  sin 2 x , then y ' 
a. 1
12.
4
3
If
b. 4
 x
1
1
a. 12
5
c. 0
d. 2
e. 6
c. 0
d. 4
e. 12
 k  dx  8 , then k 
b. 4

14.
3
0
x  1 dx 
a. 0
b.
3
2
c. 2
d.
5
2
e. 6
1
3
The volume of a cone of radius r and height h is given by V   r 2 h . If the radius and the height both increase
15.
at a constant rate of
1
centimeter per second, at what rate, in cubic centimeters per second, is the volume
2
increasing when the height is 9 cm and the radius is 6 cm?
a.
16.
1

2
b. 10
d. 54
e. 108
If the position of a particle on the x-axis at time t is 5t 2 , then the average velocity of the particle for 0  t  3 is
a. 45
17.
c. 24


3
0
a. 2
b. 30
c. 15
d. 10
c. 0
d.
e. 5
sin  3x  dx 
b. 
2
3
2
3
e. 2
18.
The graph of the derivative of f is shown in the figure below. Which of the following could be the graph of f ?
a.
b.
d.
19.
c.
e.
If f is a continuous function defined for all real numbers x and if the maximum value of f ( x ) is 5 and the
minimum value of f ( x ) is -7, then which of the following must be true?
I.
The maximum value of f
 x  is 5.
II. The maximum value of f ( x) is 7.
III. The minimum value of f
a. I only
20.
b. II only
 x  is 0.
c. I and II only
d. II and III only
e. I, II, and III
Let f and g have continuous first and second derivatives everywhere. If f ( x)  g ( x) for all real x,
which of the following must be true?
a. none
I.
f '( x)  g '( x) for all real x
II.
f ''( x)  g ''( x) for all real x
III.

3
1
3
f ( x)   g ( x)
1
b. I only
c. III only
d. I and II only
e. I, II, and III
Let f be a continuous function on the closed interval 0,5 . If 3  f ( x)  6 , then the greatest possible value
21.

of
5
0
f ( x) dx is
a. 0
c. 5
a.
f '(a ) exists
b.
d.
f (a)  L
e.
f ( x) is continuous at x  a
c.
f ( x) is defined at x  a
none of these
d x
1  t 2 dt 
dx 2
a.
x
1  x2
b.
1  x2  5
c.
1  x2
d.
x
1  x2

1
5
e.
1
2 1  x2

1
2 5
An equation of the line tangent to y  2 x3  6 x 2  5 at its point of inflection is
a. y  6 x  7
b. y  x
c. y  2 x  5
d. y  3x  1
The average value of f ( x)  3x 2 x3  9 on the closed interval 0,3 is
a. 21
26.
e. 30
x a
23.
25.
d. 15
If lim f ( x)  L , where L is a real number, which of the following must be true?
22.
24.
b. 3

5
3
b. 42
c. 84
d. 126
e. 140
b. 2
c. 12
d. 21
e. nonexistent
x2  4
dx 
x2
a. 1
e. y  4 x  1
27.


sin 
d 
1  cos 
2
0

a. 2 1  2
28.

b. 2 2
c. 2 2
d. 2


2 1
e. 2
The graph of y  f ( x) is shown in the figure below. On which of the following intervals are
dy
d2y
 0 and
 0?
dx
dx 2
I. a  x  b
II. b  x  c
III. c  x  d
a
a. I only
29.
If

12
5
b. II only
f ( x) dx  8 and
a. -15
30.

1
12
b. -1
c. III only
f ( x) dx  7 , then
c. 1
b
d. I and II only

5
1
c
d
e. II and III only
f ( x) dx 
d. 12
e. 15
Which of the following is the correct sketch of the function y  3x 2  2 x3 ?
a.
e. none of these
b.
c.
d.


2 1
31.
Julie wants to put in a 200 square foot garden behind her garage. If she wants to fence in the garden to
keep out the rabbits and uses the garage as one side, find the least amount of fencing that must be used
to enclose the remaining 3 sides of the garden.
a. 10
32.
b. 30
c. 40
d. 50
e. 60
Which of the following definite integrals represents the area of the shaded region below?
6
2
1
1
a.
33.
 x
2
0
2
 2
2
b.
 x
6
2
0
 2 dx
c.
 x
6
0
2
 2  d.
 x
2
2
0
 2 dx
e.
 x
6
2
2
 2 dx
The flow of water, in gallons per hour, through a hose is given by the graph below. Which best approximates
the total number of gallons of water that passed through the hose in the given 24 hour period?
gal
15
10
5
0
hrs
0
a. 10
b. 20
4
c. 100
8
12
d. 200
16
20
24
e. 2000
AP Calculus AB
Sem I MC Review 2 – NO Calculator
  4x
2
1.
3
1
 6 x  dx
a. 2
b. 4
3x  3
2x  3
a.
If

b
a
If f ( x )   x 3  x 
a. 3
b.
f ( x) dx  a  2b , then
a. a  2b  5
4.
c. 6
d. 36
e. 42
If f ( x)  x 2 x  3 , then f '( x) 
2.
3.
Name____________________________
Date__________________Period______
x
2x  3
c.
1
2x  3
d.
x  3
2x  3
e.
5x  6
2 2x  3
  f ( x)  5 dx 
b
a
b. 5b  5a
c. 7b  4a
d. 7b  5a
e. 7b  6a
c. -1
d. -3
e. -5
1
, then f '(1) 
x
b. 1
5.
The graph of y  3x 4  16 x3  24 x 2  48 is concave down for
a. x  0
6.
b. x  0
d. 6 x 2
   
sin  x  cos  x 
3
e.
2
x2
3
 
2sin  x  cos  x 
 
e.
3
c. sin 2 x 3
3
A bug to crawl up a vertical wire at time t  0 . The velocity v of the bug at time t, 0  t  8 , is given by the
function whose graph is shown below. At what value of t does the bug change
direction?
b. 4
c. 6
d. 7
e. 8
d. 8
e. 6
What is the total distance the bug traveled from t  0 to t  8 ?
b. 13
a. 14
9.
2
or x  2
3
b. 6 x 2 cos x 3
3
a. 2
8.
d. x 
d
cos 2  x3  
dx
a. 6 x 2 sin x3 cos x3
7.
2
3
c. x  2 or x  
c. 11
An equation of the line tangent to the graph of y  cos  2 x  at x 


a. y  1    x 


d. y    x 



4

4


b. y  1  2  x 


e. y  2  x 



4

4

4
is


c. y  2  x 


4
At what point on the graph of y 
10.
1
2
1
2
a.  ,  
1 2
x is the tangent line parallel to the line 2 x  4 y  3 ?
2
1 1
 2 8
b.  , 


1
4
Let f be a function defined for all real numbers x. If f '( x) 
11.
a.
12.
 , 2
b.
 ,  
 1
 2
c. 1,  
c.
 2, 4
d. 1, 
4  x2
x2
e.
 2, 2 
, then f is decreasing on the interval
d.
 2,  
e.
 2,  
The graph of the derivative of f is shown in the figure below. Which of the following could be the graph of f ?
a.
b.
d.
e.
c.
Let f be a differentiable function such that f (3)  2 and f '(3)  5 . If the tangent line to the graph of f at
13.
x  3 is used to find an approximation to a zero of f , that approximation is
a. 0.4
b. 0.5
c. 2.6
If x 2  y 2  41 , what is the value of
14.
a. 
15.
25
8
b. 
d. 3.4
e. 5.5
d2y
at the point  5, 2  ?
dx 2
1
2
c. 
29
8
d.
25
8
e.
5
2
The graph of the function f is shown in the figure below. Which of the following statements about f is true?
4
3
2
1
a
b
a. lim f ( x)  lim f ( x)
b. lim f ( x )  2
d. lim f ( x)  1
e. lim f ( x)  DNE
x a
x b
x b
c. lim f ( x)  2
xa
x b
x a
The average value of cos x on the interval  3,5 is
16.
a.
sin 5  sin 3
8
b.
sin 5  sin 3
2
c.
sin 3  sin 5
2
d.
sin 3  sin 5
2
e.
sin 3  sin 5
8
17.
The graph of the function y  x3  6 x 2  7 x  2cos x changes concavity at x  (use your calculator)
a. 1.58
18.
b. 1.63
c. 1.67
The graph of f is shown in the figure below. If

3
1
d. 1.89
e. 2.33
f ( x) dx  2.3 and F '( x)  f ( x) , then F (3)  F (0) 
3
2
1
1
a. 0.3
19.
2
3
4
5
6
b. 1.3
c. 3.3
Let f be a function such that lim
h 0
d. 4.3
e. 5.3
f (2  h)  f (2)
 5 . Which of the following must be true?
h
I.
f is continuous at x  2 .
II.
f is differentiable at x  2 .
III. the derivative of f is continuous at x  2 .
a. I only
b. II only
c. I and II only
d. I and III only
e. II and III only
20.
A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the
crossing and watches an eastbound train traveling at 60 meters per second. At how many meters per second
is the train moving away from the observer 4 seconds after it passes through the intersection?
(this is a calculator problem)
a. 57.60
b. 57.88
c. 59.20
d. 60.00
e. 67.4
x . If the rate of change of f at x  c is twice its rate of change at x  1 , find the rate of change
Let f ( x) 
21.
at c.
a.
1
4
b. 1
Let f ( x) 
22.
a

x
a
c. 4
1
2
c
b
b.
a
e. 2
h(t ) dt , where h has the graph below. Which of the following could be the graph of f ?
a.
c.
c
b
d.
a
b
c
a
b
c
a
b
c
e.
a
23.
d.
c
b
A table of values for a continuous function f is shown below. If four equal subintervals are used,
of the following is the trapezoidal approximation of
x
f ( x)
0
3
a. 8
b. 12

2
0
0.5
3
c. 16
which
f ( x) dx ?
1.0
5
1.5
8
d. 24
2.0
13
e. 32
24.
Which of the following are antiderivatives of f ( x)  sin x cos x ?
sin 2 x
2
cos 2 x
F ( x) 
2
 cos  2 x 
F ( x) 
4
F ( x) 
I.
II.
III.
a. I only
25.
b. II only
d. I and III only
If the function g has a continuous derivative on 1, p  , then
a. g ( p)  g (1)
26.
c. III only
b.
g ( p)  g (1)
c. g ( p)

p
1
e. II and III only
g '( x) dx 
d. g ( x )  p
e. g ''( p)  g ''(1)
Let g be a polynomial function with degree of at least 3. If a  b and g (a )  g (b)  3 , which of the following
must be true for at least one value of x between a and b?
g ( x)  0
g '( x)  0
g ''( x)  0
I.
II.
III.
a. I only
27.
c. I and II only
d. I, II, and III
e. none of these
b. csc x  C
c.  csc x  C
d.  cot x  C
e. tan x  C
 csc x cot x 
a. cot x  C
28.
b. II only
The graph of a piece wise linear function f , for 2  x  5 , is shown below. What is the value of

5
2
a. -9
f ( x) dx ?
b. -3
c. 0
d. 3
e. 6
29.
If f is continuous for a  x  b and differentiable for a  x  b , which of the following must be true?
I.
f has a minimum value on a  x  b
II.

III.
b
a
f ( x) dx exists
f '(c )  0 for some c such that a  c  b
a. all
30.
If F ( x) 
b. I only
 t
x
0
a. not possible
2
c. II only
d. I and II only
e. none of these
 2t  1 dt , then F '(3) 
b. -2
c. 0
d. 4
e. 16
31. The radius of a circle is decreasing at a constant rate of 0.3 inches per minute. In terms of the circumference C,
what is the rate of change of the area of the circle, in square inches per minute?
a. 0.3 C
b. 
0.3C
2
c. 0.3C
d. .09C
e. .09 C
AP Calculus AB
Sem I MC Review 3
Name____________________________
Date__________________Period______
Part I – NO CALCULATOR
3x 2  4
1. lim
is
x  2  7 x  x 2
(A) 3
2. If, for all x, f '( x)   x  2 
(A)
(B)
(C)
(D)
(E)
(B) 1
4
 x  1
(C) -3
3
(D) 
(E) 0
, it follows that the function f has
a relative minimum at x = 1
a relative maximum at x = 1
both a relative maximum at x = 1 and a relative minimum at x = 2
neither a relative maximum nor a relative minimum
relative minima at x = 1 and x = 2
3. If f is differentiable, we can use the line tangent to f at x  a to approximate values of f near x  a . Suppose
this method always underestimates the correct values. If so, then at x  a , f must be
(A)
(B)
(C)
(D)
(E)
positive
increasing
decreasing
concave upward
concave downward
 
4. If f ( x)  cos x sin 3 x , then f '   is equal to
6
(A)
1
2
(B) 
3
2
(C) 0
(D) 1
(E)

1
2
5. The graph of f '' is shown below. If f '(1)  0 , then f '( x)  0 at x 
(A) 0
(B) 2
(C) 3
(D) 4
(E) 7
6. If P( x)  g 2 ( x) , then P '(3) equals
(A)
(B)
(C)
(D)
(E)
4
6
9
12
18
f'
g
g'
2
1
2
-3
5
2
3
1
0
4
3
4
2
2
3
4
6
4
3
1
2
x
f
1
7. The area of the region bounded by the graph of y  x 1  x ,the x-axis, x  0 , and x  1 is
2
(A)
1
3
8. The curve of y 
(A) x  3
(B)
1
2
3
(C)
1
2
(D)
2
3
(E)1
1 x
is concave upward when
x 3
(B) 1  x  3
(C) x  1
(D) x  1
(E) x  3
(D) 8.02
(E) 8.08
9. If f (3)  8 and f '(3)  4 then f (3.02) is approximately
(A) -8.08
(B) 7.92
(C) 7.98
 x 2 for x  1
If f ( x)  
10.
2 x  1 for x  1
, then
(A) f ( x ) is not continuous at x  1
(B) f ( x ) is continuous at x  1 but f '(1) does not exist
(C) f '(1) exists and equals 1
(D) f '(1)  2
(E) lim f ( x ) does not exist
x 1
11. In the following, L(n), R(n), M (n), and T (n) denote, respectively, left, right, midpoint, and trapezoid sums
with n subdivisions. Which of the following is equal exactly to
(A) L (1)
12.

x
a
(B) L(2)

1
1
x dx ?
(C) R (1)
(D) M (1)
(E) none of these
g (t )dt
(E) g (b)  g (a)
x
g (t )dt   g (t )dt is equal to the constant
(A) 0
b
(B) b  a
(C) a  b
(D)

b
a
Part II – your calculator may be used.
13. An object moving along a line has velocity v(t )  t cos t  ln(t  2) , where 0  t  10 . How many times
does the object reverse direction?
(A) none
(B) one
(C) two
(D) three
(E) four
14. The acceleration of a particle moving along a straight line is given by a  6t . If when t  0 , its
velocity, v, is 1 and its position, s, is 3, then at any time t
(A) s  t  3
3
(B) s  t  3t  1
3
 
15. If y  f x 2 and f '( x)  5 x  1 then
(A) 2 x 5x2  1
(B)
(C) s  t  t  3
3
t
(D) s   t  3
3
dy
is equal to
dx
(C) 2 x 5 x  1
5x  1
(D)
16. If the substitution x  2t  1 is used, which of the following is equivalent to
(A)

3
4
0
x dx
(B)
1 34
x dx
2 0
t3 t2
(E) s    3
3 2
(C)
1 1 4
x dx
1
2  3
(D)

3
4
0
5x 1
2x
(E) none of these
2t  1 dt ?
1 74
x dx
2 1
(E) 2

7
1
4
x dx
17. A 26-foot ladder leans against a building so that its foot moves away from the building at the
rate of 3 feet per second. When the foot of the ladder is 10 feet from the building, the top is moving down
at the rate of r feet per second, where r is
(A)
46
3
(B)
3
4
(C) -
5
4
(D)
5
2
(E)
4
5
18. If F ( x) 
(A)

2x
1
1
dt , then F '( x) 
1 t3
1
1  x3
(B)
1
1  2x3
(C)
2
1  2x3
(D)
1
1  8x 3
(E)
2
1  8x 3
t
19. Water is leaking from a tank at the rate of R (t )  5arctan   gallons per hour, where t is the number of
5
hours since the leak began. How many gallons will leak out during the first day?
(USE CALC and remember that arctan is tan 1 )
(A) 7
(B) 82
(C) 124
(D) 141
(E) 164
a (ft/sec2)
20.
2
t (sec)
1
2
3
4
5
-2
The graph shows an object’s acceleration (in ft/sec2). It consists of a quarter-circle and two line segments. If the
object was at rest at t = 5 seconds, what was its initial velocity?
(A) -2 ft/sec
(B)
3   ft/sec
(C) 0 ft/sec
(D)   3 ft/sec
(E)   3 ft/sec