-
AP Calculus BC2004 Free-Response Questions
The College Board is a not-for-profit membership association
whose mission is to connect students to college success and
opportunity. Founded in 1900, the association is composed of more
than 4,500 schools, colleges, universities, and other educational
organizations. Each year, the College Board serves over three
million students and their parents, 23,000 high schools, and 3,500
colleges through major programs and services in college admissions,
guidance, assessment, financial aid, enrollment, and teaching and
learning. Among its best-known programs are the SAT, the
PSAT/NMSQT, and the Advanced Placement Program (AP). The College
Board is committed to the principles of excellence and equity, and
that commitment is embodied in all of its programs, services,
activities, and concerns.
For further information, visit www.collegeboard.com
Copyright 2004 College Entrance Examination Board. All rights
reserved. College Board, Advanced Placement Program, AP, AP
Central,
AP Vertical Teams, APCD, Pacesetter, Pre-AP, SAT, Student Search
Service, and the acorn logo are registered trademarks of the
College Entrance Examination Board. PSAT/NMSQT is a registered
trademark jointly owned by the
College Entrance Examination Board and the National Merit
Scholarship Corporation. Educational Testing Service and ETS are
registered trademarks of Educational Testing Service.
Other products and services may be trademarks of their
respective owners.
For the College Boards online home for AP professionals, visit
AP Central at apcentral.collegeboard.com.
The materials included in these files are intended for
noncommercial use by AP teachers for course and exam preparation;
permission for any other use
must be sought from the Advanced Placement Program. Teachers may
reproduce them, in whole or in part, in limited quantities, for
face-to-face
teaching purposes but may not mass distribute the materials,
electronically or otherwise. This permission does not apply to
any
third-party copyrights contained herein. These materials and any
copies made of them may not be resold, and the copyright
notices
must be retained as they appear here.
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2004 AP CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
GO ON TO THE NEXT PAGE.
2
CALCULUS BC SECTION II, Part A
Time45 minutes Number of problems3
A graphing calculator is required for some problems or parts of
problems.
1. Traffic flow is defined as the rate at which cars pass
through an intersection, measured in cars per minute. The
traffic flow at a particular intersection is modeled by the
function F defined by
( ) ( )82 4sin 2tF t = + for 0 30,t
where ( )F t is measured in cars per minute and t is measured in
minutes.
(a) To the nearest whole number, how many cars pass through the
intersection over the 30-minute period?
(b) Is the traffic flow increasing or decreasing at 7 ?t = Give
a reason for your answer.
(c) What is the average value of the traffic flow over the time
interval 10 15 ?t Indicate units of measure.
(d) What is the average rate of change of the traffic flow over
the time interval 10 15 ?t Indicate units of measure.
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2004 AP CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
GO ON TO THE NEXT PAGE.
3
2. Let f and g be the functions given by ( ) ( )2 1f x x x= -
and ( ) ( )3 1g x x x= - for 0 1.x The graphs of f and g are shown
in the figure above.
(a) Find the area of the shaded region enclosed by the graphs of
f and g.
(b) Find the volume of the solid generated when the shaded
region enclosed by the graphs of f and g is revolved about the
horizontal line 2.y =
(c) Let h be the function given by ( ) ( )1h x kx x= - for 0 1.x
For each 0,k > the region (not shown) enclosed by the graphs of
h and g is the base of a solid with square cross sections
perpendicular to the x-axis. There is a value of k for which the
volume of this solid is equal to 15. Write, but do not solve, an
equation involving an integral expression that could be used to
find the value of k.
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2004 AP CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
4
3. An object moving along a curve in the xy-plane has position (
) ( )( ),x t y t at time t 0 with ( )23 cos .dx tdt = +
The derivative dydt
is not explicitly given. At time t 2,= the object is at position
( )1, 8 .
(a) Find the x-coordinate of the position of the object at time
4.t =
(b) At time 2,t = the value of dydt
is 7.- Write an equation for the line tangent to the curve at
the point
( ) ( )( )2 , 2 .x y
(c) Find the speed of the object at time 2.t =
(d) For 3,t the line tangent to the curve at ( ) ( )( ),x t y t
has a slope of 2 1.t + Find the acceleration vector of the object
at time 4.t =
END OF PART A OF SECTION II
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2004 AP CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
GO ON TO THE NEXT PAGE.
5
CALCULUS BC SECTION II, Part B
Time45 minutes Number of problems3
No calculator is allowed for these problems.
4. Consider the curve given by 2 24 7 3 .x y xy+ = +
(a) Show that 3 2
.8 3
dy y xdx y x
-
=
-
(b) Show that there is a point P with x-coordinate 3 at which
the line tangent to the curve at P is horizontal. Find the
y-coordinate of P.
(c) Find the value of 2
2d y
dx at the point P found in part (b). Does the curve have a local
maximum, a local
minimum, or neither at the point P ? Justify your answer.
5. A population is modeled by a function P that satisfies the
logistic differential equation
( )1 .5 12dP P Pdt
= -
(a) If ( )0 3,P = what is ( )lim ?t
P t
If ( )0 20,P = what is ( )lim ?t
P t
(b) If ( )0 3,P = for what value of P is the population growing
the fastest?
(c) A different population is modeled by a function Y that
satisfies the separable differential equation
( )1 .5 12dY Y tdt
= -
Find ( )Y t if ( )0 3.Y =
(d) For the function Y found in part (c), what is ( )lim ?t
Y t
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2004 AP CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
6
6. Let f be the function given by ( ) ( )sin 5 ,4f x xp
= + and let ( )P x be the third-degree Taylor polynomial
for f about 0.x =
(a) Find ( ).P x
(b) Find the coefficient of 22x in the Taylor series for f about
0.x =
(c) Use the Lagrange error bound to show that ( ) ( )1 1 1 .10
10 100f P- <
(d) Let G be the function given by ( ) ( )0
.x
G x f t dt= Write the third-degree Taylor polynomial for G about
0.x =
END OF EXAMINATION
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AP Calculus BC2004 Scoring Guidelines
The College Board is a not-for-profit membership association
whose mission is to connect students to college success and
opportunity. Founded in 1900, the association is composed of more
than 4,500 schools, colleges, universities, and other educational
organizations. Each year, the College Board serves over three
million students and their parents, 23,000 high schools, and 3,500
colleges through major programs and services in college admissions,
guidance, assessment, financial aid, enrollment, and teaching and
learning. Among its best-known programs are the SAT, the
PSAT/NMSQT, and the Advanced Placement Program (AP). The College
Board is committed to the principles of excellence and equity, and
that commitment is embodied in all of its programs, services,
activities, and concerns.
For further information, visit www.collegeboard.com
Copyright 2004 College Entrance Examination Board. All rights
reserved. College Board, Advanced Placement Program, AP, AP
Central,
AP Vertical Teams, APCD, Pacesetter, Pre-AP, SAT, Student Search
Service, and the acorn logo are registered trademarks of the
College Entrance Examination Board. PSAT/NMSQT is a registered
trademark of the
College Entrance Examination Board and National Merit
Scholarship Corporation. Educational Testing Service and ETS are
registered trademarks of Educational Testing Service.
Other products and services may be trademarks of their
respective owners.
For the College Boards online home for AP professionals, visit
AP Central at apcentral.collegeboard.com.
The materials included in these files are intended for
noncommercial use by AP teachers for course and exam preparation;
permission for any other use
must be sought from the Advanced Placement Program. Teachers may
reproduce them, in whole or in part, in limited quantities, for
face-to-face
teaching purposes but may not mass distribute the materials,
electronically or otherwise. This permission does not apply to
any
third-party copyrights contained herein. These materials and any
copies made of them may not be resold, and the copyright
notices
must be retained as they appear here.
www.oneplusone.cn
-
AP CALCULUS BC 2004 SCORING GUIDELINES
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
2
Question 1
Traffic flow is defined as the rate at which cars pass through
an intersection, measured in cars per minute. The traffic flow at a
particular intersection is modeled by the function F defined by
( ) ( )82 4sin 2tF t = + for 0 30,t where ( )F t is measured in
cars per minute and t is measured in minutes.
(a) To the nearest whole number, how many cars pass through the
intersection over the 30-minute period?
(b) Is the traffic flow increasing or decreasing at 7 ?t = Give
a reason for your answer.
(c) What is the average value of the traffic flow over the time
interval 10 15 ?t Indicate units of measure.
(d) What is the average rate of change of the traffic flow over
the time interval 10 15 ?t Indicate units of measure.
(a) ( )30
02474F t dt = cars 3 :
1 : limits1 : integrand1 : answer
(b) ( )7 1.872 or 1.873F = Since ( )7 0,F < the traffic flow
is decreasing
at 7.t =
1 : answer with reason
(c) ( )15
101 81.899 cars min5 F t dt = 3 :
1 : limits1 : integrand1 : answer
(d) ( ) ( )15 10 1.51715 10F F
=
or 21.518 cars min
1 : answer
Units of cars min in (c) and 2cars min in (d)
1 : units in (c) and (d)
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AP CALCULUS BC 2004 SCORING GUIDELINES
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
3
Question 2
Let f and g be the functions given by ( ) ( )2 1f x x x= and
( ) ( )3 1g x x x= for 0 1.x The graphs of f and g are shown in
the figure above.
(a) Find the area of the shaded region enclosed by the graphs of
f and g.
(b) Find the volume of the solid generated when the shaded
region enclosed by the graphs of f and g is revolved about the
horizontal line 2.y =
(c) Let h be the function given by ( ) ( )1h x k x x= for 0 1.x
For each 0,k > the region (not shown) enclosed by the graphs of
h and g is the
base of a solid with square cross sections perpendicular to the
x-axis. There is a value of k for which the volume of this solid is
equal to 15. Write, but do not solve, an equation involving an
integral expression that could be used to find the value of k.
(a) Area ( ) ( )( )
( ) ( )( )
1
01
02 1 3 1 1.133
f x g x dx
x x x x dx
=
= =
2 : { 1 : integral1 : answer
(b) Volume ( )( ) ( )( )( )1 2 20 2 2g x f x dx= ( )( ) ( )( )(
)1 2 20 2 3 1 2 2 1
16.179
x x x x dx=
=
4 :
( ) ( )( )2 2
1 : limits and constant 2 : integrand 1 each error Note: 0 2 if
integral not of form
1 : answer
b
ac R x r x dx
(c) Volume ( ) ( )( )1 20h x g x dx=
( ) ( )( )1 20
1 3 1 15k x x x x dx =
3 : { 2 : integrand1 : answer
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AP CALCULUS BC 2004 SCORING GUIDELINES
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
4
Question 3
An object moving along a curve in the xy-plane has position ( )
( )( ),x t y t at time 0t with
( )23 cos .dx tdt = + The derivative dydt is not explicitly
given. At time 2,t = the object is at position
( )1, 8 . (a) Find the x-coordinate of the position of the
object at time 4.t =
(b) At time 2,t = the value of dydt is 7. Write an equation for
the line tangent to the curve at the point
( ) ( )( )2 , 2 .x y (c) Find the speed of the object at time
2.t =
(d) For 3,t the line tangent to the curve at ( ) ( )( ),x t y t
has a slope of 2 1.t + Find the acceleration vector of the object
at time 4.t =
(a) ( ) ( ) ( )( )( )( )
4 22
4 22
4 2 3 cos
1 3 cos 7.132 or 7.133
x x t dt
t dt
= + +
= + + =
3 : ( )( )4 22 1 : 3 cos
1 : handles initial condition 1 : answer
t dt +
(b)
22
7 2.9833 cos 4tt
dydy dtdx dx
dt=
=
= = = +
( )8 2.983 1y x =
2 : 2 1 : finds
1 : equationt
dydx =
(c) The speed of the object at time 2t = is
( )( ) ( )( )2 22 2 7.382 or 7.383.x y + =
1 : answer
(d) ( )4 2.303x =
( ) ( ) ( )( )22 1 3 cosdy dy dxy t t tdt dx dt = = = + + ( )4
24.813 or 24.814y =
The acceleration vector at 4t = is 2.303, 24.813 or 2.303,
24.814 .
3 :
( )1 : 4
1 :
1 : answer
xdydt
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AP CALCULUS BC 2004 SCORING GUIDELINES
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
5
Question 4
Consider the curve given by 2 24 7 3 .x y x y+ = +
(a) Show that 3 2 .8 3dy y xdx y x
=
(b) Show that there is a point P with x-coordinate 3 at which
the line tangent to the curve at P is horizontal. Find the
y-coordinate of P.
(c) Find the value of 2
2d ydx
at the point P found in part (b). Does the curve have a local
maximum, a
local minimum, or neither at the point P ? Justify your
answer.
(a) ( )
2 8 3 38 3 3 2
3 28 3
x y y y x yy x y y x
y xy y x
+ = + =
=
2 : 1 : implicit differentiation
1 : solves for y
(b) 3 2 0; 3 2 08 3y x y xy x
= =
When 3,x = 3 6
2yy
==
2 23 4 2 25+ = and 7 3 3 2 25+ =
Therefore, ( )3, 2P = is on the curve and the slope is 0 at this
point.
3 : ( )( )
1 : 0
1 : shows slope is 0 at 3, 21 : shows 3, 2 lies on curve
dydx
=
(c) ( )( ) ( )( )( )
2
2 28 3 3 2 3 2 8 3
8 3y x y y x yd y
dx y x =
At ( )3, 2 ,P = ( )( )( )
2
2 216 9 2 2 .716 9
d ydx
= =
Since 0y = and 0y < at P, the curve has a local maximum at
P.
4 : ( )
2
2
2
2
2 :
1 : value of at 3, 2
1 : conclusion with justification
d ydx
d ydx
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AP CALCULUS BC 2004 SCORING GUIDELINES
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
6
Question 5
A population is modeled by a function P that satisfies the
logistic differential equation
( )1 .5 12dP P Pdt = (a) If ( )0 3,P = what is ( )lim ?
tP t
If ( )0 20,P = what is ( )lim ?t
P t
(b) If ( )0 3,P = for what value of P is the population growing
the fastest? (c) A different population is modeled by a function Y
that satisfies the separable differential equation
( )1 .5 12dY Y tdt = Find ( )Y t if ( )0 3.Y = (d) For the
function Y found in part (c), what is ( )lim ?
tY t
(a) For this logistic differential equation, the carrying
capacity is 12. If ( )0 3,P = ( )lim 12.
tP t
=
If ( )0 20,P = ( )lim 12.t
P t
=
2 : 1 : answer1 : answer
(b) The population is growing the fastest when P is half the
carrying capacity. Therefore, P is growing the fastest when 6.P
=
1 : answer
(c) ( ) ( )1 1 115 12 5 60t tdY dt dtY = = 2
ln 5 120t tY C= +
( )2
5 120t t
Y t Ke
= 3K =
( )2
5 1203t t
Y t e
=
5 :
1 : separates variables 1 : antiderivatives 1 : constant of
integration 1 : uses initial condition 1 : solves for 0 1 if is not
exponential
YY
Note: max 2 5 [1-1-0-0-0] if no constant of integration Note: 0
5 if no separation of variables
(d) ( )lim 0t
Y t
=
1 : answer 0 1 if Y is not exponential
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AP CALCULUS BC 2004 SCORING GUIDELINES
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
7
Question 6
Let f be the function given by ( ) ( )sin 5 ,4f x x = + and let
( )P x be the third-degree Taylor polynomial for f about 0.x =
(a) Find ( ).P x
(b) Find the coefficient of 22x in the Taylor series for f about
0.x =
(c) Use the Lagrange error bound to show that ( ) ( )1 1 1 .10
10 100f P < (d) Let G be the function given by ( ) ( )
0.
xG x f t dt= Write the third-degree Taylor polynomial
for G about 0.x =
(a) ( ) ( ) 20 sin 4 2f = = ( ) ( ) 5 20 5cos 4 2f = = ( ) ( )
25 20 25sin 4 2f = = ( ) ( ) 125 20 125cos 4 2f = =
( ) ( ) ( )2 32 5 2 25 2 125 2
2 2 2 2! 2 3!P x x x x= +
4 : ( )P x
1 each error or missing term
deduct only once for ( )4sin evaluation error
deduct only once for ( )4cos evaluation error
1 max for all extra terms, ,+ misuse of equality
(b) ( )225 2
2 22!
2 : 1 : magnitude
1 : sign
(c) ( ) ( ) ( ) ( ) ( )( )( )1
10
44
0
4
1 1 1 1max10 10 4! 10
625 1 1 14! 10 384 100
cf P f c
= 1, as shown above.
(a) Find x dxn0
1
z in terms of n.
(b) Let T be the triangular region bounded by , the x-axis, and
the line x = 1. Show that the area of T is 12n
.
(c) Let S be the region bounded by the graph of y xn= , the line
, and the x-axis. Express the area of S in terms of n and determine
the value of n that maximizes the area of S.
END OF EXAMINATION
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AP Calculus BC2004 Scoring Guidelines
Form B
The College Board is a not-for-profit membership association
whose mission is to connect students to college success and
opportunity. Founded in 1900, the association is composed of more
than 4,500 schools, colleges, universities, and other educational
organizations. Each year, the College Board serves over three
million students and their parents, 23,000 high schools, and 3,500
colleges through major programs and services in college admissions,
guidance, assessment, financial aid, enrollment, and teaching and
learning. Among its best-known programs are the SAT, the
PSAT/NMSQT, and the Advanced Placement Program (AP). The College
Board is committed to the principles of excellence and equity, and
that commitment is embodied in all of its programs, services,
activities, and concerns.
For further information, visit www.collegeboard.com
Copyright 2004 College Entrance Examination Board. All rights
reserved. College Board, Advanced Placement Program, AP, AP
Central,
AP Vertical Teams, APCD, Pacesetter, Pre-AP, SAT, Student Search
Service, and the acorn logo are registered trademarks of the
College Entrance Examination Board. PSAT/NMSQT is a registered
trademark of the
College Entrance Examination Board and National Merit
Scholarship Corporation. Educational Testing Service and ETS are
registered trademarks of Educational Testing Service.
Other products and services may be trademarks of their
respective owners.
For the College Boards online home for AP professionals, visit
AP Central at apcentral.collegeboard.com.
The materials included in these files are intended for
noncommercial use by AP teachers for course and exam preparation;
permission for any other use
must be sought from the Advanced Placement Program. Teachers may
reproduce them, in whole or in part, in limited quantities, for
face-to-face
teaching purposes but may not mass distribute the materials,
electronically or otherwise. This permission does not apply to
any
third-party copyrights contained herein. These materials and any
copies made of them may not be resold, and the copyright
notices
must be retained as they appear here.
www.oneplusone.cn
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AP CALCULUS BC 2004 SCORING GUIDELINES (Form B)
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
2
Question 1
A particle moving along a curve in the plane has position ( ) (
)( ),x t y t at time t, where
4 9dx tdt = + and 2 5t tdy e edt
= +
for all real values of t. At time 0,t = the particle is at the
point (4, 1). (a) Find the speed of the particle and its
acceleration vector at time 0.t = (b) Find an equation of the line
tangent to the path of the particle at time 0.t = (c) Find the
total distance traveled by the particle over the time interval 0
3.t (d) Find the x-coordinate of the position of the particle at
time 3.t = (a) At time 0:t = Speed 2 2 2 2(0) (0) 3 7 58x y = + = +
= Acceleration vector ( ) ( )0 , 0 0, 3x y = =
2 : 1 : speed1 : acceleration vector
(b) ( )( )0 7
30ydy
dx x
= =
Tangent line is ( )7 4 13y x= +
2 : 1 : slope1 : tangent line
(c) Distance ( ) ( )3 2 240
9 2 5
45.226 or 45.227
t tt e e dt= + + +
=
3 :
2 : distance integral 1 each integrand error
1 error in limits 1 : answer
(d) ( )3 40
3 4 9
17.930 or 17.931
x t dt= + +
= 2 : 1 : integral1 : answer
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AP CALCULUS BC 2004 SCORING GUIDELINES (Form B)
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
3
Question 2
Let f be a function having derivatives of all orders for all
real numbers. The third-degree Taylor polynomial for f about 2x =
is given by ( ) ( ) ( )2 37 9 2 3 2 .T x x x= (a) Find ( )2f and (
)2 .f (b) Is there enough information given to determine whether f
has a critical point at 2 ?x = If not, explain why not. If so,
determine whether ( )2f is a relative maximum, a relative
minimum,
or neither, and justify your answer. (c) Use ( )T x to find an
approximation for ( )0 .f Is there enough information given to
determine
whether f has a critical point at 0 ?x = If not, explain why
not. If so, determine whether ( )0f is a relative maximum, a
relative minimum, or neither, and justify your answer.
(d) The fourth derivative of f satisfies the inequality ( ) ( )4
6f x for all x in the closed interval [ ]0, 2 . Use the Lagrange
error bound on the approximation to ( )0f found in part (c) to
explain why
( )0f is negative. (a) ( ) ( )2 2 7f T= =
( )2 92!f
= so ( )2 18f =
2 : ( )( )
1 : 2 71 : 2 18ff
= =
(b) Yes, since ( ) ( )2 2 0,f T = = f does have a critical point
at 2.x =
Since ( )2 18 0,f = < ( )2f is a relative maximum value.
2 : ( )
( )( )
1 : states 2 01 : declares 2 as a relative maximum because 2
0
ff
f
=
-
AP CALCULUS BC 2004 SCORING GUIDELINES (Form B)
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
4
Question 3
A test plane flies in a straight line with positive velocity ( )
,v t in miles per minute at time t minutes, where v is a
differentiable function of t. Selected values of ( )v t for 0 40t
are shown in the table above. (a) Use a midpoint Riemann sum with
four subintervals of equal length and values from the table to
approximate ( )40
0.v t dt Show the computations that lead to your answer. Using
correct units,
explain the meaning of ( )40
0v t dt in terms of the planes flight.
(b) Based on the values in the table, what is the smallest
number of instances at which the acceleration of the plane could
equal zero on the open interval 0 40?t< < Justify your
answer.
(c) The function f, defined by ( ) ( ) ( )76 cos 3sin ,10 40t tf
t = + + is used to model the velocity of the plane, in miles per
minute, for 0 40.t According to this model, what is the
acceleration of the plane at 23 ?t = Indicates units of
measure.
(d) According to the model f, given in part (c), what is the
average velocity of the plane, in miles per minute, over the time
interval 0 40?t
(a) Midpoint Riemann sum is
( ) ( ) ( ) ( )[ ][ ]
10 5 15 25 3510 9.2 7.0 2.4 4.3 229
v v v v + + += + + + =
The integral gives the total distance in miles that the plane
flies during the 40 minutes.
3 : ( ) ( ) ( ) ( )1 : 5 15 25 35
1 : answer 1 : meaning with units
v v v v+ + +
(b) By the Mean Value Theorem, ( ) 0v t = somewhere in the
interval ( )0, 15 and somewhere in the interval ( )25, 30 .
Therefore the acceleration will equal 0 for at least two values of
t.
2 : 1 : two instances1 : justification
(c) ( )23 0.407 or 0.408f = miles per minute2
1 : answer with units
(d) Average velocity ( )40
01405.916 miles per minute
f t dt=
=
3 : 1 : limits1 : integrand1 : answer
t (min) 0 5 10 15 20 25 30 35 40 ( )v t (mpm) 7.0 9.2 9.5 7.0
4.5 2.4 2.4 4.3 7.3
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AP CALCULUS BC 2004 SCORING GUIDELINES (Form B)
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
5
Question 4
The figure above shows the graph of ,f the derivative of the
function f, on the closed interval 1 5.x The graph of f has
horizontal tangent lines at 1x = and 3.x = The function f is twice
differentiable with
( )2 6.f = (a) Find the x-coordinate of each of the points of
inflection of the graph
of f. Give a reason for your answer. (b) At what value of x does
f attain its absolute minimum value on the
closed interval 1 5 ?x At what value of x does f attain its
absolute maximum value on the closed interval 1 5 ?x Show the
analysis that leads to your answers.
(c) Let g be the function defined by ( ) ( ).g x x f x= Find an
equation for the line tangent to the graph of g at 2.x =
(a) 1x = and 3x = because the graph of f changes from
increasing to decreasing at 1,x = and changes from decreasing to
increasing at 3.x =
2 : 1 : 1, 3
1 : reasonx x= =
(b) The function f decreases from 1x = to 4,x = then increases
from 4x = to 5.x = Therefore, the absolute minimum value for f is
at 4.x =
The absolute maximum value must occur at 1x = or at 5.x =
( ) ( ) ( )5
15 1 0f f f t dt
=
-
AP CALCULUS BC 2004 SCORING GUIDELINES (Form B)
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
6
Question 5
Let g be the function given by ( ) 1 .g xx
=
(a) Find the average value of g on the closed interval [ ]1, 4
.
(b) Let S be the solid generated when the region bounded by the
graph of ( ) ,y g x= the vertical lines 1x = and 4,x = and the
x-axis is revolved about the x-axis. Find the volume of S.
(c) For the solid S, given in part (b), find the average value
of the areas of the cross sections perpendicular to the x-axis.
(d) The average value of a function f on the unbounded interval
[ , )a is defined to be
( )lim .b
ab
f x dx
b a
Show that the improper integral ( )
4g x dx
is divergent, but the average value of g on the interval [4, )
is finite.
(a) 44
1 1
1 1 1 4 2 223 3 3 3 3dx xx= = =
2 : 1 : integral1 : antidifferentiation
and evaluation
(b) Volume 4 4
11
1 ln ln 4dx xx = = =
2 : 1 : integral1 : antidifferentiation
and evaluation
(c) The cross section at x has area ( )21 xx = Average value
4
1
1 1 ln 43 3dxx = =
1 : answer
(d) ( ) ( )4 4
1lim lim 2 4b
b bg x dx dx b
x
= = =
This limit is not finite, so the integral is divergent.
( )
4
4
1 1 2 44 4 4
bbg x dx bdxb b bx
= =
2 4lim 04bbb
=
4 :
( )
( )
4
4
1 : 2 4
1 : indicates integral diverges1 2 4 1 : 4 4
1 : finite limit as
b
b
g x dx b
bg x dxb bb
= =
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AP CALCULUS BC 2004 SCORING GUIDELINES (Form B)
Copyright 2004 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
7
Question 6
Let be the line tangent to the graph of ny x= at the point (1,
1), where 1,n > as shown above.
(a) Find 1
0nx dx in terms of n.
(b) Let T be the triangular region bounded by , the x-axis, and
the
line 1.x = Show that the area of T is 1 .2n
(c) Let S be the region bounded by the graph of ,ny x= the line
, and the x-axis. Express the area of S in terms of n and determine
the value of n that maximizes the area of S.
(a) 111
00
11 1
nn xx dx n n
+= =+ +
2 : 1 : antiderivative of 1 : answer
nx
(b) Let b be the length of the base of triangle T.
1b is the slope of line , which is n
( ) ( )1 1Area 12 2T b n= =
3 :
1 : slope of line is 1 1 : base of is
1 1 : shows area is 2
n
T n
n
(c)
( ) ( )1
0Area Area
1 11 2
nS x dx T
n n
=
= +
( ) 2 21 1Area 0
( 1) 2d Sdn n n
= + =+
( )222 1n n= + ( )2 1n n= +
1 1 22 1
n = = +
4 :
1 : area of in terms of 1 : derivative1 : sets derivative equal
to 0
1 : solves for
S n
n
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AP Calculus BC 2005 Free-Response Questions
The College Board: Connecting Students to College Success
The College Board is a not-for-profit membership association
whose mission is to connect students to college success and
opportunity. Founded in 1900, the association is composed of more
than 4,700 schools, colleges, universities, and other educational
organizations. Each year, the College Board serves over three and a
half million students and their parents, 23,000 high schools, and
3,500 colleges through major programs and services in college
admissions, guidance, assessment, financial aid, enrollment, and
teaching and learning. Among its best-known programs are the SAT,
the PSAT/NMSQT, and the Advanced Placement Program (AP). The
College Board is committed to the principles of excellence and
equity, and that commitment is embodied in all of its programs,
services, activities, and concerns.
Copyright 2005 by College Board. All rights reserved. College
Board, AP Central, APCD, Advanced Placement Program, AP, AP
Vertical Teams, Pre-AP, SAT, and the acorn logo are registered
trademarks of the College Entrance Examination Board. Admitted
Class Evaluation Service, CollegeEd, Connect to college success,
MyRoad, SAT Professional Development, SAT Readiness Program, and
Setting the Cornerstones are trademarks owned by the College
Entrance Examination Board. PSAT/NMSQT is a registered trademark of
the College Entrance Examination Board and National Merit
Scholarship Corporation. Other products and services may be
trademarks of their respective owners. Permission to use
copyrighted College Board materials may be requested online at:
http://www.collegeboard.com/inquiry/cbpermit.html. Visit the
College Board on the Web: www.collegeboard.com. AP Central is the
official online home for the AP Program and Pre-AP:
apcentral.collegeboard.com.
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2005 AP CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright 2005 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
GO ON TO THE NEXT PAGE. 2
CALCULUS BC SECTION II, Part A
Time45 minutes Number of problems3
A graphing calculator is required for some problems or parts of
problems.
1. Let f and g be the functions given by ( ) ( )1 sin4
f x xp= + and ( ) 4 .xg x -= Let R be the shaded region in the
first quadrant enclosed by the y-axis and the graphs of f and g,
and let S be the shaded region in the first
quadrant enclosed by the graphs of f and g, as shown in the
figure above.
(a) Find the area of R.
(b) Find the area of S.
(c) Find the volume of the solid generated when S is revolved
about the horizontal line y 1.= -
WRITE ALL WORK IN THE TEST BOOKLET.
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2005 AP CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright 2005 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
GO ON TO THE NEXT PAGE. 3
2. The curve above is drawn in the xy-plane and is described by
the equation in polar coordinates ( )sin 2r q q= + for 0 ,q p where
r is measured in meters and q is measured in radians. The
derivative of r with respect
to q is given by ( )1 2cos 2 .drd qq = +
(a) Find the area bounded by the curve and the x-axis.
(b) Find the angle q that corresponds to the point on the curve
with x-coordinate 2.-
(c) For 2 ,3 3p p
q< < drdq is negative. What does this fact say about r ?
What does this fact say about the
curve?
(d) Find the value of q in the interval 02p
q that corresponds to the point on the curve in the first
quadrant with greatest distance from the origin. Justify your
answer.
WRITE ALL WORK IN THE TEST BOOKLET.
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2005 AP CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright 2005 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
4
Distance x (cm)
0 1 5 6 8
Temperature ( )T x ( )C 100 93 70 62 55
3. A metal wire of length 8 centimeters (cm) is heated at one
end. The table above gives selected values of the
temperature ( ),T x in degrees Celsius ( )C , of the wire x cm
from the heated end. The function T is decreasing and twice
differentiable.
(a) Estimate ( )7 .T Show the work that leads to your answer.
Indicate units of measure.
(b) Write an integral expression in terms of ( )T x for the
average temperature of the wire. Estimate the average temperature
of the wire using a trapezoidal sum with the four subintervals
indicated by the data in the table. Indicate units of measure.
(c) Find ( )8
0,T x dx and indicate units of measure. Explain the meaning of (
)
8
0T x dx in terms of the
temperature of the wire.
(d) Are the data in the table consistent with the assertion that
( ) 0T x > for every x in the interval 0 8 ?x< < Explain
your answer.
WRITE ALL WORK IN THE TEST BOOKLET.
END OF PART A OF SECTION II
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2005 AP CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright 2005 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
GO ON TO THE NEXT PAGE. 5
CALCULUS BC SECTION II, Part B
Time45 minutes Number of problems3
No calculator is allowed for these problems.
4. Consider the differential equation 2 .dy x ydx = -
(a) On the axes provided, sketch a slope field for the given
differential equation at the twelve points indicated, and sketch
the solution curve that passes through the point ( )0, 1 .
(Note: Use the axes provided in the pink test booklet.)
(b) The solution curve that passes through the point ( )0, 1 has
a local minimum at ( )3ln .2x = What is the y-coordinate of this
local minimum?
(c) Let ( )y f x= be the particular solution to the given
differential equation with the initial condition ( )0 1.f = Use
Eulers method, starting at 0x = with two steps of equal size, to
approximate ( )0.4 .f -
Show the work that leads to your answer.
(d) Find 2
2d ydx
in terms of x and y. Determine whether the approximation found
in part (c) is less than or
greater than ( )0.4 .f - Explain your reasoning.
WRITE ALL WORK IN THE TEST BOOKLET.
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2005 AP CALCULUS BC FREE-RESPONSE QUESTIONS
Copyright 2005 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
6
5. A car is traveling on a straight road. For 0 24t seconds, the
cars velocity ( ),v t in meters per second, is modeled by the
piecewise-linear function defined by the graph above.
(a) Find ( )24
0.v t dt Using correct units, explain the meaning of ( )
24
0.v t dt
(b) For each of ( )4v and ( )20 ,v find the value or explain why
it does not exist. Indicate units of measure.
(c) Let ( )a t be the cars acceleration at time t, in meters per
second per second. For 0 24,t< < write a piecewise-defined
function for ( ).a t
(d) Find the average rate of change of v over the interval 8
20.t Does the Mean Value Theorem guarantee a value of c, for 8
20,c< < such that ( )v c is equal to this average rate of
change? Why or why not?
6. Let f be a function with derivatives of all orders and for
which ( )2 7.f = When n is odd, the nth derivative
of f at 2x = is 0. When n is even and 2,n the nth derivative of
f at 2x = is given by ( ) ( ) ( )1 !2 .3
nn
nf
-=
(a) Write the sixth-degree Taylor polynomial for f about 2.x
=
(b) In the Taylor series for f about 2,x = what is the
coefficient of ( )22 nx - for 1 ?n (c) Find the interval of
convergence of the Taylor series for f about 2.x = Show the work
that leads to your
answer.
WRITE ALL WORK IN THE TEST BOOKLET.
END OF EXAM
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AP Calculus BC 2005 Scoring Guidelines
The College Board: Connecting Students to College Success
The College Board is a not-for-profit membership association
whose mission is to connect students to college success and
opportunity. Founded in 1900, the association is composed of more
than 4,700 schools, colleges, universities, and other educational
organizations. Each year, the College Board serves over three and a
half million students and their parents, 23,000 high schools, and
3,500 colleges through major programs and services in college
admissions, guidance, assessment, financial aid, enrollment, and
teaching and learning. Among its best-known programs are the SAT,
the PSAT/NMSQT, and the Advanced Placement Program (AP). The
College Board is committed to the principles of excellence and
equity, and that commitment is embodied in all of its programs,
services, activities, and concerns.
Copyright 2005 by College Board. All rights reserved. College
Board, AP Central, APCD, Advanced Placement Program, AP, AP
Vertical Teams, Pre-AP, SAT, and the acorn logo are registered
trademarks of the College Entrance Examination Board. Admitted
Class Evaluation Service, CollegeEd, Connect to college success,
MyRoad, SAT Professional Development, SAT Readiness Program, and
Setting the Cornerstones are trademarks owned by the College
Entrance Examination Board. PSAT/NMSQT is a registered trademark of
the College Entrance Examination Board and National Merit
Scholarship Corporation. Other products and services may be
trademarks of their respective owners. Permission to use
copyrighted College Board materials may be requested online at:
http://www.collegeboard.com/inquiry/cbpermit.html. Visit the
College Board on the Web: www.collegeboard.com. AP Central is the
official online home for the AP Program and Pre-AP:
apcentral.collegeboard.com.
www.oneplusone.cn
-
AP CALCULUS BC 2005 SCORING GUIDELINES
Copyright 2005 by College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
2
Question 1
Let f and g be the functions given by ( ) ( )1 sin4f x x= + and
( ) 4 .xg x = Let
R be the shaded region in the first quadrant enclosed by the
y-axis and the graphs of f and g, and let S be the shaded region in
the first quadrant enclosed by the graphs of f and g, as shown in
the figure above. (a) Find the area of R. (b) Find the area of S.
(c) Find the volume of the solid generated when S is revolved about
the horizontal
line 1.y =
( ) ( )f x g x= when ( )1 sin 44xx + = .
f and g intersect when 0.178218x = and when 1.x = Let 0.178218.a
=
(a) ( ) ( )( )0
0.064a
g x f x dx = or 0.065
3 : 1 : limits1 : integrand1 : answer
(b) ( ) ( )( )1
0.410a
f x g x dx =
3 : 1 : limits1 : integrand1 : answer
(c) ( )( ) ( )( )( )1 2 21 1 4.558a f x g x dx + + = or
4.559
3 : { 2 : integrand1 : limits, constant, and answer
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AP CALCULUS BC 2005 SCORING GUIDELINES
Copyright 2005 by College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
3
Question 2
The curve above is drawn in the xy-plane and is described by the
equation in polar coordinates ( )sin 2r = + for 0 , where r is
measured in meters and is measured in radians. The derivative of r
with respect to is
given by ( )1 2cos 2 .drd = +
(a) Find the area bounded by the curve and the x-axis. (b) Find
the angle that corresponds to the point on the curve with
x-coordinate 2.
(c) For 2 ,3 3 < < drd is negative. What does this fact
say about r ? What does this fact say about the curve?
(d) Find the value of in the interval 0 2 that corresponds to
the point on the curve in the first quadrant
with greatest distance from the origin. Justify your answer.
(a) Area
( )( )
20
20
121 sin 2 4.3822
r d
d
=
= + =
3 : 1 : limits and constant
1 : integrand 1 : answer
(b) ( ) ( )( ) ( )2 cos sin 2 cosr = = + 2.786 =
2 : { 1 : equation1 : answer
(c) Since 0drd < for 2 ,3 3
< < r is decreasing on this
interval. This means the curve is getting closer to the
origin.
2 : { 1 : information about 1 : information about the curver
(d) The only value in 0, 2
where 0drd = is .3
=
r 0 0
3 1.913
2 1.571
The greatest distance occurs when .3 =
2 : 1 : or 1.04731 : answer with justification
=
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AP CALCULUS BC 2005 SCORING GUIDELINES
Copyright 2005 by College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
4
Question 3
Distance x (cm) 0 1 5
6 8
Temperature ( )T x ( )C 100 93 70 62 55
A metal wire of length 8 centimeters (cm) is heated at one end.
The table above gives selected values of the temperature ( ) ,T x
in degrees Celsius ( )C , of the wire x cm from the heated end. The
function T is decreasing and twice
differentiable. (a) Estimate ( )7 .T Show the work that leads to
your answer. Indicate units of measure. (b) Write an integral
expression in terms of ( )T x for the average temperature of the
wire. Estimate the average temperature
of the wire using a trapezoidal sum with the four subintervals
indicated by the data in the table. Indicate units of measure.
(c) Find ( )8
0,T x dx and indicate units of measure. Explain the meaning of (
)
8
0T x dx in terms of the temperature of the
wire. (d) Are the data in the table consistent with the
assertion that ( ) 0T x > for every x in the interval 0 8 ?x<
< Explain
your answer.
(a) ( ) ( )8 6 55 62 7 C cm8 6 2 2T T = =
1 : answer
(b) ( )8
018 T x dx
Trapezoidal approximation for ( )8
0:T x dx
100 93 93 70 70 62 62 551 4 1 22 2 2 2A+ + + += + + +
Average temperature 1 75.6875 C8 A =
3 : ( )
8
01 1 : 8
1 : trapezoidal sum 1 : answer
T x dx
(c) ( ) ( ) ( )8
08 0 55 100 45 CT x dx T T = = =
The temperature drops 45 C from the heated end of the wire to
the other end of the wire.
2 : { 1 : value1 : meaning
(d) Average rate of change of temperature on [ ]1, 5 is 70 93
5.75.5 1 =
Average rate of change of temperature on [ ]5, 6 is 62 70 8.6 5
=
No. By the MVT, ( )1 5.75T c = for some 1c in the interval ( )1,
5 and ( )2 8T c = for some 2c in the interval ( )5, 6 . It follows
that T must decrease somewhere in the interval ( )1 2, .c c
Therefore T is not positive for every x in [ ]0, 8 .
2 : { 1 : two slopes of secant lines1 : answer with
explanation
Units of C cm in (a), and C in (b) and (c) 1 : units in (a),
(b), and (c)
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AP CALCULUS BC 2005 SCORING GUIDELINES
Copyright 2005 by College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
5
Question 4
Consider the differential equation 2 .dy x ydx =
(a) On the axes provided, sketch a slope field for the given
differential equation at the twelve points indicated, and sketch
the solution curve that passes through the point ( )0, 1 . (Note:
Use the axes provided in the pink test booklet.)
(b) The solution curve that passes through the point ( )0, 1 has
a local minimum at ( )3ln .2x = What is the y-coordinate of this
local minimum?
(c) Let ( )y f x= be the particular solution to the given
differential equation with the initial condition ( )0 1.f = Use
Eulers method, starting at 0x = with two steps of equal size, to
approximate ( )0.4 .f
Show the work that leads to your answer.
(d) Find 2
2d ydx
in terms of x and y. Determine whether the approximation found
in part (c) is less than or
greater than ( )0.4 .f Explain your reasoning.
(a) 3 :
( )
1 : zero slopes 1 : nonzero slopes1 : curve through 0, 1
(b) 0dydx = when 2x y=
The y-coordinate is ( )32ln .2 2 : 1 : sets 0
1 : answer
dydx
=
(c) ( ) ( ) ( ) ( )( ) ( )
0.2 0 0 0.21 1 0.2 1.2
f f f + = + =
( ) ( ) ( )( )( )( )
0.4 0.2 0.2 0.21.2 1.6 0.2 1.52
f f f + + =
2 : ( )1 : Euler's method with two steps 1 : Euler approximation
to 0.4f
(d) 2
2 2 2 2d y dy x ydxdx
= = +
2
2d ydx
is positive in quadrant II because 0x < and 0.y >
( )1.52 0.4f< since all solution curves in quadrant II are
concave up.
2 :
2
2 1 :
1 : answer with reason
d ydx
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AP CALCULUS BC 2005 SCORING GUIDELINES
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apcentral.collegeboard.com (for AP professionals) and
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6
Question 5
A car is traveling on a straight road. For 0 24t seconds, the
cars velocity ( ) ,v t in meters per second, is modeled by the
piecewise-linear function defined by the graph above.
(a) Find ( )24
0.v t dt Using correct units, explain the meaning of ( )
24
0.v t dt
(b) For each of ( )4v and ( )20 ,v find the value or explain why
it does not exist. Indicate units of measure.
(c) Let ( )a t be the cars acceleration at time t, in meters per
second per second. For 0 24,t< < write a piecewise-defined
function for ( ).a t
(d) Find the average rate of change of v over the interval 8
20.t Does the Mean Value Theorem guarantee a value of c, for 8
20,c< < such that ( )v c is equal to this average rate of
change? Why or why not?
(a) ( ) ( )( ) ( )( ) ( )( )24
01 14 20 12 20 8 20 3602 2v t dt = + + =
The car travels 360 meters in these 24 seconds.
2 : { 1 : value1 : meaning with units
(b) ( )4v does not exist because ( ) ( ) ( ) ( )
4 4
4 4lim 5 0 lim .4 4t tv t v v t v
t t + = =
( ) 220 0 520 m sec16 24 2v = =
3 : ( )( )
1 : 4 does not exist, with explanation 1 : 20 1 : units
vv
(c)
( )
5 if 0 4 0 if 4 16
5 if 16 242
tta tt
<
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AP CALCULUS BC 2005 SCORING GUIDELINES
Copyright 2005 by College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
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7
Question 6
Let f be a function with derivatives of all orders and for which
( )2 7.f = When n is odd, the nth derivative
of f at 2x = is 0. When n is even and 2,n the nth derivative of
f at 2x = is given by ( ) ( ) ( )1 !2 .3
nn
nf =
(a) Write the sixth-degree Taylor polynomial for f about 2.x
=
(b) In the Taylor series for f about 2,x = what is the
coefficient of ( )22 nx for 1 ?n (c) Find the interval of
convergence of the Taylor series for f about 2.x = Show the work
that leads to your
answer.
(a) ( ) ( ) ( ) ( )2 4 66 2 4 61! 1 3! 1 5! 17 2 2 22! 4! 6!3 3
3
P x x x x= + + +
3 : ( )6
1 : polynomial about 2 2 : 1 each incorrect term 1 max for all
extra terms, , misuse of equality
xP x
=
+
(b) ( ) ( ) ( )2 22 1 ! 1 1
2 !3 3 2n nn
n n
=
1 : coefficient
(c) The Taylor series for f about 2x = is
( ) ( )221
17 2 .2 3
nn
nf x x
n=
= +
( ) ( ) ( )( )
( )
( ) ( )( )
2 12 1
22
222
2 2
1 1 22 1 3lim 1 1 22 322 3lim 2 92 1 3 3
nn
n nn
n
nn
xnLxn
xn xn
++
+
=
= =
+
1L < when 2 3.x < Thus, the series converges when 1 5.x
< <
When 5,x = the series is 2
21 1
3 1 17 7 ,22 3
n
nn n nn= =
+ = +
which diverges, because 1
1 ,n n=
the harmonic series, diverges.
When 1,x = the series is 2
21 1
( 3) 1 17 7 ,22 3
n
nn n nn= =
+ = +
which diverges, because 1
1 ,n n=
the harmonic series, diverges.
The interval of convergence is ( )1, 5 .
5 :
1 : sets up ratio1: computes limit of ratio
1: identifies interior of interval of convergence1 : considers
both endpoints1 : analysis/conclusion for
both endpoints
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AP Calculus BC 2005 Free-Response Questions
Form B
The College Board: Connecting Students to College Success
The College Board is a not-for-profit membership association
whose mission is to connect students to college success and
opportunity. Founded in 1900, the association is composed of more
than 4,700 schools, colleges, universities, and other educational
organizations. Each year, the College Board serves over three and a
half million students and their parents, 23,000 high schools, and
3,500 colleges through major programs and services in college
admissions, guidance, assessment, financial aid, enrollment, and
teaching and learning. Among its best-known programs are the SAT,
the PSAT/NMSQT, and the Advanced Placement Program (AP). The
College Board is committed to the principles of excellence and
equity, and that commitment is embodied in all of its programs,
services, activities, and concerns.
Copyright 2005 by College Board. All rights reserved. College
Board, AP Central, APCD, Advanced Placement Program, AP, AP
Vertical Teams, Pre-AP, SAT, and the acorn logo are registered
trademarks of the College Entrance Examination Board. Admitted
Class Evaluation Service, CollegeEd, Connect to college success,
MyRoad, SAT Professional Development, SAT Readiness Program, and
Setting the Cornerstones are trademarks owned by the College
Entrance Examination Board. PSAT/NMSQT is a registered trademark of
the College Entrance Examination Board and National Merit
Scholarship Corporation. Other products and services may be
trademarks of their respective owners. Permission to use
copyrighted College Board materials may be requested online at:
http://www.collegeboard.com/inquiry/cbpermit.html. Visit the
College Board on the Web: www.collegeboard.com. AP Central is the
official online home for the AP Program and Pre-AP:
apcentral.collegeboard.com.
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-
2005 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
Copyright 2005 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
GO ON TO THE NEXT PAGE. 2
CALCULUS BC SECTION II, Part A
Time45 minutes Number of problems3
A graphing calculator is required for some problems or parts of
problems.
1. An object moving along a curve in the xy-plane has position (
) ( )( ),x t y t at time t 0 with
212 3dx t tdt
= - and ( )( )4ln 1 4 .dy tdt = + -
At time t 0,= the object is at position ( )13, 5 .- At time 2,t
= the object is at point P with x-coordinate 3.
(a) Find the acceleration vector at time 2t = and the speed at
time t 2.=
(b) Find the y-coordinate of P.
(c) Write an equation for the line tangent to the curve at
P.
(d) For what value of t, if any, is the object at rest? Explain
your reasoning.
WRITE ALL WORK IN THE TEST BOOKLET.
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2005 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
Copyright 2005 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
GO ON TO THE NEXT PAGE. 3
2. A water tank at Camp Newton holds 1200 gallons of water at
time 0.t = During the time interval 0 18t hours, water is pumped
into the tank at the rate
( ) ( )295 sin 6tW t t= gallons per hour.
During the same time interval, water is removed from the tank at
the rate
( ) ( )2275sin 3tR t = gallons per hour.
(a) Is the amount of water in the tank increasing at time 15 ?t
= Why or why not?
(b) To the nearest whole number, how many gallons of water are
in the tank at time 18 ?t =
(c) At what time t, for t0 18, is the amount of water in the
tank at an absolute minimum? Show the work that leads to your
conclusion.
(d) For 18,t > no water is pumped into the tank, but water
continues to be removed at the rate ( )R t until the tank becomes
empty. Let k be the time at which the tank becomes empty. Write,
but do not solve, an equation involving an integral expression that
can be used to find the value of k.
WRITE ALL WORK IN THE TEST BOOKLET.
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2005 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
Copyright 2005 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
4
3. The Taylor series about 0x = for a certain function f
converges to ( )f x for all x in the interval of convergence. The
nth derivative of f at 0x = is given by
( )( )( ) ( )
( )
1
2
1 10
5 1
nn
n
nf
n
+- + !
=
-
for 2.n
The graph of f has a horizontal tangent line at 0,x = and ( )0
6.f =
(a) Determine whether f has a relative maximum, a relative
minimum, or neither at 0.x = Justify your answer.
(b) Write the third-degree Taylor polynomial for f about 0.x
=
(c) Find the radius of convergence of the Taylor series for f
about 0.x = Show the work that leads to your answer.
WRITE ALL WORK IN THE TEST BOOKLET.
END OF PART A OF SECTION II
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2005 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
Copyright 2005 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
GO ON TO THE NEXT PAGE. 5
CALCULUS BC SECTION II, Part B
Time45 minutes Number of problems3
No calculator is allowed for these problems.
4. The graph of the function f above consists of three line
segments.
(a) Let g be the function given by ( ) ( )4
.x
g x f t dt-
= For each of ( )1 ,g - ( )1 ,g - and ( )1 ,g - find the value
or state that it does not exist.
(b) For the function g defined in part (a), find the
x-coordinate of each point of inflection of the graph of g on
the open interval 4 3.x- < < Explain your reasoning.
(c) Let h be the function given by ( ) ( )3
.x
h x f t dt= Find all values of x in the closed interval 4 3x-
for which ( ) 0.h x =
(d) For the function h defined in part (c), find all intervals
on which h is decreasing. Explain your reasoning.
WRITE ALL WORK IN THE TEST BOOKLET.
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2005 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
Copyright 2005 by College Entrance Examination Board. All rights
reserved. Visit apcentral.collegeboard.com (for AP professionals)
and www.collegeboard.com/apstudents (for AP students and
parents).
6
5. Consider the curve given by 2 2 .y xy= +
(a) Show that .2
dy ydx y x
=
-
(b) Find all points ( ),x y on the curve where the line tangent
to the curve has slope 1 .2
(c) Show that there are no points ( ),x y on the curve where the
line tangent to the curve is horizontal.
(d) Let x and y be functions of time t that are related by the
equation 2 2 .y xy= + At time 5,t = the value
of y is 3 and 6.dydt
= Find the value of dxdt
at time 5.t =
6. Consider the graph of the function f given by ( )1
2f x
x=
+ for 0,x as shown in the figure above. Let R be
the region bounded by the graph of f, the x- and y-axes, and the
vertical line ,x k= where 0.k
(a) Find the area of R in terms of k.
(b) Find the volume of the solid generated when R is revolved
about the x-axis in terms of k.
(c) Let S be the unbounded region in the first quadrant to the
right of the vertical line x k= and below the graph of f, as shown
in the figure above. Find all values of k such that the volume of
the solid generated when S is revolved about the x-axis is equal to
the volume of the solid found in part (b).
WRITE ALL WORK IN THE TEST BOOKLET.
END OF EXAM
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AP Calculus BC 2005 Scoring Guidelines
Form B
The College Board: Connecting Students to College Success
The College Board is a not-for-profit membership association
whose mission is to connect students to college success and
opportunity. Founded in 1900, the association is composed of more
than 4,700 schools, colleges, universities, and other educational
organizations. Each year, the College Board serves over three and a
half million students and their parents, 23,000 high schools, and
3,500 colleges through major programs and services in college
admissions, guidance, assessment, financial aid, enrollment, and
teaching and learning. Among its best-known programs are the SAT,
the PSAT/NMSQT, and the Advanced Placement Program (AP). The
College Board is committed to the principles of excellence and
equity, and that commitment is embodied in all of its programs,
services, activities, and concerns.
Copyright 2005 by College Board. All rights reserved. College
Board, AP Central, APCD, Advanced Placement Program, AP, AP
Vertical Teams, Pre-AP, SAT, and the acorn logo are registered
trademarks of the College Entrance Examination Board. Admitted
Class Evaluation Service, CollegeEd, Connect to college success,
MyRoad, SAT Professional Development, SAT Readiness Program, and
Setting the Cornerstones are trademarks owned by the College
Entrance Examination Board. PSAT/NMSQT is a registered trademark of
the College Entrance Examination Board and National Merit
Scholarship Corporation. Other products and services may be
trademarks of their respective owners. Permission to use
copyrighted College Board materials may be requested online at:
http://www.collegeboard.com/inquiry/cbpermit.html. Visit the
College Board on the Web: www.collegeboard.com. AP Central is the
official online home for the AP Program and Pre-AP:
apcentral.collegeboard.com.
www.oneplusone.cn
-
AP CALCULUS BC 2005 SCORING GUIDELINES (Form B)
Copyright 2005 by College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
2
Question 1
An object moving along a curve in the xy-plane has position ( )
( )( ),x t y t at time 0t with
212 3dx t tdt = and ( )( )4ln 1 4 .dy tdt = +
At time 0,t = the object is at position ( )13, 5 . At time 2,t =
the object is at point P with x-coordinate 3.
(a) Find the acceleration vector at time 2t = and the speed at
time 2.t =
(b) Find the y-coordinate of P.
(c) Write an equation for the line tangent to the curve at
P.
(d) For what value of t, if any, is the object at rest? Explain
your reasoning.
(a) ( ) ( ) 322 0, 2 1.88217x y = = =
( )2 0, 1.882a = Speed ( )( )2212 ln 17 12.329 or 12.330= +
=
2 : 1 : acceleration vector
1 : speed
(b) ( ) ( ) ( )( )400 ln 1 4t
y t y u du= + +
( ) ( )( )2 402 5 ln 1 4 13.671y u du= + + = 3 :
( )( )2 40 1 : ln 1 41 : handles initial condition
1 : answer
u du +
(c) At 2,t = slope ( )ln 17 0.23612
dydtdxdt
= = =
( )13.671 0.236 3y x =
2 :
1 : slope1 : equation
(d) ( ) 0x t = if 0, 4t = ( ) 0y t = if 4t =
4t =
2 : 1 : reason1 : answer
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AP CALCULUS BC 2005 SCORING GUIDELINES (Form B)
Copyright 2005 by College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
3
Question 2
A water tank at Camp Newton holds 1200 gallons of water at time
0.t = During the time interval 0 18t hours, water is pumped into
the tank at the rate
( ) ( )295 sin 6tW t t= gallons per hour. During the same time
interval, water is removed from the tank at the rate
( ) ( )2275sin 3tR t = gallons per hour. (a) Is the amount of
water in the tank increasing at time 15 ?t = Why or why not?
(b) To the nearest whole number, how many gallons of water are
in the tank at time 18 ?t =
(c) At what time t, for 0 18,t is the amount of water in the
tank at an absolute minimum? Show the work that leads to your
conclusion.
(d) For 18,t > no water is pumped into the tank, but water
continues to be removed at the rate ( )R t until the tank becomes
empty. Let k be the time at which the tank becomes empty. Write,
but do not solve, an equation involving an integral expression that
can be used to find the value of k.
(a) No; the amount of water is not increasing at 15t = since ( )
( )15 15 121.09 0.W R = <
1 : answer with reason
(b) ( ) ( )( )18
01200 1309.788W t R t dt+ = 1310 gallons
3 : 1 : limits1 : integrand1 : answer
(c) ( ) ( ) 0W t R t = 0, 6.4948, 12.9748t =
t (hours) gallons of water 0 1200
6.495 525 12.975 1697
18 1310 The values at the endpoints and the critical points show
that the absolute minimum occurs when
6.494 or 6.495. t =
3 :
1 : interior critical points 1 : amount of water is least at
6.494 or 6.4951 : analysis for absolute minimum
t
=
(d) ( )18
1310k
R t dt =
2 : 1 : limits1 : equation
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AP CALCULUS BC 2005 SCORING GUIDELINES (Form B)
Copyright 2005 by College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
4
Question 3
The Taylor series about 0x = for a certain function f converges
to ( )f x for all x in the interval of convergence. The nth
derivative of f at 0x = is given by
( ) ( ) ( ) ( )( )
1
21 105 1
nn
nnf
n
+ + !=
for 2.n
The graph of f has a horizontal tangent line at 0,x = and ( )0
6.f =
(a) Determine whether f has a relative maximum, a relative
minimum, or neither at 0.x = Justify your answer.
(b) Write the third-degree Taylor polynomial for f about 0.x
=
(c) Find the radius of convergence of the Taylor series for f
about 0.x = Show the work that leads to your answer.
(a) f has a relative maximum at 0x = because ( )0 0f = and ( )0
0.f < 2 :
1 : answer1 : reason
(b) ( ) ( )0 6, 0 0f f = =
( ) ( )2 2 3 23! 6 4!0 , 0255 1 5 2
f f = = =
( )2 3
2 32 3 2
3! 4! 3 16 6 25 1255 2! 5 2 3!x xP x x x= + = +
3 : ( )P x 1 each incorrect term
Note: 1 max for use of extra terms
(c) ( ) ( ) ( ) ( )
( )
1
20 1 1
! 5 1
nnn n
n nf nu x xn n
+ += =
( ) ( )
( ) ( )( )
( )( )
21
1 211
2
2
1 251 15 1
2 1 11 5
nn
nnnn n
n
n xu nu n x
n
n n xn n
++
+++
+
= +
+ =+
1 1 1lim 5n
nn
u xu+
= < if 5.x <
The radius of convergence is 5.
4 :
1 : general term 1 : sets up ratio 1 : computes limit1 : applies
ratio test to get radius of convergence
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AP CALCULUS BC 2005 SCORING GUIDELINES (Form B)
Copyright 2005 by College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
5
Question 4
The graph of the function f above consists of three line
segments.
(a) Let g be the function given by ( ) ( )4
.x
g x f t dt
= For each of ( )1 ,g ( )1 ,g and ( )1 ,g find the value or
state that it does not exist.
(b) For the function g defined in part (a), find the
x-coordinate of each point of inflection of the graph of g on the
open interval 4 3.x < < Explain your reasoning.
(c) Let h be the function given by ( ) ( )3
.x
h x f t dt= Find all values of x in the closed interval 4 3x for
which ( ) 0.h x =
(d) For the function h defined in part (c), find all intervals
on which h is decreasing. Explain your reasoning.
(a) ( ) ( ) ( )( )1
41 151 3 52 2g f t dt
= = = ( ) ( )1 1 2g f = = ( )1g does not exist because f is not
differentiable
at 1.x =
3 : ( )( )( )
1 : 11 : 11 : 1
ggg
(b) 1x = g f = changes from increasing to decreasing at 1.x
=
2 : 1 : 1 (only)
1 : reasonx =
(c) 1, 1, 3x = 2 : correct values 1 each missing or extra
value
(d) h is decreasing on [ ]0, 2 0h f = < when 0f >
2 : 1 : interval1 : reason
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AP CALCULUS BC 2005 SCORING GUIDELINES (Form B)
Copyright 2005 by College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
6
Question 5
Consider the curve given by 2 2 .y xy= +
(a) Show that .2dy ydx y x=
(b) Find all points ( ),x y on the curve where the line tangent
to the curve has slope 1 .2
(c) Show that there are no points ( ),x y on the curve where the
line tangent to the curve is horizontal.
(d) Let x and y be functions of time t that are related by the
equation 2 2 .y xy= + At time 5,t = the
value of y is 3 and 6.dydt = Find the value of dxdt at time 5.t
=
(a) 2y y y x y = + ( )2y x y y =
2yy y x
=
2 : 1 : implicit differentiation
1 : solves for y
(b) 12 2y
y x =
2 2y y x= 0x =
2y = ( ) ( )0, 2 , 0, 2
2 : 1 1 : 2 2
1 : answer
yy x
=
(c) 02y
y x =
0y = The curve has no horizontal tangent since
20 2 0x + for any x.
2 : 1 : 01 : explanation
y =
(d) When 3,y = 23 2 3x= + so 7 .3x =
2dy dy ydx dxdt dx dt y x dt= =
At 5,t = 3 96 7 116 3
dx dxdt dt= =
5
223t
dxdt =
=
3 : 1 : solves for 1 : chain rule
1 : answer
x
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AP CALCULUS BC 2005 SCORING GUIDELINES (Form B)
Copyright 2005 by College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
7
Question 6
Consider the graph of the function f given by
( ) 1 2f x x= + for 0,x as shown in the figure above. Let R be
the region bounded by the graph of f, the x- and y-axes, and the
vertical line ,x k= where 0.k
(a) Find the area of R in terms of k.
(b) Find the volume of the solid generated when R is revolved
about the x-axis in terms of k.
(c) Let S be the unbounded region in the first quadrant to the
right of the vertical line x k= and below
the graph of f, as shown in the figure above. Find all values of
k such that the volume of the solid generated when S is revolved
about the x-axis is equal to the volume of the solid found in part
(b).
(a) Area of R ( ) ( )0
1 ln 2 ln 22k
dx kx= = + +
2 : 1 : integral1 : antidifferentiation and
evaluation
(b) ( )20
0
12
2 2 2
k
R
k
V dxx
x k
=+
= = + +
3 :
1 : limits 1 : integrand1 : antidifferentiation and
evaluation
(c) ( )2
12
lim 2 2
Sk
n
n k
V dxx
x k
=+
= =+ +
S RV V=
2 2 2k k = + +
2 12 2k =+
2k =
4 :
1 : improper integral1 : antidifferentiation and
evaluation 1 : equation 1 : answer
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AP Calculus BC 2006 Scoring Guidelines
The College Board: Connecting Students to College Success
The College Board is a not-for-profit membership association
whose mission is to connect students to college success and
opportunity. Founded in 1900, the association is composed of more
than 5,000 schools, colleges, universities, and other educational
organizations. Each year, the College Board serves seven million
students and their parents, 23,000 high schools, and 3,500 colleges
through major programs and services in college admissions,
guidance, assessment, financial aid, enrollment, and teaching and
learning. Among its best-known programs are the SAT, the
PSAT/NMSQT, and the Advanced Placement Program (AP). The College
Board is committed to the principles of excellence and equity, and
that commitment is embodied in all of its programs, services,
activities, and concerns.
2006 The College Board. All rights reserved. College Board, AP
Central, APCD, Advanced Placement Program, AP, AP Vertical Teams,
Pre-AP, SAT, and the acorn logo are registered trademarks of the
College Board. Admitted Class Evaluation Service, CollegeEd,
connect to college success, MyRoad, SAT Professional Development,
SAT Readiness Program, and Setting the Cornerstones are trademarks
owned by the College Board. PSAT/NMSQT is a registered trademark of
the College Board and National Merit Scholarship Corporation. All
other products and services may be trademarks of their respective
owners. Permission to use copyrighted College Board materials may
be requested online at: www.collegeboard.com/inquiry/cbpermit.html.
Visit the College Board on the Web: www.collegeboard.com. AP
Central is the official online home for the AP Program:
apcentral.collegeboard.com.
www.oneplusone.cn
-
AP CALCULUS BC 2006 SCORING GUIDELINES
2006 The College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
2
Question 1
Let R be the shaded region bounded by the graph of lny x= and
the line
2,y x= as shown above. (a) Find the area of R. (b) Find the
volume of the solid generated when R is rotated about the
horizontal
line 3.y =
(c) Write, but do not evaluate, an integral expression that can
be used to find the volume of the solid generated when R is rotated
about the y-axis.
( )ln 2x x= when 0.15859x = and 3.14619.
Let 0.15859S = and 3.14619T =
(a) Area of ( ) ( )( )ln 2 1.949T
SR x x dx= =
3 : 1 : integrand
1 : limits1 : answer
(b) Volume ( )( ) ( )( )2 2ln 3 2 334.198 or 34.199
T
Sx x dx= + +
=
3 : { 2 : integrand1 : limits, constant, and answer
(c) Volume ( )( )2 222
( 2)T
y
Sy e dy
= +
3 : { 2 : integrand1 : limits and constant
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AP CALCULUS BC 2006 SCORING GUIDELINES
2006 The College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
3
Question 2
At an intersection in Thomasville, Oregon, cars turn
left at the rate ( ) ( )260 sin 3tL t t= cars per hour over the
time interval 0 18t hours. The graph of
( )y L t= is shown above. (a) To the nearest whole number, find
the total
number of cars turning left at the intersection over the time
interval 0 18t hours.
(b) Traffic engineers will consider turn restrictions when ( )
150L t cars per hour. Find all values of t for which ( ) 150L t and
compute the average value of L over this time interval. Indicate
units of measure.
(c) Traffic engineers will install a signal if there is any
two-hour time interval during which the product of the total number
of cars turning left and the total number of oncoming cars
traveling straight through the intersection is greater than
200,000. In every two-hour time interval, 500 oncoming cars travel
straight through the intersection. Does this intersection require a
traffic signal? Explain the reasoning that leads to your
conclusion.
(a) ( )18
01658L t dt cars
2 : { 1 : setup 1 : answer (b) ( ) 150L t = when 12.42831,t =
16.12166
Let 12.42831R = and 16.12166S = ( ) 150L t for t in the interval
[ ],R S
( )1 199.426S
RL t dtS R = cars per hour
3 : ( )1 : -interval when 150
1 : average value integral 1 : answer with units
t L t
(c) For the product to exceed 200,000, the number of cars
turning left in a two-hour interval must be greater than 400.
( )15
13431.931 400L t dt = >
OR The number of cars turning left will be greater than 400
on a two-hour interval if ( ) 200L t on that interval. ( ) 200L
t on any two-hour subinterval of
[ ]13.25304, 15.32386 . Yes, a traffic signal is required.
4 : [ ]
( )2
1 : considers 400 cars1 : valid interval , 2
1 : value of
1 : answer and explanation
h
h
h h
L t dt+
+
OR
4 : ( )
1 : considers 200 cars per hour 1 : solves 2001 : discusses 2
hour interval
1 : answer and explanation
L t
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AP CALCULUS BC 2006 SCORING GUIDELINES
2006 The College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
4
Question 3
An object moving along a curve in the xy-plane is at position (
) ( )( ),x t y t at time t, where
( )1sin 1 2 tdx edt = and 34
1dy tdt t
=+
for 0.t At time 2,t = the object is at the point ( )6, 3 .
(Note: 1sin arcsinx x = )
(a) Find the acceleration vector and the speed of the object at
time 2.t = (b) The curve has a vertical tangent line at one point.
At what time t is the object at this point? (c) Let ( )m t denote
the slope of the line tangent to the curve at the point ( ) ( )( ),
.x t y t Write an expression for
( )m t in terms of t and use it to evaluate ( )lim .t
m t
(d) The graph of the curve has a horizontal asymptote .y c=
Write, but do not evaluate, an expression involving an improper
integral that represents this value c.
(a) ( )2 0.395 or 0.396, 0.741 or 0.740 a = Speed ( ) ( )2 22 2
1.207x y = + = or 1.208
2 : { 1 : acceleration 1 : speed
(b) ( )1sin 1 2 0te = 1 2 0te =
ln 2 0.693t = = and 0dydt when ln 2t =
2 : ( )1 : 01 : answer
x t =
(c) ( ) ( )3 14 1
1 sin 1 2 ttm tt e
= +
( ) ( )
( )
3 1
1
4 1lim lim1 sin 1 2
10 0sin 1
tt ttm tt e
= +
= =
2 : ( ) 1 : 1 : limit value
m t
(d) Since ( )lim ,t
x t
=
( ) 324lim 3
1ttc y t dtt
= = ++
3 :
1: integrand 1: limits
1: initial value consistent
with lower limit
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AP CALCULUS BC 2006 SCORING GUIDELINES
2006 The College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
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5
Question 4
t
(seconds) 0 10 20 30 40 50 60 70 80
( )v t (feet per second)
5 14 22 29 35 40 44 47 49
Rocket A has positive velocity ( )v t after being launched
upward from an initial height of 0 feet at time 0t = seconds. The
velocity of the rocket is recorded for selected values of t over
the interval 0 80t seconds, as shown in the table above. (a) Find
the average acceleration of rocket A over the time interval 0 80t
seconds. Indicate units of
measure.
(b) Using correct units, explain the meaning of ( )70
10v t dt in terms of the rockets flight. Use a midpoint
Riemann sum with 3 subintervals of equal length to approximate (
)70
10.v t dt
(c) Rocket B is launched upward with an acceleration of ( )
31
a tt
=+
feet per second per second. At time
0t = seconds, the initial height of the rocket is 0 feet, and
the initial velocity is 2 feet per second. Which of the two rockets
is traveling faster at time 80t = seconds? Explain your answer.
(a) Average acceleration of rocket A is
( ) ( ) 280 0 49 5 11 ft sec80 0 80 20v v = =
1 : answer
(b) Since the velocity is positive, ( )70
10v t dt represents the
distance, in feet, traveled by rocket A from 10t = seconds to
70t = seconds.
A midpoint Riemann sum is
( ) ( ) ( )[ ][ ]
20 20 40 6020 22 35 44 2020 ft
v v v+ += + + =
3 : ( ) ( ) ( ) 1 : explanation1 : uses 20 , 40 , 60
1 : valuev v v
(c) Let ( )Bv t be the velocity of rocket B at time t.
( ) 3 6 11B
v t dt t Ct
= = + ++
( )2 0 6Bv C= = + ( ) 6 1 4Bv t t= + ( ) ( )80 50 49 80Bv v=
> = Rocket B is traveling faster at time 80t = seconds.
4 : ( ) ( )
1 : 6 1 1 : constant of integration 1 : uses initial condition1
: finds 80 , compares to 80 ,
and draws a conclusionB
t
v v
+
Units of 2ft sec in (a) and ft in (b) 1 : units in (a) and
(b)
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AP CALCULUS BC 2006 SCORING GUIDELINES
2006 The College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
6
Question 5
Consider the differential equation 2 65 2dy xdx y= for 2.y Let (
)y f x= be the particular solution to this
differential equation with the initial condition ( )1 4.f =
(a) Evaluate dydx and 2
2d ydx
at ( )1, 4 .
(b) Is it possible for the x-axis to be tangent to the graph of
f at some point? Explain why or why not. (c) Find the second-degree
Taylor polynomial for f about 1.x = (d) Use Eulers method, starting
at 1x = with two steps of equal size, to approximate ( )0 .f Show
the work
that leads to your answer.
(a) ( )1, 4
6dydx =
( )2
22 10 6 2
d y dyx y dxdx= +
( ) ( )
2
2 21, 4
110 6 6 96
d ydx
= + =
3 :
( )
( )
1, 42
2
2
21, 4
1 :
1 :
1 :
dydx
d ydxd ydx
(b) The x-axis will be tangent to the graph of f if ( ), 0
0.k
dydx =
The x-axis will never be tangent to the graph of f because
( )
2
, 05 3 0
k
dy kdx = + > for all k.
2 : 1 : 0 and 0
1 : answer and explanation
dy ydx = =
(c) ( ) ( ) ( )294 6 1 12P x x x= + + +
2 : { 1 : quadratic and centered at 1 1 : coefficients x =
(d) ( )1 4f =
( ) ( )1 14 6 12 2f + = ( ) ( )1 5 50 1 22 4 8f + + =
2 : ( )1 : Euler's method with 2 steps1 : Euler's approximation
to 0f
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AP CALCULUS BC 2006 SCORING GUIDELINES
2006 The College Board. All rights reserved. Visit
apcentral.collegeboard.com (for AP professionals) and
www.collegeboard.com/apstudents (for AP students and parents).
7
Question 6
The function f is defined by the power series
( ) ( )2 3 12 3
2 3 4 1L Ln nnxx x xf x n
= + + + ++
for all real numbers x for which the series converges. The
function g is defined by the power series
( ) ( )( )2 3 11 2! 4! 6! 2 !L L
n nxx x xg x n= + + + +
for all real numbers x for which the series converges. (a) Find
the interval of convergence of the power series for f. Justify your
answer. (b) The graph of ( ) ( )y f x g x= passes through the point
( )0, 1 . Find ( )0y and ( )0 .y Determine whether y
has a relative minimum, a relative maximum, or neither at 0.x =
Give a reason for your answer.
(a) ( ) ( )( )
( )( )( )
1 211 1 112 21
n n
n nn x nn xn n nnx
+ + + ++ = + +
( )( )( )
21lim 2nn x xn n
+ =
+
The series converges when 1 1.x < <
When 1,x = the series is 1 2 32 3 4 + +L
This series does not converge, because the limit of the
individual terms is not zero.
When 1,x = the series is 1 2 32 3 4+ + +L
This series does not converge, because the limit of the
individual terms is not zero. Thus, the interval of convergence is
1 1.x < <
5 :
1 : sets up ratio 1 : computes limit of ratio1 : identifies
radius of convergence
1 : considers both endpoints 1 : analysis/conclusion for both
endpoints
(b) ( ) 21 4 92 3 4f x x x = + +L a