AP® Calculus AB
2002 Scoring Guidelines
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AP® CALCULUS AB 2002 SCORING GUIDELINES
Question 1
Let B and C be the functions given by B N A N and C (N ) ln N .
(a) Find the area of the region enclosed by the graphs of B and C between N and N 1.
(b) Find the volume of the solid generated when the region enclosed by the graphs of B and C between
N and N is revolved about the line O 4.
Let D be the function given by D(N ) B (N ) C(N ). Find the absolute minimum value of DN on the
(c)
closed interval
1
> N > 1, and find the absolute maximum value of DN on the closed interval
2
1
> N > 1. Show the analysis that leads to your answers.
2
(a) Area =
¨
A N ln N @N = 1.222 or 1.223
(b) Volume = 3 ¨ 4 ln N 4 A N @N
= 7.5153 or 23.609
(c) D N B N C N A N N 0.567143
£
¦ 1 : integral
2 ¦¤
¦
1 : answer
¦
¥
1 : limits and constant
2 : integrand
1 each error
4 Note: 0 / 2 if not of the form
>
k ¨ R(x ) r (x ) dx
=
1
:
answer
£
1 : considers D a(N ) 0
¦
¦
¦
¦
1 : identifies critical point
¦
¦
3 ¤
¦
and endpoints as candidates
¦
¦
¦
¦
1 : answers
¦
¦
¥
N
Absolute minimum value and absolute
maximum value occur at the critical point or
at the endpoints.
Note: Errors in computation come off
the third point.
D(0.567143) 2.330
D(0.5) 2.3418
D(1) 2.718
The absolute minimum is 2.330.
The absolute maximum is 2.718.
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
AP® CALCULUS AB 2002 SCORING GUIDELINES
Question 2
The rate at which people enter an amusement park on a given day is modeled by the function E defined by
E t 15600
t 24t 160 .
The rate at which people leave the same amusement park on the same day is modeled by the function L defined by
L t 9890
.
t 38t 370 Both - J and L t are measured in people per hour and time t is measured in hours after midnight. These
functions are valid for 9 > J > 23, the hours during which the park is open. At time J 9, there are no people in
the park.
(a) How many people have entered the park by 5:00 P.M. ( J % )? Round answer to the nearest whole number.
(b) The price of admission to the park is $15 until 5:00 P.M. ( J % ). 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 J
¨' E x Lx dx
for 9 > J > 23. The value of 0 % to the nearest whole number is 3725.
Find the value of 0 = % and explain the meaning of 0 % and 0 = % in the context of the park.
(d) At what time t, for 9 > J > 23, does the model predict that the number of people in the park is a maximum?
(a)
%
¨'
£
¦
1 : limits
¦
¦
¦
3 ¤ 1 : integrand
¦
¦
¦
1 : answer
¦
¦
¥
- (J ) @J 6004.270
6004 people entered the park by 5 pm.
(b) 15 ¨
%
'
- (J ) @J 11¨
!
%
- (J ) @J 104048.165
1 : setup
The amount collected was $104,048.
or
!
¨%
- (J ) @J 1271.283
1271 people entered the park between 5 pm and
11 pm, so the amount collected was
$15 (6004) $11 (1271) $104, 041.
(c) H (17) E (17) L(17) 380.281
There were 3725 people in the park at t = 17.
The number of people in the park was decreasing
at the rate of approximately 380 people/hr at
time t = 17.
1 : value of 0 (17)
2 : meanings
1 : meaning of 0 (17)
3 1 : meaning of 0 (17)
1 if no reference to J 17
1 : E (t ) L(t ) 0
2 1 : answer
(d) H t E t Lt t = 15.794 or 15.795
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Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
3
AP® CALCULUS AB 2002 SCORING GUIDELINES
Question 3
An object moves along the N-axis with initial position N 0 2. The velocity of the object at time J F is given
3
by L J sin
J .
3
(a) What is the acceleration of the object at time J 4 ?
(b) Consider the following two statements.
Statement I:
For 3 J 4.5, the velocity of the object is decreasing.
Statement II:
For 3 J 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.
What is the total distance traveled by the object over the time interval 0 > J > 4 ?
(c)
(d) What is the position of the object at time J 4 ?
1 : answer
3
43
cos
3
3
3
= or 0.523 or 0.524
$
(a) =(4) L a(4) (b) On 3 J 4.5 :
3
3
=(J ) L a(J ) cos J 0
3
3
Statement I is correct since =(J ) 0.
Statement II is correct since LJ and
=(J ) 0.
£
¦
1 : I correct, with reason
¦
¦
¦
3 ¤ 1 : II correct
¦
¦
¦
1 : reason for II
¦
¦
¥
(c) Distance =
"
¨
£
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
3 ¦
¤
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¦
¥
¦
£
¦
2 ¦¤
¦
¦
¥
L(J ) @J 2.387
OR
3
3
3
N (J ) cos J 2
3
3
3
N 9
N (4) 2 3.43239
23
LJ when J = 3
6
N (3) 2 3.90986
3
N ! N N " N ! #
= 2.387
3
(d) N (4) N (0) "
¨ L(J )@J 3.432
OR
3
3
3
N (J ) cos J 2
3
3
3
9
N (4) 2 3.432
23
£
¦
limits of 0 and 4 on an integral
¦
¦
¦
¦
of L(J ) or L(J )
¦
¦
¦
¦
1: ¤
or
¦
¦
¦
uses N (0) and N (4) to compute
¦
¦
¦
¦
distance
¦
¦
¥
1 : handles change of direction at
student’s turning point
1 : answer
0/1 if incorrect turning point or
no turning point
1 : integral
1 : answer
OR
3
3
1 : N (J ) cos J +
3
3
2 1 : answer
0/1 if no constant of integration
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Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
4
AP® CALCULUS AB 2002 SCORING GUIDELINES
Question 4
The graph of the function B shown above consists of two line segments. Let C be the
function given by C (N ) N
¨
B J @J.
(a) Find C 1 , C 1 , and C 1 .
(b) For what values of N in the open interval 2,2 is C increasing? Explain your
reasoning.
For what values of N in the open interval 2,2 is the graph of C concave
(c)
down? Explain your reasoning.
(d) On the axes provided, sketch the graph of C on the closed interval 2,2 .
(a) C ¨
B J @J ¨
B J @J 1 : C (1)
3 1 : C (1)
1 : C (1)
!
C B C B !
£
¦ 1 : interval
2 ¦¤
¦
1 : reason
¦
¥
(b) C is increasing on N because
C N B N on this interval.
(c) The graph of C is concave down on N because C N B N on this interval.
or
because C N B N is decreasing on this
£
¦ 1 : interval
2 ¦¤
¦
1 : reason
¦
¥
interval.
(d)
1 : C (2) C (0) C(2) 0
1 : appropriate increasing/decreasing
2 and concavity behavior
1 vertical asymptote
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5
AP® CALCULUS AB 2002 SCORING GUIDELINES
Question 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 D is changing at the
constant rate of
!
cm/hr.
1
(The volume of a cone of height D and radius H is given by 8 3H D. )
3
(a) Find the volume 8 of water in the container when D # cm. Indicate units
of measure.
(b) Find the rate of change of the volume of water in the container, with respect to time, when D # 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?
(a) When D = 5, H (b)
#
; 8 (5) 3 cm
5 125
12
1 5
3
3 2
!
H
#
1
, so H D
2
D
@8
@D
3D
8 3 D D 3D ! ;
@J
"
@J
! "
!
@8
!
#
3 # 3 cm hr
@J D # "
&
5 OR
@H
@D
@8
@D
@H
3 H
HD
;
@J
@J
@J
!
@J
@J
@8
1
25
3
5
3
5 3
2
5
@J D 5,H 3
4
10
2
20
2
(c)
15 cm 3
3
hr
8
1
D in (a) or (b)
2
8 as a function of one variable
in (a) or (b)
1: OR
@H
@J
2:
@8
@J
2 chain rule or product rule error
1 : evaluation at D 5
1 : correct units in (a) and (b)
!
units of cm in (a) and cm hr in (b)
!
1: H dV
£
¦
1 : shows
k ¸ area
¦
¦
dt
¦
¦
2 ¤ 1 : identifies constant of
¦
¦
¦
proportionality
¦
¦
¦
¥
@8
@D
!
3D
3D
@J
"
@J
"
!
!
!
3 H 3H =HA=
"
3
The constant of proportionality is .
10
1 : 8 when D = 5
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Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
6
AP® CALCULUS AB 2002 SCORING GUIDELINES
Question 6
1 4 6 % % 5 3 0
1.5 1.0
N
B (N ) B aN 0.5
0.5 1.0 1.5 6 4 1 3
5
7
Let B be a function that is differentiable for all real numbers. The table above gives the values of B and its
derivative B a for selected points N in the closed interval 1.5 N 1.5. The second derivative of B has the
property that B aaN for 1.5 N 1.5.
(a) Evaluate
1.5
¨0
3 B (N ) 4 @N . Show the work that leads to your answer.
(b) Write an equation of the line tangent to the graph of B at the point where N 1. Use this line to
approximate the value of B (1.2). Is this approximation greater than or less than the actual value of B (1.2)?
Give a reason for your answer.
(c)
Find a positive real number H having the property that there must exist a value ? with 0 ? 0.5 and
B aa(?) H . Give a reason for your answer.
2N N 7 for N 0
(d) Let C be the function given by C (N ) 2N N 7 for N 0.
The graph of C passes through each of the points N, B (N ) given in the table above. Is it possible that B and
C are the same function? Give a reason for your answer.
(a)
1.5
¨0
! B N " @N ! ¨
3 B (N ) "N
(b)
1.5
0
1.5
0
B N @N 1.5
¨0
£
¦ 1 : antiderivative
2 ¦
¤
¦
1 : answer
¦
¥
" @N
3 1 (7) 4 1.5 24
O # N "
£
¦
1 : tangent line
¦
¦
¦
3 ¤ 1 : computes O on tangent line at N 1.2
¦
¦
¦
1 : answer with reason
¦
¦
¥
B (1.2) 5 0.2 4 3
The approximation is less than B (1.2) because the
graph of B is concave up on the interval
(c)
1 N 1.2 .
£ 1 : reference to MVT for B a (or differentiability
¦
¦
¦
¦
2 ¤
of B a )
¦
¦
¦
1 : value of H for interval 0 b N b 0.5
¦
¦
¥
By the Mean Value Theorem there is a ? with
0 ? 0.5 such that
B (0.5) B (0)
30
B (?) 6H
0.5 0
0.5
(d)
lim C (N ) lim 4N 1 1
N N £
1 : answers “no” with reference to
¦
¦
¦
¦
2 ¤
C a or C aa
¦
¦
¦
1 : correct reason
¦
¦
¥
lim C (N ) lim 4N 1 1
N N Thus C a is not continuous at N = 0, but B a is
continuous at N = 0, so B L C .
OR
C aaN " for all N L , but it was shown in part
(c) that B aa? $ for some ? L , so B L C .
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7