Unformatted text preview: AP® Calculus AB
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Educational Testing Service and ETS are registered trademarks of Educational Testing Service. 2002 AP® CALCULUS AB FREERESPONSE QUESTIONS CALCULUS AB
SECTION II, Part A
Time—45 minutes
Number of problems—3
A graphing calculator is required for some problems or parts of problems. 05 05 1. Let f and g be the functions given by f x = e x and g x = ln x.
(a) Find the area of the region enclosed by the graphs of f and g between x = 1
and x = 1.
2 (b) Find the volume of the solid generated when the region enclosed by the graphs of f and g between x =
and x = 1 is revolved about the line y = 4. 05 1
2 05 05 (c) Let h be the function given by h x = f x  g x . Find the absolute minimum value of h( x ) on the
1
1
closed interval x 1, and find the absolute maximum value of h( x ) on the closed interval x 1.
2
2
Show the analysis that leads to your answers. Copyright © 2002 by College Entrance Examination Board. All rights reserved.
Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 2 GO ON TO THE NEXT PAGE. 2002 AP® CALCULUS AB FREERESPONSE QUESTIONS
2. The rate at which people enter an amusement park on a given day is modeled by the function E defined by
E (t ) = 3t 2 15600
.
 24t + 160 8 The rate at which people leave the same amusement park on the same day is modeled by the function L
defined by
L( t ) = 05 05 3t 2 9890
.
 38t + 370 8 Both E t and L t are measured in people per hour and time t is measured in hours after midnight. These
functions are valid for 9 t 23, the hours during which the park is open. At time t = 9, there are no people
in the park.
(a) How many people have entered the park by 5:00 P.M. ( t = 17)? Round your answer to the nearest whole
number.
(b) The price of admission to the park is $15 until 5:00 P.M. ( t = 17 ). After 5:00 P.M., the price of admission to
the park is $11. How many dollars are collected from admissions to the park on the given day? Round your
answer to the nearest whole number.
(c) Let H (t ) = I0
t 9 5 E ( x )  L( x ) dx for 9 t 23. The value of H (17) to the nearest whole number is 3725. Find the value of H (17), and explain the meaning of H (17) and H (17) in the context of the amusement
park.
(d) At what time t, for 9 t 23, does the model predict that the number of people in the park is a maximum? Copyright © 2002 by College Entrance Examination Board. All rights reserved.
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3. An object moves along the xaxis with initial position x( 0) = 2. The velocity of the object at time t 0
p
is given by v(t ) = sin t .
3 (a) What is the acceleration of the object at time t = 4 ?
(b) Consider the following two statements.
Statement I: For 3 < t < 4.5, the velocity of the object is decreasing.
Statement II: For 3 < t < 4.5, the speed of the object is increasing.
Are either or both of these statements correct? For each statement provide a reason why it is correct or not
correct.
(c) What is the total distance traveled by the object over the time interval 0 t 4 ?
(d) What is the position of the object at time t = 4 ? END OF PART A OF SECTION II Copyright © 2002 by College Entrance Examination Board. All rights reserved.
Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 4 2002 AP® CALCULUS AB FREERESPONSE QUESTIONS
CALCULUS AB
SECTION II, Part B
Time—45 minutes
Number of problems—3
No calculator is allowed for these problems. I 4. The graph of the function f shown above consists of two line segments. Let g be the function given by 05 gx = x 0 05 f t dt . (a) Find g ( 1), g ( 1), and g ( 1). 05
(c) For what values of x in the open interval 0 2, 25 is the graph of g concave down? Explain your reasoning. (b) For what values of x in the open interval 2, 2 is g increasing? Explain your reasoning.
(d) On the axes provided, sketch the graph of g on the closed interval 2, 2 .
(Note: The axes are provided in the pink test booklet only.) Copyright © 2002 by College Entrance Examination Board. All rights reserved.
Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 5 GO ON TO THE NEXT PAGE. 2002 AP® CALCULUS AB FREERESPONSE QUESTIONS 5. A container has the shape of an open right circular cone, as shown in the figure above. The height of the
container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that
its depth h is changing at the constant rate of  3 cm/hr.
10
1
(Note: The volume of a cone of height h and radius r is given by V = pr 2 h. )
3
(a) Find the volume V of water in the container when h = 5 cm. Indicate units of measure.
(b) Find the rate of change of the volume of water in the container, with respect to time, when h = 5 cm.
Indicate units of measure.
(c) Show that the rate of change of the volume of water in the container due to evaporation is directly
proportional to the exposed surface area of the water. What is the constant of proportionality? Copyright © 2002 by College Entrance Examination Board. All rights reserved.
Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 6 GO ON TO THE NEXT PAGE. 2002 AP® CALCULUS AB FREERESPONSE QUESTIONS x 1.5 1.0  0.5 0 0.5 1.0 1.5 fx 1 4 6 7 6 4 1 7 5 3 0 3 5 7 05
f 0 x 5 6. Let f be a function that is differentiable for all real numbers. The table above gives the values of f and its
derivative f for selected points x in the closed interval 1.5 x 1.5. The second derivative of f has 05 the property that f x > 0 for 1.5 x 1.5.
(a) Evaluate I0
1.5 0 5 3 f ( x ) + 4 dx. Show the work that leads to your answer. (b) Write an equation of the line tangent to the graph of f at the point where x = 1. Use this line to approximate
the value of f 1.2 . Is this approximation greater than or less than the actual value of f 1.2 ? Give a reason
for your answer. 05 05 (c) Find a positive real number r having the property that there must exist a value c with 0 < c < 0.5
and f (c) = r . Give a reason for your answer.
(d) Let g be the function given by g ( x ) = %2 x
&2 x
K
' 2
2  x  7 for x < 0 + x  7 for x 0 .
The graph of g passes through each of the points x, f ( x ) given in the table above. Is it possible that
f and g are the same function? Give a reason for your answer. 0 5 END OF EXAMINATION Copyright © 2002 by College Entrance Examination Board. All rights reserved.
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 Spring '10
 Zalla
 Calculus, Topology, College Entrance Examination Board

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