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AP
Calculus BC
2002 Scoring Guidelines
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AP
CALCULUS BC 2002 SCORING GUIDELINES
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
Question 1
Let B and C be the functions given by NB N A and ( ) ln .C N N
(a) Find the area of the region enclosed by the graphs ofB and C between
N and 1.N
(b) Find the volume of the solid generated when the region enclosed by the graphs ofB and C between
N and N is revolved about the line 4.O
(c) Let D be the function given by ( ) ( ) ( ).D N B N C N Find the absolute minimum value of D N on the
closed interval1
1,2
N> > and find the absolute maximum value of D N on the closed interval
11.
2N> > Show the analysis that leads to your answers.
(a) Area =
lnN
A N @N = 1.222 or 1.223 21 : integral
1 : answer
(b) Volume =
4 ln 4N
N A @N 3
= 7.5153 or 23.609
4
1 : limits and constant
2 : integrand
1 each error
Note: 0/2 if not of the form
( ) ( )
1 : answer
>
=
k R x r x dx
(c)
ND N B N C N AN
a a a
0.567143N
Absolute minimum value and absolute
maximum value occur at the critical point or
at the endpoints.
(0.567143) 2.330D
(0.5) 2.3418D
(1) 2.718D
The absolute minimum is 2.330.
The absolute maximum is 2.718.
3
1 : considers ( ) 0
1 : identifies critical point
and endpoints as candidates
1 : answers
D Na
Note: Errors in computation come off
the third point.
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AP
CALCULUS BC 2002 SCORING GUIDELINES
Copyright 2002 by College Entrance Examination Board. All rights reserved.
Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
3
Question 2
The rate at which people enter an amusement park on a given day is modeled by the function E defined by
15600.
24 160E t
t t
The rate at which people leave the same amusement park on the same day is modeled by the function L defined by
9890.
38 370L t
t t
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 23,J> > the hours during which the park is open. At time 9,J 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 '
J
H t E x L x dx for 9 23.J> > 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 23,J> > does the model predict that the number of people in the park is a maximum?
(a)%
'
( ) 6004.270- J @ J 6004 people entered the park by 5 pm. 3
1 : limits
1 : integrand
1 : answer
(b)% !
' %
15 ( ) 11 ( ) 104048.165- J @J - J @J
The amount collected was $104,048.
or!
%
( ) 1271.283- J @ J 1271 people entered the park between 5 pm and
11 pm, so the amount collected was
$15 (6004) $11 (1271) $104, 041.
1 : setup
(c) (17) (17) (17) 380.281H E La
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 (17)
2 : meanings
1 : meaning of (17)3
1 : meaning of (17)
1 if no reference to 17
0
0
0
J
a
a
(d) H t E t L t a
t= 15.794 or 15.7952
1 : ( ) ( ) 0
1 : answer
E t L t
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AP
CALCULUS BC 2002 SCORING GUIDELINES
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
Question 3The figure above shows the path traveled by a roller coaster
car over the time interval &J> > seconds. The position
of the car at time J seconds can be modeled parametrically
by 10 4 sin , 20 1 cos ,N J J J O J J J
where N and O are measured in meters. The derivatives of these functions are given by
10 4 cos , 20 sin cos 1.N J J O J J J J a a
(a) Find the slope of the path at time 2.J Show the computations that lead to your answer.(b) Find the acceleration vector of the car at the time when the cars horizontal position is N= 140.
(c) Find the time J at which the car is at its maximum height, and find the speed, in m/sec, of the car at
this time.
(d) For 0 18,J there are two times at which the car is at ground level 0 .O Find these two times
and write an expression that gives the average speed, in m/sec, of the car between these two times. Do
not evaluate the expression.
(a) Slope =
(2) 18 sin 2 cos 2 1
10 4 cos 2(2)J
O@O
@N N
a
a
= 1.793 or 1.794
1 : answer using@O @O @N
@J @J @N
(b)
( ) 10 4s in 140; 13.647083N J J J J
3.529,N J O J aa aa 1.225 or 1.226
Acceleration vector is 3.529,1.225
or 3.529,1.226
2
1 : identifies acceleration vector
as derivative of velocity vector
1 : computes acceleration vector
when 140N
(c) ( ) 20 sin cos 1 0O J J J J a
J = 3.023 or 3.024 at maximum height
Speed =
N J O J N J a a a
= 6.027 or 6.028
3
1 : sets ( ) 0
1 : selects first 0
1 : speed
O J
J
a
(d) O J when 2J 3 and 4J 3
Average speed = "
N J O J @J
3
33
= =
"
110 4 cos 20 sin cos 1
2 J J J J @J
3
33
3
1 : 2 , 4
1 : limits and constant
1 : integrand
J J3 3
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AP
CALCULUS BC 2002 SCORING GUIDELINES
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
Question 4The graph of the function B shown above consists of two line segments. Let C be the
function given by
( ) .N
C N B J @J (a) Find 1 ,C 1 ,C= and 1 .C==
(b) For what values of N in the open interval 2, 2 is C increasing? Explain yourreasoning.
(c) For what values of N in the open interval 2, 2 is the graph of C concave
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
C Ba
!C Baa a
3
1 : ( 1)
1 : ( 1)1 : ( 1)
C
C
C
a aa
(b) C is increasing on N because
C N B N a on this interval.2
1 : interval
1 : reason
(c) The graph ofC is concave down on N
because C N B N aa a
on this interval.or
because C N B N a is decreasing on this
interval.
21 : interval
1 : reason
(d)
2
1 : ( 2) (0) (2) 0
1 : appropriate increasing/decreasing
and concavity behavior1 vertical asymptote
C C C
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AP
CALCULUS BC 2002 SCORING GUIDELINES
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
Question 5
Consider the differential equation 2 4 .@O
O N@N
(a) The slope field for the given differential equation is provided. Sketch the solution curve that passes through
the point (0,1) and sketch the solution curve that passes through the point (0, 1) .
(b) Let B be the function that satisfies the given differential equation with the initial condition 0 1.B Use
Eulers method, starting at N with a step size of 0.1, to approximate 0.2 .B Show the work that leads
to your answer.
(c) Find the value of > for which O N > is a solution to the given differential equation. Justify your
answer.
(d) Let C be the function that satisfies the given differential equation with the initial condition 0 0.C Does
the graph of C have a local extremum at the point (0, 0)? If so, is the point a local maximum or a local
minimum? Justify your answer.
(a)
2
1 : solution curve through (0,1)
1 : solution curve through (0, 1)
Curves must go through the indicated
points, follow the given slope lines, and
extend to the boundary of the slope field.
(b) (0.1) (0) (0)(0.1)B B B=N
= 1 2 0 (0.1) 1.2
(0.2) (0.1) (0.1)(0.1)
1.2 2.4 0.4 (0.1) 1.4
B B Bax
x
2
1 : Eulers method equations or
equivalent table applied to (at least)
two iterations
1 : Euler approximation to (0.2)
(not eligible without first point)
B
(c) Substitute O N > in the DE:
" N > N > , so > = 1
ORGuess 1,> O N
Verify: " " " @O
O N N N @N
.
2 1 : uses 2 2 in DE
1 : 1
@N >
@N
>
(d) Chas local maximum at (0, 0).
(0,0)
(0) 2(0) 4(0) 0,@O
C@N
a and
( ) 2 4,
@ O @O C N
@N@Naa so
(0) 2 (0) 4 4 0.C Caa a
3
1 : (0) 0
1 : shows (0) 4
1 : conclusion
C
C
a aa
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AP
CALCULUS BC 2002 SCORING GUIDELINES
Copyright 2002 by College Entrance Examination Board. All rights reserved.
Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
7
Question 6
The Maclaurin series for the function B is given by
"
!
$ " &
! "
n n
n
xx x x x f x x
n n
@
L L
on its interval of convergence.(a) Find the interval of convergence of the Maclaurin series for B. Justify your answer.
(b) Find the first four terms and the general term for the Maclaurin series for ( ).B N=
(c) Use the Maclaurin series you found in part (b) to find the value of 1 .3
B=
(a)
2( 1)2lim lim 2 2( 2)2
1
n
nn n
xnn x xnx
n
l l@ @
N for1 1
2 2x
At
N , the series is
n
n
@
which diverges since
this is the harmonic series.
At
N , the series is
n
nn
@
which
converges by the Alternating Series Test.
Hence, the interval of convergence is
1 1
2 2x b
.
1 : sets up ratio
1 : computes limit of ratio
1 : identifies interior of interval
of convergence
2 : analysis/conclusion at endpoints5
1 : right endpoint
1 : left endpoint
1 if endpoints n 1
ot2
1 if multiple intervals
N
o
(b) ! " & $ nB N N N N Na K K2
1 : first 4 terms
1 : general term
(c) The series in (b) is a geometric series.
" & ! ! ! !
" & $
! ' % !
$
#
!
n
n
Ba
K K
K K
OR
2( )
1 2f x
xa
for
1 1
2 2x . Therefore,
$! #
!
Ba
2
11 : substitutes into infinite
3series from (b) or expresses series
from (b) in closed form
1 : answer for student's series
N