APfi Calculus BC 2002 Free-Response Questions Calculus BC... · APfi Calculus BC 2002 Free-Response Questions These materials were produced by Educational Testing Service fi (ETS

AP® Calculus BC2002 Free-Response Questions

These materials were produced by Educational Testing Service® (ETS®), which develops and administers the examinations of the Advanced Placement Program for the College Board. The College Board and Educational Testing Service (ETS) are dedicated to the principle of equal opportunity, and their

programs, services, and employment policies are guided by that principle.

The College Board is a national nonprofit membership association dedicated to preparing, inspiring, and connecting students to college and opportunity. Founded in 1900, the association is composed of more than 4,200 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 22,000 high schools, and 3,500 colleges, through major programs and services in

college admission, 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 equity and

excellence, and that commitment is embodied in all of its programs, services, activities, and concerns.

Copyright © 2002 by College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, SAT, and the acorn logo are registered trademarks of the College Entrance Examination Board. APIEL is a trademark owned by 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.

The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; 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. These materials and

any copies made of them may not be resold, and the copyright notices must be retained as they appear here. This permission does not apply to any third-party

copyrights contained herein.

2002 AP® CALCULUS BC FREE-RESPONSE QUESTIONS

Copyright © 2002 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.

GO ON TO THE NEXT PAGE. 2

CALCULUS BC

SECTION II, Part A Time—45 minutes

Number of problems—3

A graphing calculator is required for some problems or parts of problems.

1. Let f and g be the functions given by f x e x� � � and g x x� � � ln .

(a) Find the area of the region enclosed by the graphs of f and g between x � 12

and x � 1.

(b) Find the volume of the solid generated when the region enclosed by the graphs of f and g between x � 12

and x � 1 is revolved about the line y � 4.

(c) Let h be the function given by h x f x g x� � � � � �� . Find the absolute minimum value of h x� � on the

closed interval 12

1� �x , and find the absolute maximum value of h x� � on the closed interval 12

1� �x .

Show the analysis that leads to your answers.




2. The rate at which people enter an amusement park on a given day is modeled by the function E defined by

E tt t

� � ��

15600

24 1602� � .

The rate at which people leave the same amusement park on the same day is modeled by the function L defined by

L tt t

� � ��

9890

38 3702� � .

Both E t� � and L t� � are measured in people per hour and time t is measured in hours after midnight. These

functions are valid for 9 23� �t , the hours during which the park is open. At time t � 9, there are no people in the park.

(a) How many people have entered the park by 5:00 P.M. (t � 17)? Round your answer to the nearest whole number.

(b) The price of admission to the park is $15 until 5:00 P.M. ( t � 17 ). After 5:00 P.M., the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number.

(c) Let H t E x L x dxt

� � � � � � � �� 9

for 9 23� �t . The value of H 17� � to the nearest whole number is 3725.

Find the value of �� H 17 , and explain the meaning of H 17� � and �� H 17 in the context of the amusement park.

(d) At what time t, for 9 23� �t , does the model predict that the number of people in the park is a maximum?



4

3. The figure above shows the path traveled by a roller coaster car over the time interval 0 18� �t seconds. The position of the car at time t seconds can be modeled parametrically by

x t t t

y t t t

� � � ��

10 4

20 1

sin

cos , where x and y are measured in meters. The derivatives of these functions are given by

��

x t t

y t t t t

10 4

20 1

cos

sin cos . (a) Find the slope of the path at time t � 2. Show the computations that lead to your answer.

(b) Find the acceleration vector of the car at the time when the car’s horizontal position is x � 140.

(c) Find the time t 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� �t , there are two times at which the car is at ground level y � 0� �. 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.

END OF PART A OF SECTION II




CALCULUS BC

SECTION II, Part B Time—45 minutes


No calculator is allowed for these problems.

4. The graph of the function f shown above consists of two line segments. Let g be the function given by

g x f t dtx� � � �� 0 .

(a) Find g �� 1 , � �� g 1 , and �� g 1 .

(b) For what values of x in the open interval �2 2,� � is g increasing? Explain your reasoning.

(c) For what values of x in the open interval �2 2,� � is the graph of g concave down? Explain your reasoning.

(d) On the axes provided, sketch the graph of g on the closed interval �2 2, . (Note: The axes are provided in the pink test booklet only.)



6

5. Consider the differential equation dydx

y x� �2 4 .

(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, .�� (Note: Use the slope field provided in the pink test booklet.)

(b) Let f be the function that satisfies the given differential equation with the initial condition f 0 1� � � .

Use Euler’s method, starting at x � 0 with a step size of 0.1, to approximate f 0 2. .� � Show the work

that leads to your answer.

(c) Find the value of b for which y x b� �2 is a solution to the given differential equation. Justify your answer.

(d) Let g be the function that satisfies the given differential equation with the initial condition g 0 0� � � .

Does the graph of g have a local extremum at the point 0 0, ?� � If so, is the point a local maximum

or a local minimum? Justify your answer.

6. The Maclaurin series for the function f is given by

f xx

nx x x x x

n

n

n

n

� � � � ��

� �+

=

+�

� 21

2 42

83

164

21

1

0

2 3 4 1

� �

on its interval of convergence.

(a) Find the interval of convergence of the Maclaurin series for f. Justify your answer.

(b) Find the first four terms and the general term for the Maclaurin series for �� f x .

(c) Use the Maclaurin series you found in part (b) to find the value of � �� f 13

.

END OF EXAMINATION


Form B



The College Board is a national nonprofit membership association dedicated to preparing, inspiring, and connecting students to college and opportunity. Founded in 1900, the association is composed of more than 4,200 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 22,000 high schools, and 3,500 colleges, through major programs and services in

college admission, 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 equity and

excellence, and that commitment is embodied in all of its programs, services, activities, and concerns.

Copyright © 2002 by College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, SAT, and the acorn logo are registered trademarks of the College Entrance Examination Board. APIEL is a trademark owned by 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.

The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; 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. These materials and

any copies made of them may not be resold, and the copyright notices must be retained as they appear here. This permission does not apply to any third-party

copyrights contained herein.

2002 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)



CALCULUS BC

SECTION II, Part A Time—45 minutes


A graphing calculator is required for some problems or parts of problems. 1. A particle moves in the xy-plane so that its position at any time t, for � � �p pt , is given by x t t� � � � �sin 3

and y t t� � � 2 .

(a) Sketch the path of the particle in the xy-plane provided. Indicate the direction of motion along the path. (Note: Use the axes provided in the test booklet.)

(b) Find the range of x t� � and the range of y t� �.

(c) Find the smallest positive value of t for which the x-coordinate of the particle is a local maximum. What is the speed of the particle at this time?

(d) Is the distance traveled by the particle from t � �p to t � p greater than 5p ? Justify your answer.



3

2. The number of gallons, P t� �, of a pollutant in a lake changes at the rate �� -P t e t1 3 0 2. gallons per day, where t is measured in days. There are 50 gallons of the pollutant in the lake at time t � 0. The lake is considered to be safe when it contains 40 gallons or less of pollutant.

(a) Is the amount of pollutant increasing at time t � 9 ? Why or why not?

(b) For what value of t will the number of gallons of pollutant be at its minimum? Justify your answer.

(c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify your answer.

(d) An investigator uses the tangent line approximation to P t� � at t = 0 as a model for the amount of pollutant

in the lake. At what time t does this model predict that the lake becomes safe?

3. Let R be the region in the first quadrant bounded by the y-axis and the graphs of y x x� � �4 13 and

y x� 34

.

(a) Find the area of R.

(b) Find the volume of the solid generated when R is revolved about the x-axis.

(c) Write an expression involving one or more integrals that gives the perimeter of R. Do not evaluate.





CALCULUS BC

SECTION II, Part B Time—45 minutes



4. The graph of a differentiable function f on the closed interval [ , ]�3 15 is shown in the figure above. The graph

of f has a horizontal tangent line at x � 6. Let g x f t dtx

( ) = + �56� � for � � �3 15x .

(a) Find g 6� �, ′g 6� �, and ′′g 6� �. (b) On what intervals is g decreasing? Justify your answer.

(c) On what intervals is the graph of g concave down? Justify your answer.

(d) Find a trapezoidal approximation of f t dt� �-

� 3

15 using six subintervals of length Dt � 3.

5. Consider the differential equation dydx

xy

� �3 .

(a) Let y f x= � � be the particular solution to the given differential equation for 1 5� �x such that the line

y � �2 is tangent to the graph of f. Find the x-coordinate of the point of tangency, and determine whether f has a local maximum, local minimum, or neither at this point. Justify your answer.

(b) Let y g x= � � be the particular solution to the given differential equation for � � �2 8x , with the initial

condition g 6 4� � � � . Find y g x= � �.



5

6. The Maclaurin series for ln 11 �� x is x

n

n

n=

∞

∑1

with interval of convergence − ≤ <1 1x .

(a) Find the Maclaurin series for ln 11 3+�� x and determine the interval of convergence.

(b) Find the value of ��

=

�

� 1

1

n

nn

.

(c) Give a value of p such that ��

=

�

� 1

1

n

pn n

converges, but 12

1 n pn=

�

� diverges. Give reasons why your value of p

is correct.

(d) Give a value of p such that 1

1 n pn=

�

� diverges, but 12

1 n pn=

�

� converges. Give reasons why your value of p is

correct.

END OF EXAMINATION




The College Board is a national nonprofit membership association whose mission is to prepare, inspire, and connect students to college and opportunity. Founded in 1900, the association is composed of more than 4,300 schools, colleges, universities, and other educational organizations. Each year, the

College Board serves over three million students and their parents, 22,000 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 equity and excellence, and that commitment is embodied in all of its programs, services, activities, and concerns.

For further information, visit www.collegeboard.com

Copyright © 2003 College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Vertical Teams, APCD, Pacesetter, Pre-AP, SAT, Student Search Service, and the acorn logo are registered trademarks of the College Entrance Examination Board.

AP Central is a trademark owned by 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 Board’s online home for AP professionals, visit AP Central at apcentral.collegeboard.com.

The materials included in these files are intended for 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 noncommercial, face-to-face teaching

purposes. This permission does not apply to any third-party copyrights contained

herein. This material may not be mass distributed, electronically or otherwise.

These materials and any copies made of them may not be resold, and the

copyright notices must be retained as they appear here.


Copyright © 2003 by College Entrance Examination Board. All rights reserved. Available to AP professionals at apcentral.collegeboard.com and to

students and parents at www.collegeboard.com/apstudents.

GO ON TO THE NEXT PAGE.

2

CALCULUS BC SECTION II, Part A

Time—45 minutes Number of problems—3


1. Let R be the shaded region bounded by the graphs of y x= and y e x= −3 and the vertical line x = 1, as shown in the figure above.


(b) Find the volume of the solid generated when R is revolved about the horizontal line y = 1.

(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid.





3

2. A particle starts at point A on the positive x-axis at time t � 0 and travels along the curve from A to B to C

to D, as shown above. The coordinates of the particle’s position x t y t� � � �,� � are differentiable functions of t,

where ��

��x t dx

dtt t9

61

2cos sin

p p and �� y tdydt

is not explicitly given. At time t � 9, the

particle reaches its final position at point D on the positive x-axis.

(a) At point C, is dydt

positive? At point C, is dxdt

positive? Give a reason for each answer.

(b) The slope of the curve is undefined at point B. At what time t is the particle at point B ?

(c) The line tangent to the curve at the point x y8 8� � � �,� � has equation y x� �59

2. Find the velocity vector

and the speed of the particle at this point.

(d) How far apart are points A and D, the initial and final positions, respectively, of the particle?




4

3. The figure above shows the graphs of the line x y� 53

and the curve C given by x y� �1 2 . Let S be the

shaded region bounded by the two graphs and the x-axis. The line and the curve intersect at point P.

(a) Find the coordinates of point P and the value of dxdy

for curve C at point P.

(b) Set up and evaluate an integral expression with respect to y that gives the area of S.

(c) Curve C is a part of the curve x y2 2 1� � . Show that x y2 2 1� � can be written as the polar equation

r 22 2

1��cos sin

.q q

(d) Use the polar equation given in part (c) to set up an integral expression with respect to the polar angle q that represents the area of S.






5

CALCULUS BC SECTION II, Part B



4. Let f be a function defined on the closed interval � � �3 4x with f 0 3� � = . The graph of ′f , the derivative

of f, consists of one line segment and a semicircle, as shown above.

(a) On what intervals, if any, is f increasing? Justify your answer.

(b) Find the x-coordinate of each point of inflection of the graph of f on the open interval � � �3 4x . Justify your answer.

(c) Find an equation for the line tangent to the graph of f at the point 0 3, .� �

(d) Find f �� 3 and f 4� �. Show the work that leads to your answers.




6

5. A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let h be the depth of the coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume V of coffee in the pot is changing at the rate of �5p h cubic inches per second. (The volume V of a cylinder with radius r and height h is V r h� p

2 .)

(a) Show that dhdt

h� �5

.

(b) Given that h = 17 at time t � 0, solve the differential equation dhdt

h= −5

for h as a function of t.

(c) At what time t is the coffeepot empty?

6. The function f is defined by the power series

f xx

nx x x x

n

n n

n

n n

� � � ��

�� =

� 1

2 11

3 5 71

2 1

2

0

2 4 6 2

! ! ! ! !� �

for all real numbers x.

(a) Find f 0� � and f 0� �. Determine whether f has a local maximum, a local minimum, or neither at x � 0.

Give a reason for your answer.

(b) Show that 1 13

�! approximates f 1� � with error less than 1

100.

(c) Show that y f x� � � is a solution to the differential equation xy y x � � cos .

END OF EXAMINATION


Form B



The College Board is a national nonprofit membership association whose mission is to prepare, inspire, and connect students to college and opportunity. Founded in 1900, the association is composed of more than 4,300 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 22,000 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 equity and excellence, and that commitment is embodied in all of its programs, services, activities, and concerns.


Copyright © 2003 College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Vertical Teams, APCD, Pacesetter, Pre-AP, SAT, Student Search Service, and the acorn logo are registered trademarks of the College Entrance Examination Board.

AP Central is a trademark owned by 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.


The materials included in these files are intended for 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 noncommercial, face-to-face teaching

purposes. This permission does not apply to any third-party copyrights contained herein. This material may not be mass distributed, electronically or otherwise.

These materials and any copies made of them may not be resold, and the copyright notices must be retained as they appear here.





2




1. Let f be the function given by f x x x� � � �4 2 3 , and let � be the line y x� �18 3 , where � is tangent to the graph of f. Let R be the region bounded by the graph of f and the x-axis, and let S be the region bounded by the graph of f, the line �, and the x-axis, as shown above.

(a) Show that � is tangent to the graph of y f x� � � at the point x � 3.

(b) Find the area of S.

(c) Find the volume of the solid generated when R is revolved about the x-axis.




3

2. The figure above shows the graphs of the circles x y2 2 2� � and x y� � �1 12 2� � . The graphs intersect at

the points 1 1,� � and 1 1, .�� Let R be the shaded region in the first quadrant bounded by the two circles and the

x-axis.

(a) Set up an expression involving one or more integrals with respect to x that represents the area of R.

(b) Set up an expression involving one or more integrals with respect to y that represents the area of R.

(c) The polar equations of the circles are r � 2 and r � 2 cos ,q respectively. Set up an expression involving one or more integrals with respect to the polar angle q that represents the area of R.

Distance

x (mm)

0 60 120 180 240 300 360

Diameter B x� � (mm)

24 30 28 30 26 24 26

3. A blood vessel is 360 millimeters (mm) long with circular cross sections of varying diameter. The table above gives the measurements of the diameter of the blood vessel at selected points along the length of the blood vessel, where x represents the distance from one end of the blood vessel and B x� � is a twice-differentiable function that represents the diameter at that point.

(a) Write an integral expression in terms of B x� � that represents the average radius, in mm, of the blood vessel between x � 0 and x � 360.

(b) Approximate the value of your answer from part (a) using the data from the table and a midpoint Riemann sum with three subintervals of equal length. Show the computations that lead to your answer.

(c) Using correct units, explain the meaning of pB x

dx� �� 2

2

125

275 in terms of the blood vessel.

(d) Explain why there must be at least one value x, for 0 360� �x , such that �� B x( ) .0






4




4. A particle moves in the xy-plane so that the position of the particle at any time t is given by

x t e et t� � � � -2 3 7 and y t e et t� � � � -3 3 2 .

(a) Find the velocity vector for the particle in terms of t, and find the speed of the particle at time t � 0.

(b) Find dydx

in terms of t, and find lim .t

dydx��

(c) Find each value t at which the line tangent to the path of the particle is horizontal, or explain why none exists.

(d) Find each value t at which the line tangent to the path of the particle is vertical, or explain why none exists.

5. Let f be a function defined on the closed interval 0 7, . The graph of f, consisting of four line segments,

is shown above. Let g be the function given by g x f t dtx

� � � � ��2 .

(a) Find g 3� �, �� g 3 , and �� g 3 .

(b) Find the average rate of change of g on the interval 0 3� �x .

(c) For how many values c, where 0 3 c , is �� g c equal to the average rate found in part (b) ? Explain your reasoning.

(d) Find the x-coordinate of each point of inflection of the graph of g on the interval 0 7 x . Justify your answer.




5

6. The function f has a Taylor series about x � 2 that converges to f x� � for all x in the interval of convergence.

The nth derivative of f at x � 2 is given by fnn

n( )� � � ��

21

3! for n 1, and f 2 1� � � .

(a) Write the first four terms and the general term of the Taylor series for f about x � 2.

(b) Find the radius of convergence for the Taylor series for f about x � 2. Show the work that leads to your answer.

(c) Let g be a function satisfying g 2 3� � � and �� g x f x for all x. Write the first four terms and the general term of the Taylor series for g about x � 2.

(d) Does the Taylor series for g as defined in part (c) converge at x � �2 ? Give a reason for your answer.

END OF EXAMINATION


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.


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.


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.


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




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 4sin2tF 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.




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.



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 =





5




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 12

dP P Pdt

= -

(a) If ( )0 3,P = what is ( )lim ?t

P tÆ•

If ( )0 20,P = what is ( )lim ?t

P tÆ•

(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 12

dY Y tdt

= -

Find ( )Y t if ( )0 3.Y =

(d) For the function Y found in part (c), what is ( )lim ?t

Y tÆ•



6

6. Let f be the function given by ( ) ( )sin 5 ,4

f x x p= + 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 10 100

f P- <

(d) Let G be the function given by ( ) ( )0

.x

G x f t dt= Ú Write the third-degree Taylor polynomial

for G about 0.x =

END OF EXAMINATION

AP® Calculus BC2004 Scoring Guidelines





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AP® CALCULUS BC 2004 SCORING GUIDELINES


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 =∫


(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)



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 dxπ= − − −∫

( )( ) ( )( )( )1 2 20

2 3 1 2 2 1

16.179

x x x x dxπ= − − − − −

=∫

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 2

01 3 1 15k x x x x dx− − − =∫

3 : { 2 : integrand1 : answer



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 2

2 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

′′



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



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 ?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 ?

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 = − = −

2ln 5 120

t 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



7

Question 6

Let f be the function given by ( ) ( )sin 5 ,4f x x π= + 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 =

(a) ( ) ( ) 20 sin 4 2f π= =

( ) ( ) 5 20 5cos 4 2f π′ = =

( ) ( ) 25 20 25sin 4 2f π′′ = − = −

( ) ( ) 125 20 125cos 4 2f π′′′ = − = −

( ) ( ) ( )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 π evaluation error

deduct only once for ( )4cos π 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 : error bound in an appropriate inequality

(d) The third-degree Taylor polynomial for G about

0x = is 2

0

2 3

2 5 2 25 22 2 4

2 5 2 25 22 4 12

xt t dt

x x x

+ −

= + −

⌠⌡

2 : third-degree Taylor polynomial for about 0

Gx =

1− each incorrect or missing term

1− max for all extra terms, ,+ misuse of equality


Form B





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College Entrance Examination Board and the National Merit Scholarship Corporation. Educational Testing Service and ETS are registered trademarks of Educational Testing Service.











2




1. A particle moving along a curve in the plane has position x t y t( ) ( ),a f at time t, where

dxdt

t= +4 9 and

dydt

e et t= +

-2 5

for all real values of t. At time t = 0, the particle is at the point 4 1, .a f

(a) Find the speed of the particle and its acceleration vector at time t = 0.

(b) Find an equation of the line tangent to the path of the particle at time t = 0.

(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 t = 3.

2. Let f be a function having derivatives of all orders for all real numbers. The third-degree Taylor polynomial

for f about x = 2 is given by

T x x x( ) = - -( ) - -( )7 9 2 3 22 3 .

(a) Find f 2( ) and ¢¢( )f 2 .

(b) Is there enough information given to determine whether f has a critical point at x = 2 ?

If not, explain why not.

If so, determine whether f 2( ) is a relative maximum, a relative minimum, or neither, and justify your answer.

(c) Use T xa f to find an approximation for f 0a f. Is there enough information given to determine whether f has

a critical point at x = 0 ?

If not, explain why not.

If so, determine whether f 0( ) is a relative maximum, a relative minimum, or neither, and justify your answer.

(d) The fourth derivative of f satisfies the inequality f x4 6( )( ) £ for all x in the closed interval 0 2, . Use

the Lagrange error bound on the approximation to f 0( ) found in part (c) to explain why f 0( ) is negative.



3

t

(minutes) 0 5 10 15 20 25 30 35 40

v t( ) (miles per minute) 7.0 9.2 9.5 7.0 4.5 2.4 2.4 4.3 7.3

3. A test plane flies in a straight line with positive velocity v ta f, in miles per minute at time t minutes, where

v is a differentiable function of t. Selected values of v ta f for 0 40£ £t 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 v t dt( )z0

40. Show the computations that lead to your answer. Using correct units, explain the

meaning of v t dt( )z0

40 in terms of the plane’s 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 f t t ta f = +FH

IK +

FH

IK6

103 7

40cos sin , 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

t = 23 ? Indicate 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 ?





4




4. The figure above shows the graph of ¢f , the derivative of the function f, on the closed interval - £ £1 5x . The graph of ¢f has horizontal tangent lines at x = 1 and x = 3. The function f is twice differentiable with f 2 6( ) = .

(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 5x ? At what value of x does f attain its absolute maximum value on the closed interval - £ £1 5x ? 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 x = 2.



5

5. Let g be the function given by g xx

( ) =1 .

(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 x = 1 and x = 4, 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,•f is defined to be lim .b

a

bf x dx

b aÆ•

( )

-

L

NMM

O

QPP

z Show

that the improper integral g x dx( )•

z4 is divergent, but the average value of g on the interval 4,•f is finite.

6. Let � be the line tangent to the graph of y x n= at the point 1 1, ,a f where n > 1, as shown above.

(a) Find x dxn

0

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

AP® Calculus BC2004 Scoring Guidelines

Form B





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.








AP® CALCULUS BC 2004 SCORING GUIDELINES (Form B)


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 24

09 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



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

′ = ′′ <

(c) ( ) ( )0 0 5f T≈ = − It is not possible to determine if f has a critical point

at 0x = because ( )T x gives exact information only at 2.x =

3 :

( ) ( )1 : 0 0 51 : declares that it is not possible to determine

1 : reason

f T≈ = −

(d) Lagrange error bound 46 0 2 44!= − =

( ) ( )0 0 4 1f T≤ + = − Therefore, ( )0f is negative.

2 : 1 : value of Lagrange error

bound 1 : explanation



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 plane’s 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=

=∫


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



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

−′− − = <∫

Since ( ) ( )5 1 ,f f< − the absolute maximum value occurs at 1.x = −

4 :

1 : indicates decreases then increases 1 : eliminates 5 for maximum 1 : absolute minimum at 4 1 : absolute maximum at 1

fx

xx

= = = −

(c) ( ) ( ) ( )g x f x x f x′ ′= + ( ) ( ) ( ) ( )2 2 2 2 6 2 1 4g f f′ ′= + = + − = ( ) ( )2 2 2 12g f= = Tangent line is ( )4 2 12y x= − +

3 : ( ) 2 :

1 : tangent lineg x′



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 xxπ π π= = =⌠⌡


and evaluation

(c) The cross section at x has area ( )21xxππ =

Average value 4

1

1 1 ln 43 3dxxπ π= =⌠

⌡

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

= − − =− − → ∞

∫

∫



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

AP® Calculus BC 2005 Free-Response Questions

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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.



(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.




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.




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¢Ú and indicate units of measure. Explain the meaning of ( )

8

0T x dx¢Ú 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.









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 .2

x = 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 Euler’s 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.




6

5. A car is traveling on a straight road. For 0 24t£ £ seconds, the car’s 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 dtÚ Using correct units, explain the meaning of ( )

24

0.v t dtÚ

(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 car’s 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 .

3n

nn

f-

=

(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.


END OF EXAM

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 xπ= + 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 44xxπ −+ = .

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


⎧⎪⎨⎪⎩

(b) ( ) ( )( )1

0.410a

f x g x dx− =∫


⎧⎪⎨⎪⎩

(c) ( )( ) ( )( )( )1 2 21 1 4.558a

f x g x dxπ + − + =∫ or 4.559

3 : { 2 : integrand1 : limits, constant, and answer



3

Question 2

The curve above is drawn in the xy-plane and is described by the equation in polar coordinates ( )sin 2r θ θ= + for 0 ,θ π≤ ≤ where r is measured in meters and θ is measured in radians. The derivative of r with respect to θ is

given by ( )1 2cos 2 .drd θθ = +

(a) Find the area bounded by the curve and the x-axis. (b) Find the angle θ that corresponds to the point on the curve with

x-coordinate 2.−

(c) For 2 ,3 3π πθ< < dr

dθ is negative. What does this fact say about r ? What does this fact say about the curve?

(d) Find the value of θ in the interval 0 2πθ≤ ≤ 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

π

π

θ

θ θ θ

=

= + =

∫

∫

3 : 1 : limits and constant

1 : integrand 1 : answer

⎧⎪⎨⎪⎩

(b) ( ) ( )( ) ( )2 cos sin 2 cosr θ θ θ θ− = = + 2.786θ =

2 : { 1 : equation1 : answer

(c) Since 0drdθ < for 2 ,3 3

π πθ< < r is decreasing on this

interval. This means the curve is getting closer to the origin.

2 : { 1 : information about 1 : information about the curve

r

(d) The only value in 0, 2π⎡ ⎤

⎢ ⎥⎣ ⎦ where 0dr

dθ = is .3πθ =

θ r 0 0

3π 1.913

2π 1.571

The greatest distance occurs when .3πθ =

2 : 1 : or 1.04731 : answer with justification

πθ⎧ =⎪⎨⎪⎩



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)



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 Euler’s 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 :


d ydx

⎧⎪⎨⎪⎩



6

Question 5

A car is traveling on a straight road. For 0 24t≤ ≤ seconds, the car’s 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 car’s 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 t

v t v v t vt 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

< <⎧⎪ < <= ⎨⎪− < <⎩

( )a t does not exist at 4t = and 16.t =

2 : 5 1 : finds the values 5, 0, 2

1 : identifies constants with correct intervals

⎧ −⎪⎨⎪⎩

(d) The average rate of change of v on [ ]8, 20 is ( ) ( ) 220 8 5 m sec .20 8 6

v v−= −

−

No, the Mean Value Theorem does not apply to v on [ ]8, 20 because v is not differentiable at 16.t =

2 : [ ]1 : average rate of change of on 8, 20 1 : answer with explanation

v⎧⎨⎩



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 6

1! 1 3! 1 5! 17 2 2 22! 4! 6!3 3 3P 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 2

2 1 ! 1 12 !3 3 2n n

nn n

−⋅ =

1 : coefficient

(c) The Taylor series for f about 2x = is

( ) ( )22

1

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

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩


Form B










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.





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 sin6tW t t= gallons per hour.

During the same time interval, water is removed from the tank at the rate

( ) ( )2275sin3tR 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.




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.









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-

= Ú 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= Ú 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.




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).


END OF EXAM


Form B






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) ( ) ( ) ( )( )40

0 ln 1 4t

y t y u du= + + −∫

( ) ( )( )2 40

2 5 ln 1 4 13.671y u du= + + − =∫ 3 :

( )( )2 40

1 : ln 1 4

1 : 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

⎧⎨⎩



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− = − <


(b) ( ) ( )( )18

01200 1309.788W t R t dt+ − =∫

1310 gallons


⎧⎪⎨⎪⎩

(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

⎧⎨⎩



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

⎧⎪⎪⎪⎨⎪⎪⎪⎩



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

⎧⎨⎩



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 dx

dt 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 2

yy 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⎧⎪⎨⎪⎩



7

Question 6

Consider the graph of the function f given by

( ) 12f 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) ( )2

0

0

12

2 2 2

k

R

k

V dxx

x k

π

π π π

=+

= − = −+ +

⌠⎮⌡

3 :

1 : limits 1 : integrand1 : antidifferentiation and

evaluation

⎧⎪⎪⎨⎪⎪⎩

(c) ( )2

12

lim 2 2

Sk

n

n k

V dxx

x k

π

π π→

∞

∞

=+

= − =+ +

⌠⎮⌡

S RV V=

2 2 2k kπ π π= −+ +

2 12 2k =

+

2k =

4 :

1 : improper integral1 : antidifferentiation and

evaluation 1 : equation 1 : answer

⎧⎪⎪⎪⎨⎪⎪⎪⎩



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© 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents).





1. Let R be the shaded region bounded by the graph of lny x= and the line 2,y x= - as shown above.


(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.

WRITE ALL WORK IN THE PINK EXAM BOOKLET.




2. At an intersection in Thomasville, Oregon, cars turn left at the rate ( ) ( )260 sin3t

L 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.




4

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 3

4

1

dy 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 tÆ•

(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.









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

4. 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 dtÚ in terms of the rocket’s flight. Use a midpoint

Riemann sum with 3 subintervals of equal length to approximate ( )70

10.v t dtÚ

(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.




6

5. Consider the differential equation 2 652

dyx

dx y= -

- for 2.y π Let ( )y f x= be the particular solution to this

differential equation with the initial condition ( )1 4.f - = -

(a) Evaluate dydx

and 2

2

d y

dx 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 Euler’s method, starting at 1x = - with two steps of equal size, to approximate ( )0 .f Show the work that leads to your answer.


( ) ( )2 3 12 32 3 4 1

n nnxx x xf xn

-= - + - + + +

+

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 !

n nxx x xg xn

-= - + - + + +

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.


END OF EXAM






© 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 3

34.198 or 34.199

T

Sx x dxπ= + − − +

=∫

3 : { 2 : integrand1 : limits, constant, and answer

(c) Volume ( )( )2 22

2( 2)

Ty

Sy e dyπ

−

−= + −⌠⎮

⌡

3 : { 2 : integrand1 : limits and constant



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


L t⎧⎪ ≥⎪⎨⎪⎪⎩



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 3

41

dy 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

⎧⎪⎪⎨⎪⎪⎩



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.


10v t dt∫ in terms of the rocket’s 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 11Bv t dt t C

t= = + +

+⌠⎮⌡

( )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)



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 Euler’s 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 2d 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


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

⎧⎨⎩



7

Question 6

The function f is defined by the power series

( ) ( )2 3 12 32 3 4 1L L

n 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 Ln 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 2n

n 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 and ( ) 10 .2f ′ = −

( ) 21 2 32! 4! 6!g x x x′ = − + − +L and ( ) 10 .2g′ = −

( ) ( ) ( )0 0 0 0y f g′ ′ ′= − =

( ) 40 3f ′′ = and ( ) 2 10 .4! 12g′′ = =

Thus, ( ) 4 10 0.3 12y′′ = − >

Since ( )0 0y′ = and ( )0 0,y′′ > y has a relative minimum at 0.x =

4 :

( )( )

1 : 0 1 : 01 : conclusion1 : reasoning

yy′⎧

⎪ ′′⎪⎨⎪⎪⎩


Form B










1. Let f be the function given by ( )3 2

3cos .4 3 2x x xf x x= - - + Let R be the shaded region in the second

quadrant bounded by the graph of f, and let S be the shaded region bounded by the graph of f and line , the

line tangent to the graph of f at 0,x = as shown above.


(b) Find the volume of the solid generated when R is rotated about the horizontal line 2.y = -

(c) Write, but do not evaluate, an integral expression that can be used to find the area of S.

WRITE ALL WORK IN THE EXAM BOOKLET.




2. An object moving along a curve in the xy-plane is at position ( ) ( )( ),x t y t at time t, where

( )tan tdx edt

-= and ( )sec tdye

dt-=

for 0.t ≥ At time 1,t = the object is at position ( )2, 3 .-

(a) Write an equation for the line tangent to the curve at position ( )2, 3 .-

(b) Find the acceleration vector and the speed of the object at time 1.t =

(c) Find the total distance traveled by the object over the time interval 1 2.t£ £

(d) Is there a time 0t ≥ at which the object is on the y-axis? Explain why or why not.




4

3. The figure above is the graph of a function of x, which models the height of a skateboard ramp. The function meets the following requirements.

(i) At 0,x = the value of the function is 0, and the slope of the graph of the function is 0.

(ii) At 4,x = the value of the function is 1, and the slope of the graph of the function is 1.

(iii) Between 0x = and 4,x = the function is increasing.

(a) Let ( ) 2 ,f x ax= where a is a nonzero constant. Show that it is not possible to find a value for a so that f meets requirement (ii) above.

(b) Let ( )2

3 ,16xg x cx= - where c is a nonzero constant. Find the value of c so that g meets requirement (ii)

above. Show the work that leads to your answer.

(c) Using the function g and your value of c from part (b), show that g does not meet requirement (iii) above.

(d) Let ( ) ,nxh x

k= where k is a nonzero constant and n is a positive integer. Find the values of k and n so

that h meets requirement (ii) above. Show that h also meets requirements (i) and (iii) above.









4. The rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the

function f. In the figure above, ( ) 3 21 3 14 2

f t t t= - + + for 0 4t£ £ and f is piecewise linear for 4 24.t£ £

(a) Find ( )22 .f ¢ Indicate units of measure.

(b) For the time interval 0 24,t£ £ at what time t is f increasing at its greatest rate? Show the reasoning that supports your answer.

(c) Find the total number of calories burned over the time interval 6 18t£ £ minutes.

(d) The setting on the machine is now changed so that the person burns ( )f t c+ calories per minute. For this setting, find c so that an average of 15 calories per minute is burned during the time interval 6 18.t£ £




6

5. Let f be a function with ( )4 1f = such that all points ( ),x y on the graph of f satisfy the differential equation

( )2 3 .dy

y xdx

= -

Let g be a function with ( )4 1g = such that all points ( ),x y on the graph of g satisfy the logistic differential equation

( )2 3 .dy

y ydx

= -

(a) Find ( ).y f x=

(b) Given that ( )4 1,g = find ( )limx

g xÆ•

and ( )lim .x

g xÆ•

¢ (It is not necessary to solve for ( )g x or to show how

you arrived at your answers.)

(c) For what value of y does the graph of g have a point of inflection? Find the slope of the graph of g at the point of inflection. (It is not necessary to solve for ( ).g x )

6. The function f is defined by ( )3

1 .1

f xx

=+

The Maclaurin series for f is given by

( )3 6 9 31 1 ,n nx x x x- + - + + - +

which converges to ( )f x for 1 1.x- < <

(a) Find the first three nonzero terms and the general term for the Maclaurin series for ( ).f x¢

(b) Use your results from part (a) to find the sum of the infinite series ( )2 5 8 3 1

3 6 9 31 .2 2 2 2

nnn-- + - + + - +

(c) Find the first four nonzero terms and the general term for the Maclaurin series representing ( )0

.x

f t dtÚ

(d) Use the first three nonzero terms of the infinite series found in part (c) to approximate ( )1 2

0.f t dtÚ What

are the properties of the terms of the series representing ( )1 2

0f t dtÚ that guarantee that this approximation

is within 110,000

of the exact value of the integral?


END OF EXAM


Form B






2

Question 1

Let f be the function given by ( )3 2

3cos .4 3 2x x xf x x= − − + Let R

be the shaded region in the second quadrant bounded by the graph of f, and let S be the shaded region bounded by the graph of f and line ,l the line tangent to the graph of f at 0,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 2.y = −

(c) Write, but do not evaluate, an integral expression that can be used to find the area of S.

For 0,x < ( ) 0f x = when 1.37312.x = − Let 1.37312.P = −

(a) Area of ( )0

2.903P

R f x dx= =∫


(b) Volume ( )( )( )0 22 4 59.361P

f x dxπ= + − =∫

4 : 1 : limits and constant

2 : integrand 1 : answer

⎧⎪⎨⎪⎩

(c) The equation of the tangent line l is 13 .2y x= −

The graph of f and line l intersect at 3.38987.A =

Area of ( ) ( )( )0

13 2

AS x f x dx= − −⌠⎮

⌡

3 : 1 : tangent line1 : integrand

1 : limits

⎧⎪⎨⎪⎩



3

Question 2

An object moving along a curve in the xy-plane is at position ( ) ( )( ),x t y t at time t, where

( )tan tdx edt−= and ( )sec tdy edt

−=

for 0.t ≥ At time 1,t = the object is at position ( )2, 3 .−

(a) Write an equation for the line tangent to the curve at position ( )2, 3 .−

(b) Find the acceleration vector and the speed of the object at time 1.t = (c) Find the total distance traveled by the object over the time interval 1 2.t≤ ≤ (d) Is there a time 0t ≥ at which the object is on the y-axis? Explain why or why not.

(a) ( )( ) ( )

sec 1tan sin

t

t t

dyedy dt

dx dx e edt

−

− −= = =

( ) ( )12, 3

1 2.780sin

dydx e−

−= = or 2.781

( ) ( )113 2

siny x

e−+ = −

2 : ( )2, 3 1 :

1 : equation of tangent line

dydx −

⎧⎪⎨⎪⎩

(b) ( )1 0.42253,x′′ = − ( )1 0.15196y′′ = − ( )1 0.423, 0.152a = − − or 0.422, 0.151 .− −

speed ( )( ) ( )( )2 21 1sec tan 1.138e e− −= + = or 1.139

2 : { 1 : acceleration vector 1 : speed

(c) ( )( ) ( )( )2 2 2

11.059x t y t dt′ ′+ =∫


(d) ( ) ( ) ( )1

00 1 2 0.775553 0x x x t dt′= − = − >∫

The particle starts to the right of the y-axis. Since ( ) 0x t′ > for all 0,t ≥ the object is always moving

to the right and thus is never on the y-axis.

3 : ( )( )

1 : 0 expression 1 : 01 : conclusion and reason

xx t

⎧⎪ ′ >⎨⎪⎩



4

Question 3

The figure above is the graph of a function of x, which models the height of a skateboard ramp. The function meets the following requirements.

(i) At 0,x = the value of the function is 0, and the slope of the graph of the function is 0. (ii) At 4,x = the value of the function is 1, and the slope of the graph of the function is 1. (iii) Between 0x = and 4,x = the function is increasing.

(a) Let ( ) 2 ,f x ax= where a is a nonzero constant. Show that it is not possible to find a value for a so that f meets requirement (ii) above.

(b) Let ( )2

3 ,16xg x cx= − where c is a nonzero constant. Find the value of c so that g meets requirement (ii)

above. Show the work that leads to your answer. (c) Using the function g and your value of c from part (b), show that g does not meet requirement (iii) above.

(d) Let ( ) ,nxh x k= where k is a nonzero constant and n is a positive integer. Find the values of k and n so that

h meets requirement (ii) above. Show that h also meets requirements (i) and (iii) above.

(a) ( )4 1f = implies that 116a = and ( ) ( )4 2 4 1f a′ = =

implies that 1 .8a = Thus, f cannot satisfy (ii).

2 : 1 1 1 : or 16 8

1 : shows does not work

a a

a

⎧ = =⎪⎨⎪⎩

(b) ( )4 64 1 1g c= − = implies that 1 .32c =

When 1 ,32c = ( ) ( ) ( ) ( )( )2 2 4 1 14 3 4 3 16 116 32 2g c′ = − = − =

1 : value of c

(c) ( ) ( )23 1 3 432 8 32xg x x x x′ = − = −

( ) 0g x′ < for 40 ,3x< < so g does not satisfy (iii).

2 : ( ) 1 : 1 : explanationg x′⎧

⎨⎩

(d) ( ) 44 1n

h k= = implies that 4 .n k=

( )1 14 44 144

n n

nn n nh k

− −′ = = = = gives 4n = and 44 256.k = =

( ) ( )4

0 0.256xh x h= ⇒ =

( ) ( )34 0 0256xh x h′ ′= ⇒ = and ( ) 0h x′ > for 0 4.x< <

4 : -1

4 1 : 1

4 1 : 1

1 : values for and 1 : verifications

n

nknk

k n

⎧ =⎪⎪⎪

=⎨⎪⎪⎪⎩



5

Question 4

The rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the function

f. In the figure above, ( ) 3 21 3 14 2f t t t= − + + for

0 4t≤ ≤ and f is piecewise linear for 4 24.t≤ ≤

(a) Find ( )22 .f ′ Indicate units of measure.

(b) For the time interval 0 24,t≤ ≤ at what time t is f increasing at its greatest rate? Show the reasoning that supports your answer.

(c) Find the total number of calories burned over the time interval 6 18t≤ ≤ minutes.

(d) The setting on the machine is now changed so that the person burns ( )f t c+ calories per minute. For this setting, find c so that an average of 15 calories per minute is burned during the time interval 6 18.t≤ ≤

(a) ( ) 15 322 320 24f −′ = = −−

calories/min/min

1 : ( )22f ′ and units

(b) f is increasing on [ ]0, 4 and on [ ]12, 16 .

On ( )12, 16 , ( ) 15 9 316 12 2f t −′ = =

− since f has

constant slope on this interval.

On ( )0, 4 , ( ) 23 34f t t t′ = − + and

( ) 3 3 02f t t′′ = − + = when 2.t = This is where f ′

has a maximum on [ ]0, 4 since 0f ′′ > on ( )0, 2 and 0f ′′ < on ( )2, 4 .

On [ ]0, 24 , f is increasing at its greatest rate when

2t = because ( ) 32 3 .2f ′ = >

4 :

( )( )

( ) ( )

1 : on 0, 41 : shows has a max at 2 on 0, 41 : shows for 12 16, 2

1 : answer

ff t

t f t f

′⎧⎪ ′ =⎪⎨ ′ ′< < <⎪⎪⎩

(c) ( ) ( ) ( )( ) ( )18

616 9 4 9 15 2 152

132 calories

f t dt = + + +

=∫

2 : { 1 : method1 : answer

(d) We want ( )( )18

61 15.12 f t c dt+ =∫

This means 132 12 15(12).c+ = So, 4.c =

OR

Currently, the average is 132 1112 = calories/min.

Adding c to ( )f t will shift the average by c. So 4c = to get an average of 15 calories/min.

2 : { 1 : setup1 : value of c



6

Question 5

Let f be a function with ( )4 1f = such that all points ( ),x y on the graph of f satisfy the differential equation

( )2 3 .dy y xdx = −

Let g be a function with ( )4 1g = such that all points ( ),x y on the graph of g satisfy the logistic differential equation

( )2 3 .dy y ydx = −

(a) Find ( ).y f x=

(b) Given that ( )4 1,g = find ( )limx

g x→∞

and ( )lim .x

g x→∞

′ (It is not necessary to solve for ( )g x or to show how

you arrived at your answers.) (c) For what value of y does the graph of g have a point of inflection? Find the slope of the graph of g at the

point of inflection. (It is not necessary to solve for ( ).g x )

(a) ( )2 3dy y xdx = −

( )1 2 3dy x dxy = −

2ln 6y x x C= − + 0 24 16 C= − +

8C = − 2ln 6 8y x x= − −

26 8x xy e − −= for x− < <∞ ∞

5 :

1 : separates variables 1 : antiderivatives1 : constant of integration1 : uses initial condition1 : solution

⎧⎪⎪⎨⎪⎪⎩

Note: max 2 5 [1-1-0-0-0] if no constant of integration Note: 0 5 if no separation of variables

(b) ( )lim 3x

g x→∞

=

( )lim 0

xg x

→∞′ =

2 : ( )( )

1 : lim 3

1 : lim 0x

x

g x

g x→

→

∞

∞

=⎧⎪⎨ ′ =⎪⎩

(c) 2

2 (6 4 )d y dyy dxdx= −

Because 0dydx ≠ at any point on the graph of g, the

concavity only changes sign at 3 ,2y = half the carrying

capacity.

( )( )3 2

3 3 92 32 2 2y

dydx =

= − =

2 :

3 2

3 1 : 2

1 : y

y

dydx =

⎧ =⎪⎪⎨⎪⎪⎩



7

Question 6

The function f is defined by ( ) 31 .

1f x

x=

+ The Maclaurin series for f is given by

( )3 6 9 31 1 ,n nx x x x− + − + + − +L L

which converges to ( )f x for 1 1.x− < <

(a) Find the first three nonzero terms and the general term for the Maclaurin series for ( ).f x′

(b) Use your results from part (a) to find the sum of the infinite series ( )2 5 8 3 13 6 9 31 .

2 2 2 2n

nn−− + − + + − +L L

(c) Find the first four nonzero terms and the general term for the Maclaurin series representing ( )0

.xf t dt∫

(d) Use the first three nonzero terms of the infinite series found in part (c) to approximate ( )1 2

0.f t dt∫ What are

the properties of the terms of the series representing ( )1 2

0f t dt∫ that guarantee that this approximation is

within 110,000 of the exact value of the integral?

(a) ( ) ( )2 5 8 3 13 6 9 3 1 n nf x x x x n x −′ = − + − + + − +L L

2 : { 1 : first three terms 1 : general term

(b) The given series is the Maclaurin series for ( )f x′ with 1 .2x =

( ) ( ) ( )23 21 3f x x x−

′ = − +

Thus, the sum of the series is ( ) ( )( )2

131 164 .2 2711 8

f ′ = − = −+

2 : ( )

( ) 1 :

1 1 : 2

f x

f

′⎧⎪⎨ ′⎪⎩

(c) ( ) 3 14 7 10

30

114 7 10 3 11

n nx xx x xdt x nt

+−= − + − + + +

++⌠⎮⌡

L L

2 : { 1 : first four terms 1 : general term

(d) ( ) ( )4 7

1 2

30

1 11 1 2 2 .2 4 71

dtt

≈ − ++

⌠⎮⌡

The series in part (c) with 12x = has terms that alternate, decrease in

absolute value, and have limit 0. Hence the error is bounded by the absolute value of the next term.

( ) ( ) ( )4 7 10

1 2

30

1 1 11 1 12 2 2 0.00012 4 7 10 102401

dtt

⎛ ⎞⎜ ⎟

− − + < = <⎜ ⎟+ ⎜ ⎟⎜ ⎟

⎝ ⎠

⌠⎮⌡

3 :

1 : approximation 1 : properties of terms 1 : absolute value of fourth term 0.0001

⎧⎪⎪⎨⎪⎪ <⎩




© 2007 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. 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.



GO ON TO THE NEXT PAGE. -2-




1. Let R be the region in the first and second quadrants bounded above by the graph of 2

20

1y

x=

+ and below

by the horizontal line 2.=y


(b) Find the volume of the solid generated when R is rotated about the x-axis.

(c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are semicircles. Find the volume of this solid.





2. The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval

0 7,£ £t where t is measured in hours. In this model, rates are given as follows:

(i) The rate at which water enters the tank is ( ) ( )2100 sin=f t t t gallons per hour for 0 7.£ £t

(ii) The rate at which water leaves the tank is

( ) { 250 for 0 32000 for 3 7

£ <=

< £t

g tt

gallons per hour.

The graphs of f and g, which intersect at 1.617=t and 5.076,=t are shown in the figure above. At time 0,t = the amount of water in the tank is 5000 gallons.

(a) How many gallons of water enter the tank during the time interval 0 7 ?£ £t Round your answer to the nearest gallon.

(b) For 0 7,t£ £ find the time intervals during which the amount of water in the tank is decreasing. Give a reason for each answer.

(c) For 0 7,£ £t at what time t is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer.




-4-

3. The graphs of the polar curves 2=r and 3 2cos q= +r are shown in the figure above. The curves intersect

when 23p

q = and 4 .3p

q =

(a) Let R be the region that is inside the graph of 2=r and also inside the graph of 3 2cos ,q= +r as shaded in the figure above. Find the area of R.

(b) A particle moving with nonzero velocity along the polar curve given by 3 2cos q= +r has position

( ) ( )( ),x t y t at time t, with 0q = when 0.=t This particle moves along the curve so that .q

=dr drdt d

Find

the value of drdt

at 3p

q = and interpret your answer in terms of the motion of the particle.

(c) For the particle described in part (b), .q

=dy dydt d

Find the value of dydt

at 3p

q = and interpret your answer

in terms of the motion of the particle.









4. Let f be the function defined for > 0,x with ( ) 2=f e and ,¢f the first derivative of f, given by

( ) 2 ln .=¢f x x x

(a) Write an equation for the line tangent to the graph of f at the point ( ), 2 .e

(b) Is the graph of f concave up or concave down on the interval 1 3 ?< <x Give a reason for your answer.

(c) Use antidifferentiation to find ( ).f x

t (minutes) 0 2 5 7 11 12

( )¢r t (feet per minute)

5.7 4.0 2.0 1.2 0.6 0.5

5. The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the

balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0 12,< <t the graph of r is concave down. The table above gives selected values of the rate of change, ( ),¢r t of the radius of the balloon over the time interval 0 12.£ £t The radius of the balloon is 30 feet when 5.=t

(Note: The volume of a sphere of radius r is given by 34 .3p=V r )

(a) Estimate the radius of the balloon when 5.4=t using the tangent line approximation at 5.=t Is your estimate greater than or less than the true value? Give a reason for your answer.

(b) Find the rate of change of the volume of the balloon with respect to time when 5.=t Indicate units of measure.

(c) Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate

( )12

0.¢Ú r t dt Using correct units, explain the meaning of ( )

12

0¢Ú r t dt in terms of the radius of the balloon.

(d) Is your approximation in part (c) greater than or less than ( )12

0?¢Ú r t dt Give a reason for your answer.




-6-

6. Let f be the function given by ( )2.-= xf x e

(a) Write the first four nonzero terms and the general term of the Taylor series for f about 0.=x

(b) Use your answer to part (a) to find ( )2

40

1lim .Æ

- -x

x f x

x

(c) Write the first four nonzero terms of the Taylor series for 2

0

-Úx te dt about 0.=x Use the first two terms of

your answer to estimate 21 2

0.-Ú te dt

(d) Explain why the estimate found in part (c) differs from the actual value of 21 2

0

-Ú te dt by less than 1 .200


END OF EXAM






Question 1


Let R be the region in the first and second quadrants bounded above by the graph of 220

1y

x=

+ and

below by the horizontal line 2.y =

(a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the x-axis. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the

x-axis are semicircles. Find the volume of this solid.

2

20 21 x

=+

when 3x = ±

1 : correct limits in an integral in (a), (b), or (c)

(a) Area 3

23

20 2 37.961 or 37.9621

dxx−

⎛ ⎞= − =⎜ ⎟⎝ ⎠+

⌠⎮⌡


(b) Volume 3 2

22

3

20 2 1871.1901

dxx

π−

⎛ ⎞⎛ ⎞= − =⎜ ⎟⎜ ⎟⎝ ⎠+⎝ ⎠

⌠⎮⌡


(c) Volume 3 2

233 2

23

1 20 22 2 1

20 2 174.2688 1

dxx

dxx

π

π−

−

⎛ ⎛ ⎞⎞= −⎜ ⎜ ⎟⎟⎝ ⎝ ⎠⎠+

⎛ ⎞= − =⎜ ⎟⎝ ⎠+

⌠⎮⌡

⌠⎮⌡



Question 2


The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval 0 7,t≤ ≤ where t is measured in hours. In this model, rates are given as follows: (i) The rate at which water enters the tank is

( ) ( )2100 sinf t t t= gallons per hour for 0 7.t≤ ≤ (ii) The rate at which water leaves the tank is

( ) 250 for 0 32000 for 3 7

tg t

t≤ <⎧= ⎨ < ≤⎩

gallons per hour.

The graphs of f and g, which intersect at 1.617t = and 5.076,t = are shown in the figure above. At time 0,t = the amount of water in the tank is 5000 gallons.

(a) How many gallons of water enter the tank during the time interval 0 7 ?t≤ ≤ Round your answer to the nearest gallon.

(b) For 0 7,t≤ ≤ find the time intervals during which the amount of water in the tank is decreasing. Give a reason for each answer.

(c) For 0 7,t≤ ≤ at what time t is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer.

(a) ( )7

08264f t dt ≈∫ gallons


(b) The amount of water in the tank is decreasing on the

intervals 0 1.617t≤ ≤ and 3 5.076t≤ ≤ because ( ) ( )f t g t< for 0 1.617t≤ < and 3 5.076.t< <

2 : { 1 : intervals1 : reason

(c) Since ( ) ( )f t g t− changes sign from positive to negative only at 3,t = the candidates for the absolute maximum are at 0, 3,t = and 7. t (hours) gallons of water

0 5000

3 ( ) ( )3

05000 250 3 5126.591f t dt+ − =∫

7 ( ) ( )7

35126.591 2000 4 4513.807f t dt+ − =∫

The amount of water in the tank is greatest at 3 hours. At that time, the amount of water in the tank, rounded to the nearest gallon, is 5127 gallons.

5 :

1 : identifies 3 as a candidate 1 : integrand 1 : amount of water at 3 1 : amount of water at 7 1 : conclusion

t

tt

=⎧⎪⎪

=⎨⎪ =⎪⎩


Question 3


The graphs of the polar curves 2r = and 3 2cosr θ= + are shown in

the figure above. The curves intersect when 23πθ = and 4 .3

πθ =

(a) Let R be the region that is inside the graph of 2r = and also inside the graph of 3 2cos ,r θ= + as shaded in the figure above. Find the area of R.

(b) A particle moving with nonzero velocity along the polar curve given by 3 2cosr θ= + has position ( ) ( )( ),x t y t at time t, with 0θ =

when 0.t = This particle moves along the curve so that .dr drdt dθ=

Find the value of drdt at 3

πθ = and interpret your answer in terms of the motion of the particle.

(c) For the particle described in part (b), .dy dydt dθ= Find the value of dy

dt at 3πθ = and interpret your

answer in terms of the motion of the particle. (a)

Area ( ) ( )4 32 22 3

2 12 3 2cos3 210.370

dπ

ππ θ θ= + +

=∫

4 :

1 : area of circular sector2 : integral for section of limaçon

1 : integrand 1 : limits and constant 1 : answer

⎧⎪⎪⎨⎪⎪⎩

(b) 3 3

1.732dr drdt dθ π θ πθ= =

= = −

The particle is moving closer to the origin, since 0drdt <

and 0r > when .3πθ =

2 : 3 1 :

1 : interpretation

drdt θ π=

⎧⎪⎨⎪⎩

(c) ( )sin 3 2cos siny r θ θ θ= = +

3 30.5dy dy

dt dθ π θ πθ= == =

The particle is moving away from the x-axis, since

0dydt > and 0y > when .3

πθ =

3 : 3

1 : expression for in terms of

1 :

1: interpretation

ydydt θ π

θ

=

⎧⎪⎪⎨⎪⎪⎩


Question 4


Let f be the function defined for 0,x > with ( ) 2f e = and ,f ′ the first derivative of f, given by ( ) 2 ln .f x x x′ =

(a) Write an equation for the line tangent to the graph of f at the point ( ), 2 .e

(b) Is the graph of f concave up or concave down on the interval 1 3 ?x< < Give a reason for your answer.

(c) Use antidifferentiation to find ( ).f x

(a) ( ) 2f e e′ = An equation for the line tangent to the graph of f at the

point ( ), 2e is ( )22 .y e x e− = −

2 : ( ) 1 : 1 : equation of tangent line

f e′⎧⎨⎩

(b) ( ) 2 ln .f x x x x′′ = + For 1 3,x< < 0x > and ln 0,x > so ( ) 0.f x′′ > Thus, the graph of f is concave up on ( )1, 3 .

3 : ( ) 2 : 1 : answer with reason

f x′′⎧⎨⎩

(c) Since ( ) ( )2 ln ,f x x x dx= ∫ we consider integration by

parts.

( )2

2 3

ln1 1

3

u x dv x dx

du dx v x dx xx

= =

= = =∫

Therefore,

( ) ( )( )

2

3 3

3 3

ln

1 1 1ln3 31 1ln .3 9

f x x x dx

x x x dxx

x x x C

=

= − ⋅

= − +

⌠⎮⌡

∫

Since ( ) 2,f e = 3 3

2 3 9e e C= − + and 322 .9C e= −

Thus, ( )3

3 31 2ln 2 .3 9 9xf x x x e= − + −

4 : ( )2 : antiderivative1 : uses 2

1 : answerf e

⎧⎪ =⎨⎪⎩


Question 5


t (minutes) 0 2 5 7 11 12

( )r t′ (feet per minute)

5.7 4.0 2.0 1.2 0.6 0.5

The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0 12,t< < the graph of r is concave down. The table above gives selected values of the rate of change,

( ) ,r t′ of the radius of the balloon over the time interval 0 12.t≤ ≤ The radius of the balloon is 30 feet when

5.t = (Note: The volume of a sphere of radius r is given by 34 .3V rπ= )

(a) Estimate the radius of the balloon when 5.4t = using the tangent line approximation at 5.t = Is your estimate greater than or less than the true value? Give a reason for your answer.

(b) Find the rate of change of the volume of the balloon with respect to time when 5.t = Indicate units of measure.

(c) Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate

( )12

0.r t dt′∫ Using correct units, explain the meaning of ( )

12

0r t dt′∫ in terms of the radius of the

balloon.

(d) Is your approximation in part (c) greater than or less than ( )12

0?r t dt′∫ Give a reason for your answer.

(a) ( ) ( ) ( ) ( )5.4 5 5 30 2 0.4 30.8r r r t′≈ + Δ = + = ft Since the graph of r is concave down on the interval 5 5.4,t< < this estimate is greater than ( )5.4 .r

2 : { 1 : estimate1 : conclusion with reason

(b) ( ) 243 3dV drrdt dtπ=

( )2 3

54 30 2 7200 ft min

t

dVdt π π

== =

3 : 2 :

1 : answer

dVdt

⎧⎪⎨⎪⎩

(c) ( ) ( ) ( ) ( ) ( ) ( )12

02 4.0 3 2.0 2 1.2 4 0.6 1 0.5r t dt′ ≈ + + + +∫

19.3= ft

( )12

0r t dt′∫ is the change in the radius, in feet, from

0t = to 12t = minutes.

2 : { 1 : approximation 1 : explanation

(d) Since r is concave down, r′ is decreasing on 0 12.t< <Therefore, this approximation, 19.3 ft, is less than

( )12

0.r t dt′∫

1 : conclusion with reason

Units of 3ft min in part (b) and ft in part (c) 1 : units in (b) and (c)


Question 6


Let f be the function given by ( )2.xf x e−=

(a) Write the first four nonzero terms and the general term of the Taylor series for f about 0.x =

(b) Use your answer to part (a) to find ( )2

40

1lim .x

x f xx→

− −

(c) Write the first four nonzero terms of the Taylor series for 2

0

x te dt−∫ about 0.x = Use the first two

terms of your answer to estimate 21 2

0.te dt−∫

(d) Explain why the estimate found in part (c) differs from the actual value of 21 2

0te dt−∫ by less than

1 .200

(a)

( ) ( ) ( ) ( )

( )

22 32 2 2 2

24 62

1 1! 2! 3! !11 2 6 !

nx

n n

x x x xe n

xx xx n

− − − − −= + + + + + +

−= − + − + + +

3 :

4 62 1 : two of 1, , ,2 6

1 : remaining terms 1 : general term

x xx⎧ − −⎪⎪⎨⎪⎪⎩

(b) ( ) ( ) 12 2 42

44

1 112 6 !

n n

n

x f x xxnx

+ −

=

∞− − −= − + + ∑

Thus, ( )2

40

1 1lim .2x

x f xx→

⎛ ⎞− −= −⎜ ⎟

⎝ ⎠

1 : answer

(c) ( )2 24 62

00

3 5 7

11 2 6 !

3 10 42

x n nx t tt te dt t dtn

x x xx

− ⎛ ⎞−= − + − + + +⎜ ⎟⎜ ⎟

⎝ ⎠

= − + − +

⌠⎮⌡

∫

Using the first two terms of this series, we estimate that

( )( )21 2

01 1 1 112 3 8 24

te dt− ≈ − =∫ .

3 : 1 : two terms1 : remaining terms

1 : estimate

⎧⎪⎨⎪⎩

(d) ( )2 51 2

011 1 1 1 1 ,24 2 10 320 200

te dt− − < ⋅ = <∫ since

( ) ( )( )

2

2 1

1 2

0 0

11 2 ,! 2 1

nn

t

ne dt n n

+

−

=

∞ −=

+∑∫ which is an alternating

series with individual terms that decrease in absolute value to 0.

2 : 1 : uses the third term as

the error bound 1 : explanation

⎧⎪⎨⎪⎩


Form B










1. Let R be the region bounded by the graph of 22 -= x xy e and the horizontal line 2,=y and let S be the region

bounded by the graph of 22 -= x xy e and the horizontal lines 1=y and 2,=y as shown above.



(c) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line 1.=y





2. An object moving along a curve in the xy-plane is at position ( ) ( )( ),x t y t at time t with

( )arctan1

=+

dx tdt t

and ( )2ln 1= +dyt

dt

for 0.≥t At time 0,=t the object is at position ( )3, 4 .- - (Note: 1tan arctan- =x x )

(a) Find the speed of the object at time 4.=t

(b) Find the total distance traveled by the object over the time interval 0 4.£ £t

(c) Find ( )4 .x

(d) For 0,>t there is a point on the curve where the line tangent to the curve has slope 2. At what time t is the object at this point? Find the acceleration vector at this point.




-4-

3. The wind chill is the temperature, in degrees Fahrenheit ( )F ,∞ a human feels based on the air temperature, in

degrees Fahrenheit, and the wind velocity v, in miles per hour ( )mph . If the air temperature is 32 F,∞ then the

wind chill is given by ( ) 0.1655.6 22.1= -W v v and is valid for 5 60.£ £v

(a) Find ( )20 .¢W Using correct units, explain the meaning of ( )20¢W in terms of the wind chill.

(b) Find the average rate of change of W over the interval 5 60.£ £v Find the value of v at which the instantaneous rate of change of W is equal to the average rate of change of W over the interval 5 60.£ £v

(c) Over the time interval 0 4£ £t hours, the air temperature is a constant 32 F.∞ At time 0,=t the wind velocity is 20=v mph. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate of change of the wind chill with respect to time at 3=t hours? Indicate units of measure.









4. Let f be a function defined on the closed interval 5 5x- £ £ with ( )1 3.f = The graph of ,f ¢ the derivative of f, consists of two semicircles and two line segments, as shown above.

(a) For 5 5,x- < < find all values x at which f has a relative maximum. Justify your answer.

(b) For 5 5,x- < < find all values x at which the graph of f has a point of inflection. Justify your answer.

(c) Find all intervals on which the graph of f is concave up and also has positive slope. Explain your reasoning.

(d) Find the absolute minimum value of ( )f x over the closed interval 5 5.x- £ £ Explain your reasoning.




-6-

5. Consider the differential equation 3 2 1.dy

x ydx

= + +

(a) Find 2

2

d y

dx in terms of x and y.

(b) Find the values of the constants m, b, and r for which rxy mx b e= + + is a solution to the differential equation.

(c) Let ( )y f x= be a particular solution to the differential equation with the initial condition ( )0 2.f = - Use

Euler’s method, starting at 0x = with a step size of 1 ,2

to approximate ( )1 .f Show the work that leads to

your answer.

(d) Let ( )y g x= be another solution to the differential equation with the initial condition ( )0 ,g k= where k

is a constant. Euler’s method, starting at 0x = with a step size of 1, gives the approximation ( )1 0.g ª Find the value of k.

6. Let f be the function given by ( ) 36 xf x e-= for all x.

(a) Find the first four nonzero terms and the general term for the Taylor series for f about 0.x =

(b) Let g be the function given by ( ) ( )0

.x

g x f t dt= Ú Find the first four nonzero terms and the general term for

the Taylor series for g about 0.x =

(c) The function h satisfies ( ) ( )h x k f ax= ¢ for all x, where a and k are constants. The Taylor series for h about 0x = is given by

( )2 3

1 .2! 3! !

nx x xh x x

n= + + + + + +

Find the values of a and k.


END OF EXAM


Form B





Question 1


Let R be the region bounded by the graph of 22x xy e −= and the

horizontal line and let S be the region bounded by the graph of 2,y =22x xy e −= and the horizontal lines and 1y = 2,y = as shown above.

(a) Find the area of R. (b) Find the area of S. (c) Write, but do not evaluate, an integral expression that gives the

volume of the solid generated when R is rotated about the horizontal line 1.y =

22 2x xe − = when 0.446057, 1.553943x =Let and 0.446057P = 1.553943Q =

(a) Area of ( )22 2 0.5Q

x x

P14dx−= − =⌠

⌡ ⎪⎩

R e

3 : ⎪⎨ 1 : integrand 1 : limits1 : answer

⎧

(b) when 2 22 1x xe − = 0,x =

Area of S e Area of R ( )222

01x x dx−= − −⌠

⌡− Area of 2.06016= 1.546R =

OR

( ) ( ) ( )2 222 2

01 1

0.219064 1.107886 0.219064 1.546

Px x x x

Qe dx Q P e d− −− + − ⋅ + −

= + + =

⌠ ⌠⌡ ⌡

1 x

⎪⎩

3 : ⎪⎨ 1 : integrand 1 : limits1 : answer

⎧

(c) Volume ( ) ( )2 2

22 1 2 1Q

x x

Pe dπ −⎛ ⎞− −⎜ ⎟

⎝ ⎠⌠⎮⌡

x= −

3 : { 2 : integrand1 : constant and limits


Question 2


An object moving along a curve in the xy-plane is at position ( ) ( )( ),x t y t at time t with

( )arctan 1dx tdt t=

+ and ( )2ln 1dy tdt = +

for At time the object is at position 0.t ≥ 0,t = ( )3, 4 .− − (Note: 1tan arctanx x− = )

(a) Find the speed of the object at time 4.t =(b) Find the total distance traveled by the object over the time interval 0 4.t≤ ≤ (c) Find ( )4 .x

(d) For there is a point on the curve where the line tangent to the curve has slope 2. At what time t is the object at this point? Find the acceleration vector at this point.

0,t >

(a) Speed ( ) ( )2 24 4 2.912x y′ ′= + =

1 : speed at 4t =

(b) Distance 4 2 2

06.423dydx dtdt dt

⎛ ⎞ ⎛ ⎞= + =⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⌠⎮⌡

2 : { 1 : integral 1 : answer

(c) ( ) ( ) ( )4

04 0

3 2.10794 0.892

x x x t dt′= +

= − + = −

∫

3 : ( ) 1 : integrand

2 : 1 : uses 0 3

1 : answerx

⎧ ⎧⎪ ⎨ = −⎨ ⎩⎪⎩

(d) The slope is 2, so 2,

dydtdxdt

= or ( ) ( )2ln 1 2arctan .1tt t+ =+

Since At this time, the acceleration is 0,t > 1.35766.t =

( ) ( ) 1.35766, 0.135, 0.955 .tx t y t =′′ ′′ =

3 : 1 : 2

1 : -value1 : values for and

dydtdxdt

tx y

⎧⎪ =⎪⎪⎨⎪⎪

′′ ′′⎪⎩


Question 3


The wind chill is the temperature, in degrees Fahrenheit ( )F ,° a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity v, in miles per hour ( )mph . If the air temperature is 32 then the

wind chill is given by and is valid for 5 6

F,°

( ) 0.1655.6 22.1W v v= − 0.v≤ ≤ (a) Find ( )20 .W ′ Using correct units, explain the meaning of ( )20W ′ in terms of the wind chill.

(b) Find the average rate of change of W over the interval 5 60.v≤ ≤ Find the value of v at which the instantaneous rate of change of W is equal to the average rate of change of W over the interval 5 60.v≤ ≤

(c) Over the time interval hours, the air temperature is a constant 32 At time the wind velocity is mph. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate of change of the wind chill with respect to time at

0 t≤ ≤ 4 F.° 0,t =20v =

3t = hours? Indicate units of measure.

(a) or ( ) 0.8420 22.1 0.16 20 0.285W −′ = − ⋅ ⋅ = − 0.286−

When mph, the wind chill is decreasing at 20v =0.286 F mph.°

2 : { 1 : value1 : explanation

(b) The average rate of change of W over the interval

is 5 60v≤ ≤( ) ( )60 5 0.25360 5

W W−= −

− or 0.254.−

( ) ( ) ( )60 560 5

W WW v −′ =−

when 23.011.v =

3 : ( ) 1 : average rate of change 1 : average rate of change 1 : value of W v

v

⎧⎪ ′ =⎨⎪⎩

(c) ( ) ( )3 3

35 5 0.892 F hrt t

dW dW dv Wdt dv dt= =′= ⋅ = ⋅ = − °

OR ( )0.1655.6 22.1 20 5W t= − +

3

0.892 F hrt

dWdt =

= − °

3 : ( )

( )

1 : 5

1 : uses 3 35, or uses 20 5 1 : answer

dvdt

v

v t t

⎧ =⎪⎪ =⎪⎨⎪

= +⎪⎪⎩

1 : units in (a) and (c) Units of F mph° in (a) and F hr° in (c)


Question 4


Let f be a function defined on the closed interval 5 5x− ≤ ≤ with ( )1 3f = . The graph of ,f ′ the derivative of f, consists of two semicircles and two line segments, as shown above.

(a) For − < find all values x at which f has a relative maximum. Justify your answer.

5 x 5,<

5,<(b) For − < find all values x at which the graph of f has a point of inflection. Justify your answer.

5 x

(c) Find all intervals on which the graph of f is concave up and also has positive slope. Explain your reasoning.

(d) Find the absolute minimum value of ( )f x over the closed interval 5 x 5.− ≤ ≤ Explain your reasoning.

(a) ( ) 0f x′ = at 1, 4 3,x = −f ′ changes from positive to negative at 3− and 4.

Thus, f has a relative maximum at 3x = − and at 4.x =

2 : { 1 : -values1 : justificationx

(b) f ′ changes from increasing to decreasing, or vice versa, at and 2. Thus, the graph of f has points of

inflection when and 2. 4,x = − 1,−

4,x = − 1,−

2 : { 1 : -values1 : justificationx

(c) The graph of f is concave up with positive slope where f ′ is increasing and positive: and 1 25 4x− < < − .x< <

2 : { 1 : intervals1 : explanation

(d) Candidates for the absolute minimum are where f ′ changes from negative to positive (at 1x = ) and at the endpoints ( ). 5, 5x = −

( ) ( )5

15 3 3 22f f x dx π π

−′− = + = − + >∫ 3

( )1 3f =

( ) ( )5

13 2 15 3 3 2 2f f x dx ⋅′= + = + − >∫ 3

The absolute minimum value of f on [ ]5, 5− is ( )1 3f .=

3 : 1 : identifies 1 as a candidate 1 : considers endpoints 1 : value and explanation

x =⎧⎪⎨⎪⎩


Question 5


Consider the differential equation 3 2 1dy x ydx = + + .

(a) Find 2

2d ydx

in terms of x and y.

(b) Find the values of the constants m, b, and r for which is a solution to the differential equation.

rxy mx b e= + +

(c) Let ( )y f x= be a particular solution to the differential equation with the initial condition ( )0f 2.= − Use

Euler’s method, starting at with a step size of 0x = 1 ,2 to approximate ( )1 .f Show the work that leads to

your answer. (d) Let ( )y g x= be another solution to the differential equation with the initial condition ( )0 ,g k= where k is

a constant. Euler’s method, starting at 0x = with a step size of 1, gives the approximation ( )1 0g .≈ Find the value of k.

2 : 1 : 3 2

1 : answer

dydx

⎧ +⎪⎨⎪⎩

(a) ( )2

2 3 2 3 2 3 2 1 6 4 5d y dy x y x ydxdx= + = + + + = + +

(b) If is a solution, then rxy mx b e= + +

( )3 2 1rx rxm re x mx b e+ = + + + + .

= + 2,r = ,

If 0 m b 0 3

so

:r ≠ 2 1, 2m= +3 ,2m = − and 2,r = 5 .4b = −

OR

If 0 m b

so

:r = 2 3,= + 0,r = ,0 3 2m= +3 ,2m = − 0,r = 9 .4b = −

3 : 1 :

1 : value for 1 : values for and

rxdy m redxrm b

⎧ = +⎪⎪⎨⎪⎪⎩

(c) ( ) ( ) ( ) ( )1 10 0 2 32 2f f f ′≈ + ⋅ = − + − ⋅ = −1 72 2

( ) ( ) ( )1 1 73 2 12 2 2f ′ ≈ + − + = − 92

( ) ( ) ( ) ( )1 1 1 7 9 11 2 2 2 2 2 2f f f ′≈ + ⋅ = − + − ⋅ = − 234

2 : ( )

1 : Euler's method with 2 steps1 : Euler's approximation for 1f

⎧⎨⎩

(d) ( )0 3 0 2 1 2g k′ = ⋅ + ⋅ + = +2 : ( ) ( )1 : 0 0 1

1 : value of g g

k′+ ⋅⎧

⎨⎩

1k

( ) ( ) ( ) ( )1 0 0 1 2 1 3 1g g g k k k′≈ + ⋅ = + + = + = 0 13k = −


Question 6


Let f be the function given by ( ) 36 xf x e−= for all x.

(a) Find the first four nonzero terms and the general term for the Taylor series for f about 0.x =

(b) Let g be the function given by ( ) ( )0

.x

g x f t= ∫ dt Find the first four nonzero terms and the general term for the Taylor

series for g about 0.x =(c) The function h satisfies ( ) ( )h x k f ax′= for all x, where a and k are constants. The Taylor series for h about 0x =

is given by

( )2 3

1 .2! 3! !

nx x xh x xn

= + + + + + +

Find the values of a and k.

(a) ( ) ( )

( )

2 3

2 3

2 3

16 1 3 2!3 3!3 !3

6 16 2 3 27 !3

n n

n

n n

n

xx x xf xn

xx xxn

⎡ ⎤−= − + − + + +⎢ ⎥

⎣ ⎦−

= − + − + + +

3 :

2 3 1 : two of 6, 2 , ,3 27 1 : remaining terms 1 : general term

x xx⎧− −⎪⎪

⎨⎪⎪⎩

1− missing factor of 6

(b) ( )0g = 0 and ( ) ( ) ,g x f x′ = so

( ) ( )

( )

( )( )( )

12 3 4

2 3

13 42

16 6 3!3 4!3 1 !3

6 16 9 4 27 1 !3

n n

n

n n

n

xx x xg x xn

xx xx xn

+

+

⎡ ⎤−= − + − + + +⎢ ⎥

+⎣ ⎦−

= − + − + + ++

3 : 1 : two terms1 : remaining terms 1 : general term

⎧⎪⎨⎪⎩

1− missing factor of 6

(c) ( ) 32 ,xf x e−′ = − so ( ) 32 axh x k e−= −

( )2 3

1 2! 3! !n

xx x xh x x en= + + + + + + =

32 ax xk e e−− =

13a− = and 2 1k− =

and 3a = − 12k = −

OR

( ) 22 ,3f x x′ = − + + so

( ) ( )

( )

22 31

h x kf ax k ak x

h x x

′= = − + +

= + +

2k− = 1 and 2 13 ak =

12k = − and 3a = −

3 :

( )( )

( )

1 : computes

1 : recognizes , or equates 2 series for 1 : values for and

x

k f ax

h x e

h xa k

′⎧⎪ =⎪⎪⎨⎪⎪⎪⎩



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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.







1. Let R be the region bounded by the graphs of siny xp and 3 4 ,y x x as shown in the figure above.


(b) The horizontal line 2y splits the region R into two parts. Write, but do not evaluate, an integral expression for the area of the part of R that is below this horizontal line.

(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of this solid.

(d) The region R models the surface of a small pond. At all points in R at a distance x from the y-axis, the depth of the water is given by 3 .h x x Find the volume of water in the pond.





t (hours) 0 1 3 4 7 8 9

L t (people) 120 156 176 126 150 80 0

2. Concert tickets went on sale at noon 0t and were sold out within 9 hours. The number of people waiting

in line to purchase tickets at time t is modeled by a twice-differentiable function L for 0 9.t Values of L t at various times t are shown in the table above.

(a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5:30 P.M. 5.5 .t Show the computations that lead to your answer. Indicate units of measure.

(b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first 4 hours that tickets were on sale.

(c) For 0 9,t what is the fewest number of times at which L t must equal 0 ? Give a reason for your answer.

(d) The rate at which tickets were sold for 0 9t is modeled by 2550 tr t te tickets per hour. Based on

the model, how many tickets were sold by 3 P.M. 3 ,t to the nearest whole number?





x h x h x h x h x 4h x

1 11 30 42 99 18

2 80 128 4883

4483

5849

3 317 7532

13834

348316

112516

3. Let h be a function having derivatives of all orders for 0.x Selected values of h and its first four derivatives

are indicated in the table above. The function h and these four derivatives are increasing on the interval 1 3.x

(a) Write the first-degree Taylor polynomial for h about 2x and use it to approximate 1.9 .h Is this

approximation greater than or less than 1.9 ?h Explain your reasoning.

(b) Write the third-degree Taylor polynomial for h about 2x and use it to approximate 1.9 .h

(c) Use the Lagrange error bound to show that the third-degree Taylor polynomial for h about 2x

approximates 1.9h with error less than 43 10 .









4. A particle moves along the x-axis so that its velocity at time t, for 0 6,t is given by a differentiable function v whose graph is shown above. The velocity is 0 at 0,t 3,t and 5,t and the graph has horizontal tangents at 1t and 4.t The areas of the regions bounded by the t-axis and the graph of v on the intervals 0, 3 , 3, 5 , and 5, 6 are 8, 3, and 2, respectively. At time 0,t the particle is at 2.x

(a) For 0 6,t find both the time and the position of the particle when the particle is farthest to the left. Justify your answer.

(b) For how many values of t, where 0 6,t is the particle at 8 ?x Explain your reasoning.

(c) On the interval 2 3,t is the speed of the particle increasing or decreasing? Give a reason for your answer.

(d) During what time intervals, if any, is the acceleration of the particle negative? Justify your answer.





5. The derivative of a function f is given by 3 xf x x e for 0,x and 1 7.f

(a) The function f has a critical point at 3.x At this point, does f have a relative minimum, a relative maximum, or neither? Justify your answer.

(b) On what intervals, if any, is the graph of f both decreasing and concave up? Explain your reasoning.

(c) Find the value of 3 .f




-7-

6. Consider the logistic differential equation 6 .8

dy yy

dt Let y f t be the particular solution to the

differential equation with 0 8.f

(a) A slope field for this differential equation is given below. Sketch possible solution curves through the points 3, 2 and 0, 8 .

(Note: Use the axes provided in the exam booklet.)

(b) Use Euler’s method, starting at 0t with two steps of equal size, to approximate 1 .f

(c) Write the second-degree Taylor polynomial for f about 0,t and use it to approximate 1 .f

(d) What is the range of f for 0 ?t


END OF EXAM




© 2008 The College Board. All rights reserved. College Board, AP Central, Advanced Placement Program, AP, SAT, and the acorn logo are registered trademarks of 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 online home for AP teachers: apcentral.collegeboard.com.


Question 1

© 2008 The College Board. All rights reserved. Visit the College Board on the Web: www.collegeboard.com.

Let R be the region bounded by the graphs of ( )siny xπ= and 3 4 ,y x x= − as shown in the figure above. (a) Find the area of R. (b) The horizontal line 2y = − splits the region R into two parts. Write, but do not evaluate, an integral

expression for the area of the part of R that is below this horizontal line. (c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a

square. Find the volume of this solid. (d) The region R models the surface of a small pond. At all points in R at a distance x from the y-axis,

the depth of the water is given by ( ) 3 .h x x= − Find the volume of water in the pond.

(a) ( ) 3sin 4x x xπ = − at 0x = and 2x =

Area ( ) ( )( )2 30

sin 4 4x x x dxπ= − − =∫


⎧⎪⎨⎪⎩

(b) 3 4 2x x− = − at 0.5391889r = and 1.6751309s =

The area of the stated region is ( )( )32 4s

rx x dx− − −∫

2 : { 1 : limits1 : integrand

(c) Volume ( ) ( )( )2 230

sin 4 9.978x x x dxπ= − − =∫ 2 : { 1 : integrand1 : answer

(d)

Volume ( ) ( ) ( )( )2 30

3 sin 4 8.369 or 8.370x x x x dxπ= − − − =∫ 2 : { 1 : integrand1 : answer


Question 2


t (hours) 0 1 3 4 7 8 9

( )L t (people) 120 156 176 126 150 80 0

Concert tickets went on sale at noon ( )0t = and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by a twice-differentiable function L for 0 9.t≤ ≤ Values of ( )L t at various times t are shown in the table above. (a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at

5:30 P.M. ( )5.5 .t = Show the computations that lead to your answer. Indicate units of measure.

(b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first 4 hours that tickets were on sale.

(c) For 0 9,t≤ ≤ what is the fewest number of times at which ( )L t′ must equal 0 ? Give a reason for your answer.

(d) The rate at which tickets were sold for 0 9t≤ ≤ is modeled by ( ) 2550 tr t te−= tickets per hour. Based on the model, how many tickets were sold by 3 P.M. ( )3 ,t = to the nearest whole number?

(a) ( ) ( ) ( )7 4 150 1265.5 87 4 3L LL − −′ ≈ = =− people per hour 2 : { 1 : estimate

1 : units

(b) The average number of people waiting in line during the first 4 hours is approximately

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 1 1 3 3 41 1 0 (3 1) 4 34 2 2 2L L L L L L+ + +⎛ ⎞− + − + −⎜ ⎟

⎝ ⎠

155.25= people

2 : { 1 : trapezoidal sum 1 : answer

(c) L is differentiable on [ ]0, 9 so the Mean Value Theorem implies ( ) 0L t′ > for some t in ( )1, 3 and some t in ( )4, 7 . Similarly, ( ) 0L t′ < for some t in ( )3, 4 and some t in ( )7, 8 . Then, since L′ is

continuous on [ ]0, 9 , the Intermediate Value Theorem implies that ( ) 0L t′ = for at least three values of t in [ ]0, 9 .

OR

The continuity of L on [ ]1, 4 implies that L attains a maximum value there. Since ( ) ( )3 1L L> and ( ) ( )3 4 ,L L> this maximum occurs on ( )1, 4 . Similarly, L attains a minimum on ( )3, 7 and a maximum on ( )4, 8 . L is differentiable, so ( ) 0L t′ = at each relative extreme point on ( )0, 9 . Therefore ( ) 0L t′ = for at least three values of t in [ ]0, 9 . [Note: There is a function L that satisfies the given conditions with

( ) 0L t′ = for exactly three values of t.]

3 :

1 : considers change in sign of 1 : analysis 1 : conclusion

L⎧⎪ ′⎪⎨⎪⎪⎩

OR

3 : ( )

1 : considers relative extrema of on 0, 9 1 : analysis 1 : conclusion

L⎧⎪⎪⎨⎪⎪⎩

(d) ( )3

0972.784r t dt =∫

There were approximately 973 tickets sold by 3 P.M.

2 : { 1 : integrand1 : limits and answer


Question 3


x ( )h x ( )h x′ ( )h x′′ ( )h x′′′ ( ) ( )4h x

1 11 30 42 99 18

2 80 128 4883 448

3 5849

3 317 7532 1383

4 348316 1125

16

Let h be a function having derivatives of all orders for 0.x > Selected values of h and its first four derivatives are indicated in the table above. The function h and these four derivatives are increasing on the interval 1 3.x≤ ≤

(a) Write the first-degree Taylor polynomial for h about 2x = and use it to approximate ( )1.9 .h Is this approximation greater than or less than ( )1.9 ?h Explain your reasoning.

(b) Write the third-degree Taylor polynomial for h about 2x = and use it to approximate ( )1.9 .h

(c) Use the Lagrange error bound to show that the third-degree Taylor polynomial for h about 2x = approximates ( )1.9h with error less than 43 10 .−×

(a) ( ) ( )1 80 128 2 ,P x x= + − so ( ) ( )11.9 1.9 67.2h P≈ =

( ) ( )1 1.9 1.9P h< since h′ is increasing on the interval 1 3.x≤ ≤

4 : ( )( )( ) ( )

1

1

1

2 : 1 : 1.91 : 1.9 1.9 with reason

P xPP h

⎧⎪⎨⎪ <⎩

(b) ( ) ( ) ( ) ( )2 33

488 44880 128 2 2 26 18P x x x x= + − + − + −

( ) ( )31.9 1.9 67.988h P≈ =

3 : ( )( )

3

3

2 : 1 : 1.9

P xP

⎧⎨⎩

(c)

The fourth derivative of h is increasing on the interval

1 3,x≤ ≤ so ( ) ( )41.9 2

584max .9xh x

≤ ≤=

Therefore, ( ) ( )4

3

4

4

1.9 25841.9 1.9 9 4!2.7037 103 10

h P−

−

−− ≤

= ×

< ×

2 : { 1 : form of Lagrange error estimate 1 : reasoning


Question 4


A particle moves along the x-axis so that its velocity at time t, for 0 6,t≤ ≤ is given by a differentiable function v whose graph is shown above. The velocity is 0 at 0,t = 3,t = and 5,t = and the graph has horizontal tangents at 1t = and 4.t = The areas of the regions bounded by the t-axis and the graph of v on the intervals [ ]0, 3 , [ ]3, 5 , and [ ]5, 6 are 8, 3, and 2, respectively. At time 0,t = the particle is at 2.x = −

(a) For 0 6,t≤ ≤ find both the time and the position of the particle when the particle is farthest to the left. Justify your answer.

(b) For how many values of t, where 0 6,t≤ ≤ is the particle at 8 ?x = − Explain your reasoning.

(c) On the interval 2 3,t< < is the speed of the particle increasing or decreasing? Give a reason for your answer.

(d) During what time intervals, if any, is the acceleration of the particle negative? Justify your answer.

(a) Since ( ) 0v t < for 0 3t< < and 5 6,t< < and ( ) 0v t > for 3 5,t< < we consider 3t = and 6.t =

( ) ( )3

03 2 2 8 10x v t dt= − + = − − = −∫

( ) ( )6

06 2 2 8 3 2 9x v t dt= − + = − − + − = −∫

Therefore, the particle is farthest left at time 3t = when its position is ( )3 10.x = −

3 : ( )6

0

1 : identifies 3 as a candidate

1 : considers

1 : conclusion

t

v t dt

=⎧⎪⎪⎨⎪⎪⎩

∫

(b)

The particle moves continuously and monotonically from ( )0 2x = − to ( )3 10.x = − Similarly, the particle moves

continuously and monotonically from ( )3 10x = − to ( )5 7x = − and also from ( )5 7x = − to ( )6 9.x = −

By the Intermediate Value Theorem, there are three values of t for which the particle is at ( ) 8.x t = −

3 :

1 : positions at 3, 5, and 6 1 : description of motion 1 : conclusion

t tt

= =⎧⎪ =⎪⎨⎪⎪⎩

(c) The speed is decreasing on the interval 2 3t< < since on this interval 0v < and v is increasing.


(d) The acceleration is negative on the intervals 0 1t< < and 4 6t< < since velocity is decreasing on these intervals. 2 : { 1 : answer

1 : justification


Question 5


The derivative of a function f is given by ( ) ( )3 xf x x e′ = − for 0,x > and ( )1 7.f =

(a) The function f has a critical point at 3.x = At this point, does f have a relative minimum, a relative maximum, or neither? Justify your answer.

(b) On what intervals, if any, is the graph of f both decreasing and concave up? Explain your reasoning. (c) Find the value of ( )3 .f

(a)

( ) 0f x′ < for 0 3x< < and ( ) 0f x′ > for 3x > Therefore, f has a relative minimum at 3.x =

2 : 1: minimum at 3

1: justificationx =⎧

⎨⎩

(b) ( ) ( ) ( )3 2x x xf x e x e x e′′ = + − = − ( ) 0f x′′ > for 2x >

( ) 0f x′ < for 0 3x< <

Therefore, the graph of f is both decreasing and concave up on the interval 2 3.x< <

3 : ( ) 2 : 1 : answer with reason

f x′′⎧⎨⎩

(c) ( ) ( ) ( ) ( )3 3

1 13 1 7 3 xf f f x dx x e dx′= + = + −∫ ∫

3 x

xu x dv e dxdu dx v e

= − =

= =

( ) ( )

( )( )

3 3

113

13

3 7 3

7 3

7 3

x x

x x

f x e e dx

x e e

e e

= + − −

= + − −

= + −

∫

4: 1 : uses initial condition2 : integration by parts

1 : answer

⎧⎪⎨⎪⎩


Question 6


Consider the logistic differential equation ( )6 .8dy y ydt = − Let ( )y f t= be the particular solution to the

differential equation with ( )0 8.f =

(a) (b)

(c)

(d)

A slope field for this differential equation is given below. Sketch possible solution curves through the points ( )3, 2 and ( )0, 8 . (Note: Use the axes provided in the exam booklet.) Use Euler’s method, starting at 0t = with two steps of equal size, to approximate ( )1 .f

Write the second-degree Taylor polynomial for f about 0,t = and use it to approximate ( )1 .f

What is the range of f for 0 ?t ≥

(a)

2 : ( )( )

1: solution curve through 0,81: solution curve through 3,2

⎧⎨⎩

(b) ( ) ( )( )1 18 2 72 2f ≈ + − =

( ) ( )( )7 1 1051 7 8 2 16f ≈ + − =

2 : ( )1 : Euler’s method with two steps

1 : approximation of 1f⎧⎨⎩

(c) ( ) ( )2

21 68 8

d y dy y dyydt dtdt= − + −

( ) ( ) ( )0

80 8; 0 6 8 2;8t

dyf f dt =′= = = − = − and

( ) ( )( ) ( )2

20

1 8 50 2 2 28 8 2t

d yfdt =

′′ = = − − + =

The second-degree Taylor polynomial for f about

0t = is ( ) 22

58 2 .4P t t t= − +

( ) ( )2291 1 4f P≈ =

4 :

( )

2

2 2 :

1 : second-degree Taylor polynomial 1 : approximation of 1

d ydt

f

⎧⎪⎪⎨⎪⎪⎩

(d) The range of f for 0t ≥ is 6 8y< ≤ . 1 : answer


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 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

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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.

© 2008 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, SAT, and the

acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and

National Merit Scholarship Corporation.

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.







1. A particle moving along a curve in the xy-plane has position ,x t y t at time 0t with

3dx tdt

and 2

3cos .2

dy tdt

The particle is at position 1, 5 at time 4.t

(a) Find the acceleration vector at time 4.t

(b) Find the y-coordinate of the position of the particle at time 0.t

(c) On the interval 0 4,t at what time does the speed of the particle first reach 3.5 ?

(d) Find the total distance traveled by the particle over the time interval 0 4.t





2. For time 0t hours, let 210120 1 tr t e represent the speed, in kilometers per hour, at which a car

travels along a straight road. The number of liters of gasoline used by the car to travel x kilometers is modeled

by 20.05 1 .xg x x e

(a) How many kilometers does the car travel during the first 2 hours?

(b) Find the rate of change with respect to time of the number of liters of gasoline used by the car when 2t hours. Indicate units of measure.

(c) How many liters of gasoline have been used by the car when it reaches a speed of 80 kilometers per hour?




-4-

Distance from the river’s edge (feet)

0 8 14 22 24

Depth of the water (feet) 0 7 8 2 0

3. A scientist measures the depth of the Doe River at Picnic Point. The river is 24 feet wide at this location. The

measurements are taken in a straight line perpendicular to the edge of the river. The data are shown in the table above. The velocity of the water at Picnic Point, in feet per minute, is modeled by 16 2sin 10v t t for

0 120t minutes.

(a) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the area of the cross section of the river at Picnic Point, in square feet. Show the computations that lead to your answer.

(b) The volumetric flow at a location along the river is the product of the cross-sectional area and the velocity of the water at that location. Use your approximation from part (a) to estimate the average value of the volumetric flow at Picnic Point, in cubic feet per minute, from 0t to 120t minutes.

(c) The scientist proposes the function f, given by 8sin ,24px

f x as a model for the depth of the water,

in feet, at Picnic Point x feet from the river’s edge. Find the area of the cross section of the river at Picnic Point based on this model.

(d) Recall that the volumetric flow is the product of the cross-sectional area and the velocity of the water at a location. To prevent flooding, water must be diverted if the average value of the volumetric flow at Picnic Point exceeds 2100 cubic feet per minute for a 20-minute period. Using your answer from part (c), find the average value of the volumetric flow during the time interval 40 60t minutes. Does this value indicate that the water must be diverted?









4. Let f be the function given by 2 3,f x kx x where k is a positive constant. Let R be the region in the first quadrant bounded by the graph of f and the x-axis.

(a) Find all values of the constant k for which the area of R equals 2.

(b) For 0,k write, but do not evaluate, an integral expression in terms of k for the volume of the solid generated when R is rotated about the x-axis.

(c) For 0,k write, but do not evaluate, an expression in terms of k, involving one or more integrals, that gives the perimeter of R.





5. Let g be a continuous function with 2 5.g The graph of the piecewise-linear function ,g the derivative of g, is shown above for 3 7.x

(a) Find the x-coordinate of all points of inflection of the graph of y g x for 3 7.x Justify your answer.

(b) Find the absolute maximum value of g on the interval 3 7.x Justify your answer.

(c) Find the average rate of change of g x on the interval 3 7.x

(d) Find the average rate of change of g x on the interval 3 7.x Does the Mean Value Theorem

applied on the interval 3 7x guarantee a value of c, for 3 7,c such that g c is equal to this average rate of change? Why or why not?




-7-

6. Let f be the function given by 2

2 .1

xf xx

(a) Write the first four nonzero terms and the general term of the Taylor series for f about 0.x

(b) Does the series found in part (a), when evaluated at 1,x converge to 1 ?f Explain why or why not.

(c) The derivative of 2ln 1 x is 2

2 .1

x

x Write the first four nonzero terms of the Taylor series for

2ln 1 x about 0.x

(d) Use the series found in part (c) to find a rational number A such that 5 1ln .4 100

A Justify your

answer.


END OF EXAM


Form B



© 2008 The College Board. All rights reserved. College Board, AP Central, Advanced Placement Program, AP, SAT, and the acorn logo are registered trademarks of 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 online home for AP teachers: apcentral.collegeboard.com.


Question 1


A particle moving along a curve in the xy-plane has position ( ) ( )( ),x t y t at time 0t ≥ with

3dx tdt = and 2

3cos .2dy tdt

⎛ ⎞= ⎜ ⎟⎝ ⎠

The particle is at position ( )1, 5 at time 4.t =

(a) Find the acceleration vector at time 4.t = (b) Find the y-coordinate of the position of the particle at time 0.t = (c) On the interval 0 4,t≤ ≤ at what time does the speed of the particle first reach 3.5 ?

(d) Find the total distance traveled by the particle over the time interval 0 4.t≤ ≤

(a) ( ) ( ) ( )4 4 , 4 0.433, 11.872a x y′′ ′′= = −

1 : answer

(b)

( )0 2

40 5 3cos 1.6002

ty dt⎛ ⎞= + =⎜ ⎟⎝ ⎠

⌠⎮⌡

or 1.601

3 : ( ) 1 : integrand1 : uses 4 5

1 : answery

⎧⎪ =⎨⎪⎩

(c) Speed ( )( ) ( )( )2 2

223 9cos 3.52

x t y t

tt

′ ′= +

⎛ ⎞= + =⎜ ⎟⎝ ⎠

The particle first reaches this speed when

2.225t = or 2.226.

3 : 1 : expression for speed

1 : equation 1 : answer

⎧⎪⎨⎪⎩

(d) 4 2

2

03 9cos 13.1822

tt dt⎛ ⎞+ =⎜ ⎟⎝ ⎠

⌠⎮⌡



Question 2


For time 0t ≥ hours, let ( ) ( )210120 1 tr t e−= − represent the speed, in kilometers per hour, at which a

car travels along a straight road. The number of liters of gasoline used by the car to travel x kilometers is modeled by ( ) ( )20.05 1 .xg x x e−= −

(a) How many kilometers does the car travel during the first 2 hours? (b) Find the rate of change with respect to time of the number of liters of gasoline used by the car when

2t = hours. Indicate units of measure. (c) How many liters of gasoline have been used by the car when it reaches a speed of 80 kilometers per

hour?

(a)

( )2

0206.370r t dt =∫ kilometers


(b) ;dg dg dxdt dx dt= ⋅ ( )dx r tdt =

( )

( )( )2 206.370

2

0.050 120 6 liters hourt x

dg dg rdt dx= == ⋅

= =

3 : { 2 : uses chain rule1 : answer with units

(c)

Let T be the time at which the car’s speed reaches 80 kilometers per hour. Then, ( ) 80r T = or 0.331453T = hours. At time T, the car has gone

( ) ( )0

10.794097T

x T r t dt= =∫ kilometers

and has consumed ( )( ) 0.537g x T = liters of gasoline.

4 : ( )1 : equation 80

2 : distance integral 1 : answer

r t =⎧⎪⎨⎪⎩


Question 3


Distance from the river’s edge (feet) 0 8 14 22 24

Depth of the water (feet) 0 7 8 2 0

A scientist measures the depth of the Doe River at Picnic Point. The river is 24 feet wide at this location. The measurements are taken in a straight line perpendicular to the edge of the river. The data are shown in the table above. The velocity of the water at Picnic Point, in feet per minute, is modeled by

( ) ( )16 2sin 10v t t= + + for 0 120t≤ ≤ minutes.

(a) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the area of the cross section of the river at Picnic Point, in square feet. Show the computations that lead to your answer.

(b) The volumetric flow at a location along the river is the product of the cross-sectional area and the velocity of the water at that location. Use your approximation from part (a) to estimate the average value of the volumetric flow at Picnic Point, in cubic feet per minute, from 0t = to 120t = minutes.

(c) The scientist proposes the function f, given by ( ) ( )8sin ,24xf x π= as a model for the depth of the

water, in feet, at Picnic Point x feet from the river’s edge. Find the area of the cross section of the river at Picnic Point based on this model.

(d) Recall that the volumetric flow is the product of the cross-sectional area and the velocity of the water at a location. To prevent flooding, water must be diverted if the average value of the volumetric flow at Picnic Point exceeds 2100 cubic feet per minute for a 20-minute period. Using your answer from part (c), find the average value of the volumetric flow during the time interval 40 60t≤ ≤ minutes. Does this value indicate that the water must be diverted?

(a) ( ) ( ) ( ) ( )

2

0 7 7 8 8 2 2 08 6 8 22 2 2 2115 ft

+ + + +⋅ + ⋅ + ⋅ + ⋅

=

1 : trapezoidal approximation

(b) ( )120

03

1 1151201807.169 or 1807.170 ft min

v t dt

=

∫

3 :

1 : limits and average value constant 1 : integrand 1 : answer

⎧⎪⎪⎨⎪⎪⎩

(c) ( )24 20

8sin 122.230 or 122.231 ft24x dxπ =∫ 2 : { 1 : integra1

1 : answer

(d) Let C be the cross-sectional area approximation from part (c). The average volumetric flow is

( )60 340

1 2181.912 or 2181.913 ft min.20 C v t dt⋅ =∫

Yes, water must be diverted since the average volumetric flow for this 20-minute period exceeds 2100 3ft min.

3 : 1 : volumetric flow integral1 : average volumetric flow


⎧⎪⎨⎪⎩


Question 4


Let f be the function given by ( ) 2 3,f x kx x= − where k is a positive constant. Let R be the region in the first quadrant bounded by the graph of f and the x-axis. (a) Find all values of the constant k for which the area of R equals 2. (b) For 0,k > write, but do not evaluate, an integral expression in terms of k for the volume of the solid

generated when R is rotated about the x-axis. (c) For 0,k > write, but do not evaluate, an expression in terms of k, involving one or more integrals,

that gives the perimeter of R.

(a) For 0,x ≥ ( ) ( )2 0f x x k x= − ≥ if 0 x k≤ ≤

( ) ( ) 42 3 3 4

0 0

13 4 12

x kk

x

k kkx x dx x x=

=− = − =∫

Area 4

2;12k= = 4 24k =

4 :

1 : integral1 : antiderivative1 : value of integral

1 : answer

⎧⎪⎪⎨⎪⎪⎩

(b)

Volume ( )22 30

kkx x dxπ= −∫ 2 : { 1 : integrand

1 : limits and constant

(c) Perimeter ( )220

1 2 3k

k kx x dx= + + −∫

3 : ( )( )

( )

20

2

1 : 1

1 : uses 2 3 in integrand 1 : answer

kf x dx

f x kx x

⎧ ′+⎪⎪⎨ ′ = −⎪⎪⎩

∫


Question 5


Let g be a continuous function with ( )2 5.g = The graph of the piecewise-linear function ,g′ the derivative of g, is shown above for 3 7.x− ≤ ≤ (a) Find the x-coordinate of all points of inflection of the

graph of ( )y g x= for 3 7.x− < < Justify your answer.

(b) Find the absolute maximum value of g on the interval 3 7.x− ≤ ≤ Justify your answer.

(c) Find the average rate of change of ( )g x on the interval 3 7.x− ≤ ≤

(d) Find the average rate of change of ( )g x′ on the interval 3 7.x− ≤ ≤ Does the Mean Value Theorem applied on the interval 3 7x− ≤ ≤ guarantee a value of c, for 3 7,c− < < such that ( )g c′′ is equal to this average rate of change? Why or why not?

(a) g ′ changes from increasing to decreasing at 1;x = g ′ changes from decreasing to increasing at 4.x = Points of inflection for the graph of ( )y g x= occur at

1x = and 4.x =

2 : { 1 : -values1 : justification

x

(b) The only sign change of g′ from positive to negative in the interval is at 2.x =

( ) ( ) ( )( )

( ) ( ) ( )

3

2

7

2

3 153 5 5 42 22 5

1 37 5 5 4 2 2

g g x dx

g

g g x dx

−′− = + = + − + =

=

′= + = + − + =

∫

∫

The maximum value of g for 3 7x− ≤ ≤ is 15 .2

3 : 1 : identifies 2 as a candidate

1 : considers endpoints1 : maximum value and justification

x =⎧⎪⎨⎪⎩

(c) ( ) ( )( )

3 157 3 32 2

10 57 3g g −− − = = −− −

2 : { 1 : difference quotient 1 : answer

(d) ( ) ( )( )

( )7 3 1 4 110 27 3

g g′ ′− − − −= =− −

No, the MVT does not guarantee the existence of a value c with the stated properties because g′ is not differentiable for at least one point in 3 7.x− < <

2 : ( ) 1 : average value of 1 : answer “No” with reason

g x′⎧⎨⎩


Question 6


Let f be the function given by ( ) 22 .

1xf xx

=+

(a) Write the first four nonzero terms and the general term of the Taylor series for f about 0.x =

(b) Does the series found in part (a), when evaluated at 1,x = converge to ( )1 ?f Explain why or why not.

(c) The derivative of ( )2ln 1 x+ is 22 .

1xx+

Write the first four nonzero terms of the Taylor series for

( )2ln 1 x+ about 0.x =

(d) Use the series found in part (c) to find a rational number A such that ( )5 1ln .4 100A − < Justify

your answer.

(a) 21 11nu u uu = + + + + +

−

( )2 4 6 22

1 11

nx x x x

x= − + − + + − +

+

3 5 7 2 12

2 2 2 2 2 ( 1) 21

n nx x x x x xx

+= − + − + + − ++

3 : 1 : two of the first four terms 1 : remaining terms 1 : general term

⎧⎪⎨⎪⎩

(b) No, the series does not converge when 1x = because when 1,x = the terms of the series do not converge to 0.


(c) ( )( )

22

0

3 5 70

2 4 6 8

2ln 11

2 2 2 2

1 1 12 3 4

x

x

tx dtt

t t t t dt

x x x x

+ =+

= − + − +

= − + − +

⌠⎮⌡

∫

2 : 1 : two of the first four terms 1 : remaining terms⎧⎨⎩

(d) ( ) ( ) ( ) ( ) ( ) ( )2 4 6 85 1 1 1 1 1 1 1 1ln ln 14 4 2 2 2 3 2 4 2= + = − + − +

Let ( ) ( )( )2 41 1 1 7 .2 2 2 32A = − =

Since the series is a converging alternating series and the absolute values of the individual terms decrease to 0,

( ) ( )65 1 1 1 1 1ln .4 3 2 3 64 100A − < = ⋅ <

3 :

1 1 : uses 21 : value of 1 : justification

x

A

⎧ =⎪⎨⎪⎩


The College Board

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 5,600 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools and 3,800 colleges through major programs and services in college readiness, 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.








1. Caren rides her bicycle along a straight road from home to school, starting at home at time 0t = minutes and arriving at school at time 12t = minutes. During the time interval 0 12t£ £ minutes, her velocity ( ),v t in miles per minute, is modeled by the piecewise-linear function whose graph is shown above.

(a) Find the acceleration of Caren’s bicycle at time 7.5t = minutes. Indicate units of measure.


0v t dtÚ in terms of Caren’s trip. Find the value of

( )12

0.v t dtÚ

(c) Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer.

(d) Larry also rides his bicycle along a straight road from home to school in 12 minutes. His velocity is modeled

by the function w given by ( ) ( )sin ,15 12

w t tp p= where ( )w t is in miles per minute for 0 12t£ £ minutes.

Who lives closer to school: Caren or Larry? Show the work that leads to your answer.





2. The rate at which people enter an auditorium for a rock concert is modeled by the function R given by

( ) 2 31380 675R t t t= - for 0 2t£ £ hours; ( )R t is measured in people per hour. No one is in the auditorium at time 0,t = when the doors open. The doors close and the concert begins at time 2.t =

(a) How many people are in the auditorium when the concert begins?

(b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer.

(c) The total wait time for all the people in the auditorium is found by adding the time each person waits, starting at the time the person enters the auditorium and ending when the concert begins. The function w models the total wait time for all the people who enter the auditorium before time t. The derivative of w is given by ( ) ( ) ( )2 .w t t R t= -¢ Find ( ) ( )2 1 ,w w- the total wait time for those who enter the auditorium after time 1.t =

(d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part (c).




-4-

3. A diver leaps from the edge of a diving platform into a pool below. The figure above shows the initial position of the diver and her position at a later time. At time t seconds after she leaps, the horizontal distance from the front edge of the platform to the diver’s shoulders is given by ( ),x t and the vertical distance from the water surface to

her shoulders is given by ( ),y t where ( )x t and ( )y t are measured in meters. Suppose that the diver’s shoulders are 11.4 meters above the water when she makes her leap and that

0.8 dxdt

= and 3.6 9.8 ,dy

tdt

= -

for 0 ,t A£ £ where A is the time that the diver’s shoulders enter the water.

(a) Find the maximum vertical distance from the water surface to the diver’s shoulders.

(b) Find A, the time that the diver’s shoulders enter the water.

(c) Find the total distance traveled by the diver’s shoulders from the time she leaps from the platform until the time her shoulders enter the water.

(d) Find the angle ,q 0 ,2p

q< < between the path of the diver and the water at the instant the diver’s

shoulders enter the water.









4. Consider the differential equation 2 26 .dy

x x ydx

= - Let ( )y f x= be a particular solution to this differential

equation with the initial condition ( )1 2.f - =

(a) Use Euler’s method with two steps of equal size, starting at 1,x = - to approximate ( )0 .f Show the work that leads to your answer.

(b) At the point ( )1, 2 ,- the value of 2

2d y

dx is 12.- Find the second-degree Taylor polynomial for

f about 1.x = -

(c) Find the particular solution ( )y f x= to the given differential equation with the initial condition ( )1 2.f - =




-6-

x 2 3 5 8 13

( )f x 1 4 –2 3 6

5. Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for

selected points in the closed interval 2 13.x£ £

(a) Estimate ( )4 .f ¢ Show the work that leads to your answer.

(b) Evaluate ( )( )13

23 5 .f x dx- ¢Ú Show the work that leads to your answer.

(c) Use a left Riemann sum with subintervals indicated by the data in the table to approximate ( )13

2.f x dxÚ

Show the work that leads to your answer.

(d) Suppose ( )5 3f =¢ and ( ) 0f x <¢¢ for all x in the closed interval 5 8.x£ £ Use the line tangent to the

graph of f at 5x = to show that ( )7 4.f £ Use the secant line for the graph of f on 5 8x£ £ to show

that ( ) 47 .3

f ≥

6. The Maclaurin series for xe is 2 3

1 .2 6 !

nx x x xe x

n= + + + + + + The continuous function f is defined by

( )( )

( )

21

21

1

xef xx

- -=-

for 1x π and ( )1 1.f = The function f has derivatives of all orders at 1.x =

(a) Write the first four nonzero terms and the general term of the Taylor series for ( )21xe - about 1.x =

(b) Use the Taylor series found in part (a) to write the first four nonzero terms and the general term of the Taylor series for f about 1.x =

(c) Use the ratio test to find the interval of convergence for the Taylor series found in part (b).

(d) Use the Taylor series for f about 1x = to determine whether the graph of f has any points of inflection.


END OF EXAM


The College Board


© 2009 The College Board. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. 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 AP teachers: apcentral.collegeboard.com.


Question 1


Caren rides her bicycle along a straight road from home to school, starting at home at time 0t = minutes and arriving at school at time 12t = minutes. During the time interval 0 12t≤ ≤ minutes, her velocity

( ) ,v t in miles per minute, is modeled by the piecewise-linear function whose graph is shown above.

(a) Find the acceleration of Caren’s bicycle at time 7.5t = minutes. Indicate units of measure.


0v t dt∫ in terms of Caren’s trip. Find the value

of ( )12

0.v t dt∫

(c) Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer.

(d) Larry also rides his bicycle along a straight road from home to school in 12 minutes. His velocity is

modeled by the function w given by ( ) ( )sin ,15 12w t tπ π= where ( )w t is in miles per minute for

0 12t≤ ≤ minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer.

(a) ( ) ( ) ( ) ( ) 28 77.5 7.5 0.1 miles minute8 7v va v −′= = = −− 2 : { 1 : answer

1 : units

(b) ( )12

0v t dt∫ is the total distance, in miles, that Caren rode

during the 12 minutes from 0t = to 12.t =

( ) ( ) ( ) ( )12 2 4 12

0 0 2 4

0.2 0.2 1.4 1.8 miles

v t dt v t dt v t dt v t dt= − +

= + + =∫ ∫ ∫ ∫

2 : { 1 : meaning of integral1 : value of integral

(c)

Caren turns around to go back home at time 2t = minutes. This is the time at which her velocity changes from positive to negative.

2 : { 1 : answer1 : reason

(d) ( )12

01.6;w t dt =∫ Larry lives 1.6 miles from school.

( )12

01.4;v t dt =∫ Caren lives 1.4 miles from school.

Therefore, Caren lives closer to school.

3 :

2 : Larry’s distance from school 1 : integral 1 : value1 : Caren’s distance from school

and conclusion

⎧⎪⎪⎨⎪⎪⎩


Question 2


The rate at which people enter an auditorium for a rock concert is modeled by the function R given by ( ) 2 31380 675R t t t= − for 0 2t≤ ≤ hours; ( )R t is measured in people per hour. No one is in the

auditorium at time 0,t = when the doors open. The doors close and the concert begins at time 2.t =

(a) How many people are in the auditorium when the concert begins? (b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer. (c) The total wait time for all the people in the auditorium is found by adding the time each person waits,

starting at the time the person enters the auditorium and ending when the concert begins. The functionw models the total wait time for all the people who enter the auditorium before time t. The derivative of w is given by ( ) ( ) ( )2 .w t t R t′ = − Find ( ) ( )2 1 ,w w− the total wait time for those who enter the auditorium after time 1.t =

(d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part (c).

(a)

( )2

0980R t dt =∫ people 2 : { 1 : integral

1 : answer

(b) ( ) 0R t′ = when 0t = and 1.36296t = The maximum rate may occur at 0, 1.36296,a = or 2.

( )0 0R = ( ) 854.527R a = ( )2 120R =

The maximum rate occurs when 1.362t = or 1.363.

3 : ( ) 1 : considers 0

1 : interior critical point1 : answer and justification

R t′ =⎧⎪⎨⎪⎩

(c)

( ) ( ) ( ) ( ) ( )2 2

1 122 1 387.5w t dt t Rw t dtw− = ′ = − =∫ ∫

The total wait time for those who enter the auditorium after time 1t = is 387.5 hours.


(d) ( ) ( ) ( )2

01 12 0.77551980 980 2w Rt t dt= =−∫

On average, a person waits 0.775 or 0.776 hour.



Question 3


A diver leaps from the edge of a diving platform into a pool below. The figure above shows the initial position of the diver and her position at a later time. At time t seconds after she leaps, the horizontal distance from the front edge of the platform to the diver’s shoulders is given by ( ) ,x t and the vertical distance from the water surface to her shoulders is given by ( ) ,y t where ( )x t and ( )y t are measured in meters. Suppose that the diver’s shoulders are 11.4 meters above the water when she makes her leap and that

0.8 dxdt = and 3.6 9.8 ,dy tdt = −

for 0 ,t A≤ ≤ where A is the time that the diver’s shoulders enter the water.

(a) Find the maximum vertical distance from the water surface to the diver’s shoulders.

(b) Find A, the time that the diver’s shoulders enter the water. (c) Find the total distance traveled by the diver’s shoulders from the time

she leaps from the platform until the time her shoulders enter the water.

(d) Find the angle ,θ 0 ,2πθ< < between the path of the diver and the

water at the instant the diver’s shoulders enter the water.

(a)

0dydt = only when 0.36735.t = Let 0.36735.b =

The maximum vertical distance from the water surface to the diver’s shoulders is

( )0

11.4 12.061b dyy b dtdt= + =⌠

⌡ meters.

Alternatively, ( ) 211.4 3.6 4.9 ,y t t t= + − so ( ) 12.061y b = meters.

3 : ( ) 1 : considers 0

1 : integral or 1 : answer

dydty t

⎧ =⎪⎪⎨⎪⎪⎩

(b) ( ) 2

011.4 11.4 3.6 4.9 0

A dyy A dt A Adt= + = + − =⌠⌡

when

1.936A = seconds.

2 : { 1 : equation1 : answer

(c) ( ) ( )22

012.946

A dydx dtdt dt+ =⌠⎮⌡

meters


(d)

At time A, 19.21913.t A

dy dtdydx dx dt =

= = −

The angle between the path of the diver and the water is

( )1tan 19.21913 1.518− = or 1.519.

2 : 1 : at time

1 : answer

dy Adx⎧⎪⎨⎪⎩


Question 4


Consider the differential equation 2 26 .dy x x ydx = − Let ( )y f x= be a particular solution to this

differential equation with the initial condition ( )1 2.f − =

(a) Use Euler’s method with two steps of equal size, starting at 1,x = − to approximate ( )0 .f Show the work that leads to your answer.

(b) At the point ( )1, 2 ,− the value of 2

2d ydx

is 12.− Find the second-degree Taylor polynomial for

f about 1.x = −

(c) Find the particular solution ( )y f x= to the given differential equation with the initial condition ( )1 2.f − =

(a) ( ) ( )( )1, 2

1 12

12 4 42

dyf f xdx −

⎛ ⎞− ≈ − + ⋅ Δ⎜ ⎟

⎝ ⎠

= + ⋅ =

( ) ( ) ( )1 , 42

10 2

1 1 174 2 2 4

dyf f xdx −

⎛ ⎞⎜ ⎟≈ − + ⋅ Δ⎜ ⎟⎝ ⎠

≈ + ⋅ =

2 : { 1 : Euler’s method with two steps 1 : answer

(b) ( ) ( ) ( )22 2 4 1 6 1P x x x= + + − +

1 : answer

(c) ( )2 6dy x ydx = −

216 dy x dxy =−

⌠⎮⌡ ∫

31ln 6 3y x C− − = +

1ln 4 3 C− = − +

1 ln 43C = −

( )31 1ln 6 ln 43 3y x− = − − −

( )31 136 4x

y e− +

− =

( )31 136 4x

y e− +

= −

6 :

1 : separation of variables 2 : antiderivatives1 : constant of integration1 : uses initial condition

1 : solves for y

⎧⎪⎪⎨⎪⎪⎩

Note: max 3 6 [1-2-0-0-0] if no constant of

integration Note: 0 6 if no separation of variables


Question 5


x 2 3 5 8 13

( )f x 1 4 –2 3 6

Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for selected points in the closed interval 2 13.x≤ ≤ (a) Estimate ( )4 .f ′ Show the work that leads to your answer.


23 5 .f x dx′−∫ Show the work that leads to your answer.

(c) Use a left Riemann sum with subintervals indicated by the data in the table to approximate ( )13

2.f x dx∫

Show the work that leads to your answer. (d) Suppose ( )5 3f ′ = and ( ) 0f x′′ < for all x in the closed interval 5 8.x≤ ≤ Use the line tangent to

the graph of f at 5x = to show that ( )7 4.f ≤ Use the secant line for the graph of f on 5 8x≤ ≤ to

show that ( ) 47 .3f ≥

(a) ( ) ( ) ( )5 34 35 3f ff −′ ≈ = −

−

1 : answer

(b) ( )( ) ( )

( ) ( )

13 13 13

2 2 23 5 3 5

3 13 2 5 (13) (2) 8

f x dx dx f x dx

f f

′ ′− = −

= − − − =∫ ∫ ∫

2 : 1 : uses Fundamental Theorem

of Calculus 1 : answer

⎧⎪⎨⎪⎩

(c) ( ) ( )( ) ( )( )

( )( ) ( )( )

13

22 3 2 3 5 3

5 8 5 8 13 8 18

f x dx f f

f f

≈ − + −

+ − + − =∫

2 : { 1 : left Riemann sum 1 : answer

(d) An equation for the tangent line is ( )2 3 5 .y x= − + −

Since ( ) 0f x′′ < for all x in the interval 5 8,x≤ ≤ the line tangent to the graph of ( )y f x= at 5x = lies above the graph for all x in the interval 5 8.x< ≤

Therefore, ( )7 2 3 2 4.f ≤ − + ⋅ =

An equation for the secant line is ( )52 5 .3y x= − + −

Since ( ) 0f x′′ < for all x in the interval 5 8,x≤ ≤ the secant line connecting ( )( )5, 5f and ( )( )8, 8f lies below the graph of ( )y f x= for all x in the interval 5 8.x< <

Therefore, ( ) 5 47 2 2 .3 3f ≥ − + ⋅ =

4 : ( )

( )

1 : tangent line1 : shows 7 4

1 : secant line4 1 : shows 7 3

f

f

⎧⎪ ≤⎪⎨⎪⎪ ≥⎩


Question 6


The Maclaurin series for xe is 2 3

1 .2 6 !n

x x x xe x n= + + + + + + The continuous function f is defined

by ( )( )

( )

21

21

1

xef xx

− −=−

for 1x ≠ and ( )1 1.f = The function f has derivatives of all orders at 1.x =

(a) Write the first four nonzero terms and the general term of the Taylor series for ( )21xe − about 1.x = (b) Use the Taylor series found in part (a) to write the first four nonzero terms and the general term of the

Taylor series for f about 1.x = (c) Use the ratio test to find the interval of convergence for the Taylor series found in part (b). (d) Use the Taylor series for f about 1x = to determine whether the graph of f has any points of

inflection.

(a) ( ) ( ) ( ) ( )4 6 22 1 1 11 1 2 6 !

nx x xx n− − −+ − + + + + +

2 : { 1 : first four terms1 : general term

(b) ( ) ( ) ( ) ( )( )

2 4 6 21 1 1 11 2 6 24 1 !

nx x x xn

− − − −+ + + + + ++

2 : { 1 : first four terms1 : general term

(c)

( )( )( )( )

( )( ) ( ) ( )

2 2

22

2

12 ! 1 ! 1lim lim 1 lim 022 !11 !

n

nn n n

xn n xx nnxn

+

→ → →∞ ∞ ∞

−+ + −= − = =++−+

Therefore, the interval of convergence is ( ), .−∞ ∞

3 : 1 : sets up ratio1 : computes limit of ratio

1 : answer

⎧⎪⎨⎪⎩

(d) ( ) ( ) ( )( ) ( )

2 4

2 2

4 3 6 51 1 16 242 2 1 1( 1)!

n

f x x x

n n xn−

⋅ ⋅′′ = + − + − +

−+ − ++

Since every term of this series is nonnegative, ( ) 0f x′′ ≥ for all x. Therefore, the graph of f has no points of inflection.

2 : ( ) 1 : 1 : answer

f x′′⎧⎨⎩


Form B

The College Board









1. A baker is creating a birthday cake. The base of the cake is the region R in the first quadrant under the graph of

( )y f x= for 0 30,x£ £ where ( ) ( )20sin .30

xf x p= Both x and y are measured in centimeters. The region R

is shown in the figure above. The derivative of f is ( ) ( )2 cos .3 30

xf x p p=¢

(a) The region R is cut out of a 30-centimeter-by-20-centimeter rectangular sheet of cardboard, and the remaining cardboard is discarded. Find the area of the discarded cardboard.

(b) The cake is a solid with base R. Cross sections of the cake perpendicular to the x-axis are semicircles. If the baker uses 0.05 gram of unsweetened chocolate for each cubic centimeter of cake, how many grams of unsweetened chocolate will be in the cake?

(c) Find the perimeter of the base of the cake.





2. A storm washed away sand from a beach, causing the edge of the water to get closer to a nearby road. The rate at which the distance between the road and the edge of the water was changing during the storm is modeled by ( ) cos 3f t t t= + - meters per hour, t hours after the storm began. The edge of the water was 35 meters from

the road when the storm began, and the storm lasted 5 hours. The derivative of ( )f t is ( ) 1 sin .2

f t tt

= -¢

(a) What was the distance between the road and the edge of the water at the end of the storm?

(b) Using correct units, interpret the value ( )4 1.007f =¢ in terms of the distance between the road and the edge of the water.

(c) At what time during the 5 hours of the storm was the distance between the road and the edge of the water decreasing most rapidly? Justify your answer.

(d) After the storm, a machine pumped sand back onto the beach so that the distance between the road and the edge of the water was growing at a rate of ( )g p meters per day, where p is the number of days since pumping began. Write an equation involving an integral expression whose solution would give the number of days that sand must be pumped to restore the original distance between the road and the edge of the water.




-4-

3. A continuous function f is defined on the closed interval 4 6.x- £ £ The graph of f consists of a line segment and a curve that is tangent to the x-axis at 3,x = as shown in the figure above. On the interval 0 6,x< < the function f is twice differentiable, with ( ) 0.f x >¢¢

(a) Is f differentiable at 0 ?x = Use the definition of the derivative with one-sided limits to justify your answer.

(b) For how many values of a, 4 6,a- £ < is the average rate of change of f on the interval [ ], 6a equal to 0 ? Give a reason for your answer.

(c) Is there a value of a, 4 6,a- £ < for which the Mean Value Theorem, applied to the interval [ ], 6 ,a

guarantees a value c, 6,a c< < at which ( ) 1 ?3

f c =¢ Justify your answer.

(d) The function g is defined by ( ) ( )0

xg x f t dt= Ú for 4 6.x- £ £ On what intervals contained in [ ]4, 6-

is the graph of g concave up? Explain your reasoning.









4. The graph of the polar curve 1 2cosr q= - for 0 q p£ £ is shown above. Let S be the shaded region in the third quadrant bounded by the curve and the x-axis.

(a) Write an integral expression for the area of S.

(b) Write expressions for dxdq

and dydq

in terms of .q

(c) Write an equation in terms of x and y for the line tangent to the graph of the polar curve at the point

where .2p

q = Show the computations that lead to your answer.





5. Let f be a twice-differentiable function defined on the interval 1.2 3.2x- < < with ( )1 2.f = The graph of ,f ¢

the derivative of f, is shown above. The graph of f ¢ crosses the x-axis at 1x = - and 3x = and has

a horizontal tangent at 2.x = Let g be the function given by ( ) ( ).f xg x e=

(a) Write an equation for the line tangent to the graph of g at 1.x =

(b) For 1.2 3.2,x- < < find all values of x at which g has a local maximum. Justify your answer.

(c) The second derivative of g is ( ) ( ) ( )( ) ( )2 .f xg x e f x f xÈ ˘= +¢¢ ¢ ¢¢Î ˚ Is ( )1g -¢¢ positive, negative, or zero?

Justify your answer.

(d) Find the average rate of change of ,g¢ the derivative of g, over the interval [ ]1, 3 .




-7-


( ) ( ) ( ) ( ) ( )2

0

1 1 1 1 1n n

n

f x x x x x=

•= + + + + + + + + = +Â

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 power series above is the Taylor series for f about 1.x = - Find the sum of the series for f.

(c) Let g be the function defined by ( ) ( )1

.x

g x f t dt-

= Ú Find the value of ( )12

g - , if it exists, or explain why

( )12

g - cannot be determined.

(d) Let h be the function defined by ( ) ( )2 1 .h x f x= - Find the first three nonzero terms and the general term

of the Taylor series for h about 0,x = and find the value of ( )1 .2

h


END OF EXAM


Form B

The College Board


© 2009 The College Board. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. 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 AP teachers: apcentral.collegeboard.com.


Question 1


A baker is creating a birthday cake. The base of the cake is the region R in the first quadrant under the graph of ( )y f x= for

0 30,x≤ ≤ where ( ) ( )20sin .30xf x π= Both x and y are

measured in centimeters. The region R is shown in the figure

above. The derivative of f is ( ) ( )2 cos .3 30xf x π π′ =

(a) The region R is cut out of a 30-centimeter-by-20-centimeter rectangular sheet of cardboard, and the remaining cardboard is discarded. Find the area of the discarded cardboard.

(b) The cake is a solid with base R. Cross sections of the cake perpendicular to the x-axis are semicircles. If the baker uses 0.05 gram of unsweetened chocolate for each cubic centimeter of cake, how many grams of unsweetened chocolate will be in the cake?

(c) Find the perimeter of the base of the cake.

(a)

Area ( )30 20

30 20 218.028 cmf x dx= ⋅ − =∫ 3 : { 2 : integral 1 : answer

(b) Volume ( )30 23

02356.194 cm2 2

f x dxπ ⎛ ⎞= =⎜ ⎟⎝ ⎠

⌠⎮⌡

Therefore, the baker needs 2356.194 0.05 117.809× = or 117.810 grams of chocolate.


(c)

Perimeter ( )( )30 20

30 1 81.803f x dx′= + + =∫ or 81.804 cm 3 : { 2 : integral 1 : answer


Question 2


A storm washed away sand from a beach, causing the edge of the water to get closer to a nearby road. The rate at which the distance between the road and the edge of the water was changing during the storm is modeled by ( ) cos 3f t t t= + − meters per hour, t hours after the storm began. The edge of the water was 35 meters from the road when the storm began, and the storm lasted 5 hours. The derivative of ( )f t

is ( ) 1 sin .2

f t tt

′ = −

(a) What was the distance between the road and the edge of the water at the end of the storm? (b) Using correct units, interpret the value ( )4 1.007f ′ = in terms of the distance between the road and

the edge of the water. (c) At what time during the 5 hours of the storm was the distance between the road and the edge of the

water decreasing most rapidly? Justify your answer. (d) After the storm, a machine pumped sand back onto the beach so that the distance between the road

and the edge of the water was growing at a rate of ( )g p meters per day, where p is the number of days since pumping began. Write an equation involving an integral expression whose solution would give the number of days that sand must be pumped to restore the original distance between the road and the edge of the water.

(a)

( )5

035 26.494f t dt+ =∫ or 26.495 meters 2 : { 1 : integral

1 : answer

(b)

Four hours after the storm began, the rate of change of the distance between the road and the edge of the water is increasing at a rate of 21.007 meters hours .

2 : ( )1 : interpretation of 4 1 : units

f ′⎧⎨⎩

(c) ( ) 0f t′ = when 0.66187t = and 2.84038t = The minimum of f for 0 5t≤ ≤ may occur at 0, 0.66187, 2.84038, or 5.

( )0 2f = − ( )0.66187 1.39760f = − ( )2.84038 2.26963f = − ( )5 0.48027f = −

The distance between the road and the edge of the water was decreasing most rapidly at time 2.840t = hours after the storm began.

3 : ( )1 : considers 0

1 : answer 1 : justification

f t′ =⎧⎪⎨⎪⎩

(d) ( ) ( )5

0 0

xf t dt g p dp− =∫ ∫

2 : { 1 : integral of 1 : answer

g


Question 3


A continuous function f is defined on the closed interval 4 6.x− ≤ ≤ The graph of f consists of a line segment and a curve that is tangent to the x-axis at 3,x = as shown in the figure above. On the interval 0 6,x< < the function f is twice differentiable, with ( ) 0.f x′′ >

(a) Is f differentiable at 0 ?x = Use the definition of the derivative with one-sided limits to justify your answer.

(b) For how many values of a, 4 6,a− ≤ < is the average rate of change of f on the interval [ ], 6a equal to 0 ? Give a reason for your answer.

(c) Is there a value of a, 4 6,a− ≤ < for which the Mean Value Theorem, applied to the interval [ ], 6 ,a

guarantees a value c, 6,a c< < at which ( ) 1 ?3f c′ = Justify your answer.

(d) The function g is defined by ( ) ( )0

xg x f t dt= ∫ for 4 6.x− ≤ ≤ On what intervals contained in

[ ]4, 6− is the graph of g concave up? Explain your reasoning.

(a) ( ) ( )0

0 2lim 3h

f h fh−→

− =

( ) ( )0

0lim 0h

f h fh+→

− <

Since the one-sided limits do not agree, f is not differentiable at 0.x =

2 : { 1 : sets up difference quotient at 0 1 : answer with justification

x =

(b) ( ) ( )6 06f f a

a− =− when ( ) ( )6 .f a f= There are

two values of a for which this is true.

2 : { 1 : expression for average rate of change 1 : answer with reason

(c) Yes, 3.a = The function f is differentiable on the interval 3 6x< < and continuous on 3 6.x≤ ≤

Also, ( ) ( )6 3 1 0 1 .6 3 6 3 3f f− −= =− −

By the Mean Value Theorem, there is a value c,

3 6,c< < such that ( ) 1 .3f c′ =

2 : { 1 : answers “yes” and identifies 3 1 : justification

a =

(d) ( ) ( ) ( ) ( ),g x f x g x f x′ ′′ ′= =

( ) 0g x′′ > when ( ) 0f x′ > This is true for 4 0x− < < and 3 6.x< <

3 : ( ) ( )

( ) 1 : 1 : considers 0

1 : answer

g x f xg x

′ =⎧⎪ ′′ >⎨⎪⎩


Question 4


The graph of the polar curve 1 2cosr θ= − for 0 θ π≤ ≤ is shown above. Let S be the shaded region in the third quadrant bounded by the curve and the x-axis.

(a) Write an integral expression for the area of S.

(b) Write expressions for dxdθ and dy

dθ in terms of .θ

(c) Write an equation in terms of x and y for the line tangent to

the graph of the polar curve at the point where .2πθ =

Show the computations that lead to your answer.

(a) ( )0 1;r = − ( ) 0r θ = when .3πθ =

Area of ( )3 2

01 1 2cos2S d

πθ θ= −∫

2 : { 1 : limits and constant 1 : integrand

(b) cosx r θ= and siny r θ=

2sindrd θθ =

cos sin 4sin cos sindx dr rd d θ θ θ θ θθ θ= − = −

( )2sin cos 2sin 1 2cos cosdy dr rd d θ θ θ θ θθ θ= + = + −

4 :

1 : uses cos and sin

1 :

2 : answer

x r y rdrd

θ θ

θ

= =⎧⎪⎨⎪⎩

(c) When ,2πθ = we have 0, 1.x y= =

2 2

2dy ddydx dx dπ πθ θ

θθ= =

= = −

The tangent line is given by 1 2 .y x= −

3 :

1 : values for and

1 : expression for

1 : tangent line equation

x ydydx

⎧⎪⎪⎨⎪⎪⎩


Question 5


Let f be a twice-differentiable function defined on the interval 1.2 3.2x− < < with ( )1 2.f = The graph of ,f ′ the derivative

of f, is shown above. The graph of f ′ crosses the x-axis at 1x = − and 3x = and has a horizontal tangent at 2.x = Let g

be the function given by ( ) ( ).f xg x e=

(a) Write an equation for the line tangent to the graph of g at 1.x =

(b) For 1.2 3.2,x− < < find all values of x at which g has a local maximum. Justify your answer.

(c) The second derivative of g is ( ) ( ) ( )( ) ( )2 .f xg x e f x f x⎡ ⎤′′ ′ ′′= +⎣ ⎦ Is ( )1g′′ − positive, negative, or

zero? Justify your answer. (d) Find the average rate of change of ,g′ the derivative of g, over the interval [ ]1, 3 .

(a) ( ) ( )1 21 fg e e= =

( ) ( ) ( ) ,f xg x e f x′ ′= ( ) ( ) ( )1 21 1 4fg e f e′ ′= = −

The tangent line is given by ( )2 24 1 .y e e x= − −

3 : ( )( ) ( )

1 : 1 : 1 and 11 : tangent line equation

g xg g

′⎧⎪ ′⎨⎪⎩

(b) ( ) ( ) ( )f xg x e f x′ ′= ( ) 0f xe > for all x

So, g′ changes from positive to negative only when f ′ changes from positive to negative. This occurs at 1x = − only. Thus, g has a local maximum at 1.x = −

2 : { 1 : answer1 : justification

(c) ( ) ( ) ( )( ) ( )211 1 1fg e f f− ⎡ ⎤′′ ′ ′′− = − + −⎣ ⎦

( )1 0fe − > and ( )1 0f ′ − =

Since f ′ is decreasing on a neighborhood of 1,− ( )1 0.f ′′ − < Therefore, ( )1 0.g′′ − <

2 : { 1 : answer1 : justification

(d) ( ) ( ) ( ) ( ) ( ) ( )3 123 1 3 1 23 1 2

f fg g e f e f e′ ′ ′ ′− −= =−

2 : { 1 : difference quotient 1 : answer


Question 6


The function f is defined by the power series

( ) ( ) ( ) ( ) ( )2

01 1 1 1 1n n

nf x x x x x

=

∞= + + + + + + + + = +∑

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 power series above is the Taylor series for f about 1.x = − Find the sum of the series for f.

(c) Let g be the function defined by ( ) ( )1

.x

g x f t dt−

= ∫ Find the value of ( )12g − , if it exists, or explain

why ( )12g − cannot be determined.

(d) Let h be the function defined by ( ) ( )2 1 .h x f x= − Find the first three nonzero terms and the general

term of the Taylor series for h about 0,x = and find the value of ( )1 .2h

(a) The power series is geometric with ratio ( )1 .x +

The series converges if and only if 1 1.x + < Therefore, the interval of convergence is 2 0.x− < < OR

( )( )

11lim 1 11

n

nn

x xx

+

→∞+ = + <+

when 2 0x− < <

At 2,x = − the series is ( )0

1 ,n

n=

∞−∑ which diverges since the

terms do not converge to 0. At 0,x = the series is 01,

n=

∞∑

which similarly diverges. Therefore, the interval of convergence is 2 0.x− < <

3 : 1 : identifies as geometric

1 : 1 11 : interval of convergence

x⎧⎪ + <⎨⎪⎩

OR

3 : 1 : sets up limit of ratio1 : radius of convergence1 : interval of convergence

⎧⎪⎨⎪⎩

(b) Since the series is geometric,

( ) ( ) ( )0

1 11 1 1n

nf x x xx=

∞= + = = −− +∑ for 2 0.x− < <

1 : answer

(c) ( )1 12 2

11

1 1 ln ln 22x

xg dx xx

− =−

=−−− = − = − =⌠

⌡

2 : { 1 : antiderivative 1 : value

(d) ( ) ( )2 2 4 21 1 nh x f x x x x= − = + + + + +

( ) ( )1 3 42 4 3h f= − =

3 :

( )1 : first three terms

1 : general term1 1 : value of 2h

⎧⎪⎪⎨⎪⎪⎩


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© 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com.





1. There is no snow on Janet’s driveway when snow begins to fall at midnight. From midnight to 9 A.M., snow

accumulates on the driveway at a rate modeled by ( ) cos7 tf t te= cubic feet per hour, where t is measured

in hours since midnight. Janet starts removing snow at 6 A.M. ( )6 .t = The rate ( ),g t in cubic feet per hour, at which Janet removes snow from the driveway at time t hours after midnight is modeled by

( )0 for 0 6

125 for 6 7

108 for 7 9 .

t

g t t

t

£ <ÏÔ= £ <ÌÔ £ £Ó

(a) How many cubic feet of snow have accumulated on the driveway by 6 A.M.?

(b) Find the rate of change of the volume of snow on the driveway at 8 A.M.

(c) Let ( )h t represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time t hours after midnight. Express h as a piecewise-defined function with domain 0 9.t£ £

(d) How many cubic feet of snow are on the driveway at 9 A.M.?





t

(hours) 0 2 5 7 8

( )E t (hundreds of

entries) 0 4 13 21 23

2. A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box

between noon ( )0t = and 8 P.M. ( )8 .t = The number of entries in the box t hours after noon is modeled by a

differentiable function E for 0 8.t£ £ Values of ( ),E t in hundreds of entries, at various times t are shown in the table above.

(a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time 6.t = Show the computations that lead to your answer.

(b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value of ( )8

0

1 .8

E t dtÚ

Using correct units, explain the meaning of ( )8

0

18

E t dtÚ in terms of the number of entries.

(c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the

function P, where ( ) 3 230 298 976P t t t t= - + - hundreds of entries per hour for 8 12.t£ £ According

to the model, how many entries had not yet been processed by midnight ( )12 ?t =

(d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer.




-4-

3. A particle is moving along a curve so that its position at time t is ( ) ( )( ), ,x t y t where ( ) 2 4 8x t t t= - + and

( )y t is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known

that 3 1.tdyte

dt-= -

(a) Find the speed of the particle at time 3t = seconds.

(b) Find the total distance traveled by the particle for 0 4t£ £ seconds.

(c) Find the time t, 0 4,t£ £ when the line tangent to the path of the particle is horizontal. Is the direction of motion of the particle toward the left or toward the right at that time? Give a reason for your answer.

(d) There is a point with x-coordinate 5 through which the particle passes twice. Find each of the following.

(i) The two values of t when that occurs

(ii) The slopes of the lines tangent to the particle’s path at that point

(iii) The y-coordinate of that point, given ( ) 12 3ye

= +









4. Let R be the region in the first quadrant bounded by the graph of 2 ,y x= the horizontal line 6,y = and the y-axis, as shown in the figure above.


(b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line 7.y =

(c) Region R is the base of a solid. For each y, where 0 6,y£ £ the cross section of the solid taken perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R. Write, but do not evaluate, an integral expression that gives the volume of the solid.




-6-

5. Consider the differential equation 1 .dy

ydx

= - Let ( )y f x= be the particular solution to this differential

equation with the initial condition ( )1 0.f = For this particular solution, ( ) 1f x < for all values of x.

(a) Use Euler’s method, starting at 1x = with two steps of equal size, to approximate ( )0 .f Show the work that leads to your answer.

(b) Find ( )

31lim .

1x

f x

xÆ - Show the work that leads to your answer.

(c) Find the particular solution ( )y f x= to the differential equation 1dy

ydx

= - with the initial condition

( )1 0.f =

( )2

cos 1for 0

1 for 02

xx

xf x

x

-Ï πÔ= ÌÔ- =Ó

6. The function f, defined above, has derivatives of all orders. Let g be the function defined by

( ) ( )0

1 .x

g x f t dt= + Ú

(a) Write the first three nonzero terms and the general term of the Taylor series for cos x about 0.x = Use this series to write the first three nonzero terms and the general term of the Taylor series for f about 0.x =

(b) Use the Taylor series for f about 0x = found in part (a) to determine whether f has a relative maximum, relative minimum, or neither at 0.x = Give a reason for your answer.

(c) Write the fifth-degree Taylor polynomial for g about 0.x =

(d) The Taylor series for g about 0,x = evaluated at 1,x = is an alternating series with individual terms that decrease in absolute value to 0. Use the third-degree Taylor polynomial for g about 0x = to estimate the

value of ( )1 .g Explain why this estimate differs from the actual value of ( )1g by less than 1 .6!


END OF EXAM


The College Board 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 College Board is composed of more than 5,700 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,800 colleges through major programs and services in college readiness, college admission, guidance, assessment, financial aid and enrollment. Among its widely recognized programs are the SAT®, the PSAT/NMSQT®, the Advanced Placement Program®



Question 1


There is no snow on Janet’s driveway when snow begins to fall at midnight. From midnight to 9 A.M., snow accumulates on the driveway at a rate modeled by ( ) cos7 tf t te= cubic feet per hour, where t is measured in hours since midnight. Janet starts removing snow at 6 A.M. ( )6 .t = The rate ( ) ,g t in cubic feet per hour, at which Janet removes snow from the driveway at time t hours after midnight is modeled by

( )0 for 0 6125 for 6 7108 for 7 9 .

tg t t

t

≤ <⎧⎪= ≤ <⎨⎪ ≤ ≤⎩

(a) How many cubic feet of snow have accumulated on the driveway by 6 A.M.? (b) Find the rate of change of the volume of snow on the driveway at 8 A.M. (c) Let ( )h t represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time

t hours after midnight. Express h as a piecewise-defined function with domain 0 9.t≤ ≤ (d) How many cubic feet of snow are on the driveway at 9 A.M.?

(a)

( )6

0142.274f t dt =∫ or 142.275 cubic feet


(b) Rate of change is ( ) ( )8 8 59.582f g− = − or 59.583− cubic feet per hour.

1 : answer

(c) ( )0 0h =

For 0 6,t< ≤ ( ) ( ) ( )0 0

0 0 0 0.t t

h t h g s ds ds= + = + =∫ ∫

For 6 7,t< ≤ ( ) ( ) ( ) ( )6 6

6 0 125 125 6 .t t

h t h g s ds ds t= + = + = −∫ ∫

For 7 9,t< ≤ ( ) ( ) ( ) ( )7 7

125 107 8 125 108 7 .t tg s ds dsh th t = + = += + −∫ ∫

Thus, ( ) ( )( )

0 for 0 6125 6 for 6 7125 108 7 for 7 9

th t t t

t t

≤ ≤⎧⎪= − < ≤⎨⎪ + − < ≤⎩

3 : ( )( )( )

1 : for 0 6 1 : for 6 7 1 : for 7 9

h t th t th t t

≤ ≤⎧⎪ < ≤⎨⎪ < ≤⎩

(d)

Amount of snow is ( ) ( )9

09 26.334f t dt h− =∫ or 26.335 cubic feet. 3 : ( )

1 : integral 1 : 9 1 : answer

h⎧⎪⎨⎪⎩


Question 2


t (hours) 0 2 5 7 8

( )E t (hundreds of

entries) 0 4 13 21 23

A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon ( )0t = and 8 P.M. ( )8 .t = The number of entries in the box t hours after noon is modeled by a differentiable function E for 0 8.t≤ ≤ Values of ( ) ,E t in hundreds of entries, at various times t are shown in the table above. (a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being

deposited at time 6.t = Show the computations that lead to your answer.

(b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value of ( )8

01 .8 E t dt∫

Using correct units, explain the meaning of ( )8

018 E t dt∫ in terms of the number of entries.

(c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the function P, where ( ) 3 230 298 976P t t t t= − + − hundreds of entries per hour for 8 12.t≤ ≤ According to the model, how many entries had not yet been processed by midnight ( )12 ?t =

(d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer.

(a) ( ) ( ) ( )7 56 47 5E EE −′ ≈ =

− hundred entries per hour

1 : answer

(b) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

8

0

0 2 2 5 5 7 7 81 2· 3· 2· 1·8 2 2 2 210.687 or 10 8

8

6

1

. 8

E t dt

E E E E E E E E

≈

+ + + +⎛ ⎞+ + +⎜ ⎟⎝ ⎠

=

∫

( )8

018 E t dt∫ is the average number of hundreds of entries in the box

between noon and 8 P.M.

3 : 1 : trapezoidal sum 1 : approximation 1 : meaning

⎧⎪⎨⎪⎩

(c)

( )12

823 23 16 7P t dt− = − =∫ hundred entries 2 : { 1 : integral

1 : answer

(d) ( ) 0P t′ = when 9.183503t = and 10.816497.t = ( )

8 09.183503 5.088662

10.816497 2.91133812 8

t P t

Entries are being processed most quickly at time 12.t =

3 : ( ) 1 : considers 0

1 : identifies candidates 1 : answer with justification

P t′ =⎧⎪⎨⎪⎩


Question 3


A particle is moving along a curve so that its position at time t is ( ) ( )( ), ,x t y t where ( ) 2 4 8x t t t= − + and ( )y t is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known

that 3 1.tdy tedt−= −

(a) Find the speed of the particle at time 3t = seconds. (b) Find the total distance traveled by the particle for 0 4t≤ ≤ seconds. (c) Find the time t, 0 4,t≤ ≤ when the line tangent to the path of the particle is horizontal. Is the direction of

motion of the particle toward the left or toward the right at that time? Give a reason for your answer. (d) There is a point with x-coordinate 5 through which the particle passes twice. Find each of the following.

(i) The two values of t when that occurs (ii) The slopes of the lines tangent to the particle’s path at that point

(iii) The y-coordinate of that point, given ( ) 12 3y e= +

(a) Speed ( )( ) ( )( )2 23 3 2.828x y′ ′= + = meters per second

1 : answer

(b) ( ) 2 4x t t′ = −

Distance ( ) ( )4 22 30

2 4 1 11.587tt te dt−= − + − =∫ or 11.588 meters


(c) 0dy dtdydx dx dt= = when 3 1 0tte − − = and 2 4 0t − ≠

This occurs at 2.20794.t =

Since ( )2.20794 0,x′ > the particle is moving toward the right at time 2.207t = or 2.208.

3 :

1 : considers 0

1 : 2.207 or 2.2081 : direction of motion with

reason

dydx

t

⎧ =⎪⎪ =⎨⎪⎪⎩

(d) ( ) 5x t = at 1t = and 3t =

At time 1,t = the slope is 1 1

0.432.t t

dy dtdydx dx dt= =

= =

At time 3,t = the slope is 3 3

1.t t

dy dtdydx dx dt= =

= =

( ) ( )3

2

11 3 3 4dyy y dte dt= = + + =⌠⌡

3 : 1 : 1 and 3

1 : slopes1 : -coordinate

t t

y

= =⎧⎪⎨⎪⎩


Question 4


Let R be the region in the first quadrant bounded by the graph of 2 ,y x= the horizontal line 6,y = and the y-axis, as shown in the figure above. (a) Find the area of R. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is

rotated about the horizontal line 7.y =

(c) Region R is the base of a solid. For each y, where 0 6,y≤ ≤ the cross section of the solid taken perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R. Write, but do not evaluate, an integral expression that gives the volume of the solid.

(a)

Area ( ) ( ) 99 3 20 0

46 2 6 183

x

xx dx x x

=

== − = − =∫

3 : 1 : integrand1 : antiderivative

1 : answer

⎧⎪⎨⎪⎩

(b)

Volume ( ) ( )( )9 2 20

7 2 7 6x dxπ= − − −∫

3 : { 2 : integrand1 : limits and constant

(c) Solving 2y x= for x yields 2

.4yx =

Each rectangular cross section has area 2 2

433 .4 4 16y y y

⎛ ⎞⎛ ⎞=⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

Volume 6

4

0

316 y dy= ⌠

⌡



Question 5


Consider the differential equation 1 .dy ydx = − Let ( )y f x= be the particular solution to this differential

equation with the initial condition ( )1 0.f = For this particular solution, ( ) 1f x < for all values of x.

(a) Use Euler’s method, starting at 1x = with two steps of equal size, to approximate ( )0 .f Show the work that leads to your answer.

(b) Find ( )31

lim .1x

f xx→ −

Show the work that leads to your answer.

(c) Find the particular solution ( )y f x= to the differential equation 1dy ydx = − with the initial condition

( )1 0.f =

(a) ( ) ( )( )

( )1, 0

1 12

1 10 1 2 2

dyf f xdx⎛ ⎞

≈ + ⋅ Δ⎜ ⎟⎝ ⎠

= + ⋅ − = −

( ) ( ) ( )( )

1 1,2 2

10 2

1 3 1 52 2 2 4

dyf f xdx −

⎛ ⎞⎜ ⎟≈ + ⋅ Δ⎜ ⎟⎝ ⎠

≈ − + ⋅ − = −

2 : { 1 : Euler’s method with two steps 1 : answer

(b) Since f is differentiable at 1,x = f is continuous at 1.x = So,

( ) ( )31 1

lim 0 lim 1x x

f x x→ →

= = − and we may apply L’Hospital’s

Rule. ( ) ( ) ( )

13 2 21 1

1

lim 1lim lim 31 3 lim 3x

x xx

f xf x f xx x x

→→ →

→

′′= = =

−

2 : { 1 : use of L’Hospital’s Rule 1 : answer

(c) 1dy ydx = −

1 11 dy dxy =−⌠ ⌠⎮⎮

⌡⌡

ln 1 y x C− − = +

ln 1 1 1C C− = + ⇒ = −

ln 1 1y x− = −

11 xy e −− =

( ) 11 xf x e −= −

5 :


1 : solves for y

⎧⎪⎪⎨⎪⎪⎩

Note: max 2 5 [1-1-0-0-0] if no

constant of integration Note: 0 5 if no separation of variables


Question 6


( )2

cos 1 for 0

1 for 02

x xxf x

x

−⎧ ≠⎪= ⎨⎪− =⎩

The function f, defined above, has derivatives of all orders. Let g be the function defined by

( ) ( )0

1 .x

g x f t dt= + ∫

(a) Write the first three nonzero terms and the general term of the Taylor series for cos x about 0.x = Use this series to write the first three nonzero terms and the general term of the Taylor series for f about 0.x =

(b) Use the Taylor series for f about 0x = found in part (a) to determine whether f has a relative maximum, relative minimum, or neither at 0.x = Give a reason for your answer.

(c) Write the fifth-degree Taylor polynomial for g about 0.x = (d) The Taylor series for g about 0,x = evaluated at 1,x = is an alternating series with individual terms that

decrease in absolute value to 0. Use the third-degree Taylor polynomial for g about 0x = to estimate the

value of ( )1 .g Explain why this estimate differs from the actual value of ( )1g by less than 1 .6!

(a) ( ) ( ) ( )2 4 2

cos 1 12 4! 2 !n

nx x xx n= − + − + − +

( ) ( ) ( )2 4 2

11 12 4! 6! 2 2 !n

nx x xf x n+= − + − + + − ++

3 :

1 : terms for cos 2 : terms for 1 : first three terms 1 : general term

xf

⎧⎪⎪⎨⎪⎪⎩

(b)

( )0f ′ is the coefficient of x in the Taylor series for f about 0,x = so ( )0 0.f ′ =

( )0 12! 4!

f ′′= is the coefficient of 2x in the Taylor series for f about

0,x = so ( ) 10 .12f ′′ =

Therefore, by the Second Derivative Test, f has a relative minimum at 0.x =

2 : ( ) 1 : determines 0 1 : answer with reason

f ′⎧⎨⎩

(c) ( )3 5

5 1 2 3·4! ·6!5x x xP x = − + −

2 : { 1 : two correct terms 1 : remaining terms

(d) ( ) 1 1 371 1 2 3·4! 72g ≈ − + =

Since the Taylor series for g about 0x = evaluated at 1x = is alternating and the terms decrease in absolute value to 0, we know

( ) 37 1 11 .72 5 6·6! !g − < <

2 : { 1 : estimate 1 : explanation


Form B

The College Board









1. In the figure above, R is the shaded region in the first quadrant bounded by the graph of ( )4 ln 3 ,= -y x the horizontal line 6,=y and the vertical line 2.=x


(b) Find the volume of the solid generated when R is revolved about the horizontal line 8.=y

(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of the solid.




-3-

2. The velocity vector of a particle moving in the xy-plane has components given by

( ) ( )214cos sin= tdx t edt

and ( )21 2sin ,= +dyt

dt for 0 1.5.£ £t

At time 0,=t the position of the particle is ( )2, 3 .-

(a) For 0 1.5,< <t find all values of t at which the line tangent to the path of the particle is vertical.

(b) Write an equation for the line tangent to the path of the particle at 1.=t

(c) Find the speed of the particle at 1.=t

(d) Find the acceleration vector of the particle at 1.=t

t 0 2 4 6 8 10 12

P(t) 0 46 53 57 60 62 63

3. The figure above shows an aboveground swimming pool in the shape of a cylinder with a radius of 12 feet and a

height of 4 feet. The pool contains 1000 cubic feet of water at time 0.=t During the time interval 0 12£ £t hours, water is pumped into the pool at the rate ( )P t cubic feet per hour. The table above gives values of ( )P t

for selected values of t. During the same time interval, water is leaking from the pool at the rate ( )R t cubic feet

per hour, where ( ) 0.0525 .-= tR t e (Note: The volume V of a cylinder with radius r and height h is given by 2 .p=V r h )

(a) Use a midpoint Riemann sum with three subintervals of equal length to approximate the total amount of water that was pumped into the pool during the time interval 0 12£ £t hours. Show the computations that lead to your answer.

(b) Calculate the total amount of water that leaked out of the pool during the time interval 0 12£ £t hours.

(c) Use the results from parts (a) and (b) to approximate the volume of water in the pool at time 12=t hours. Round your answer to the nearest cubic foot.

(d) Find the rate at which the volume of water in the pool is increasing at time 8=t hours. How fast is the water level in the pool rising at 8=t hours? Indicate units of measure in both answers.









4. A squirrel starts at building A at time 0=t and travels along a straight, horizontal wire connected to building B. For 0 18,£ £t the squirrel’s velocity is modeled by the piecewise-linear function defined by the graph above.

(a) At what times in the interval 0 18,< <t if any, does the squirrel change direction? Give a reason for your answer.

(b) At what time in the interval 0 18£ £t is the squirrel farthest from building A ? How far from building A is the squirrel at that time?

(c) Find the total distance the squirrel travels during the time interval 0 18.£ £t

(d) Write expressions for the squirrel’s acceleration ( ),a t velocity ( ),v t and distance ( )x t from building A that are valid for the time interval 7 10.< <t




-5-

5. Let f and g be the functions defined by ( ) 1=f xx

and ( )2

4 ,1 4

=+

xg xx

for all 0.>x

(a) Find the absolute maximum value of g on the open interval ( )0, • if the maximum exists. Find the

absolute minimum value of g on the open interval ( )0, • if the minimum exists. Justify your answers.

(b) Find the area of the unbounded region in the first quadrant to the right of the vertical line 1,=x below the graph of f, and above the graph of g.

6. The Maclaurin series for the function f is given by ( ) ( ) ( )2

1 21=

• -= -Â

n n

n

xf x

n on its interval of convergence.

(a) Find the interval of convergence for the Maclaurin series of f. Justify your answer.

(b) Show that ( )=y f x is a solution to the differential equation 24

1 2- =¢ +

xxy yx

for ,<x R where R is the

radius of convergence from part (a).


END OF EXAM


Form B

The College Board




Question 1


In the figure above, R is the shaded region in the first quadrant bounded by the graph of ( )4ln 3 ,y x= − the horizontal line 6,y = and the vertical line 2.x =

(a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the

horizontal line 8.y =

(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of the solid.

1 : Correct limits in an integral in (a), (b), or (c)

(a) ( )( )2

06 4ln 3 6.816x dx− − =∫ or 6.817


(b) ( )( ) ( )( )2 2 20

8 4ln 3 8 6x dxπ − − − −∫

168.179= or 168.180


(c) ( )( )2 20

6 4ln 3 26.266x dx− − =∫ or 26.267



Question 2


The velocity vector of a particle moving in the plane has components given by

( ) ( )214cos sin tdx t edt = and ( )21 2sin ,dy tdt = + for 0 1.5.t≤ ≤

At time 0,t = the position of the particle is ( )2, 3 .−

(a) For 0 1.5,t< < find all values of t at which the line tangent to the path of the particle is vertical. (b) Write an equation for the line tangent to the path of the particle at 1.t = (c) Find the speed of the particle at 1.t = (d) Find the acceleration vector of the particle at 1.t =

(a) The tangent line is vertical when ( ) 0x t′ = and ( ) 0.y t′ ≠

On 0 1.5,t< < this happens at 1.253t = and 1.144t = or 1.145.

2 : 1 : sets 0

1 : answer

dxdy

⎧ =⎪⎨⎪⎩

(b) ( )( )1

1 0.8634471t

ydydx x=

′= =′

( ) ( )1

01 2 9.314695x x t dt′= − + =∫

( ) ( )1

01 3 4.620537y y t dt′= + =∫

The line tangent to the path of the particle at 1t = has equation ( )4.621 0.863 9.315 .y x= + −

4 : ( )( )

1 1 :

1 : 1 1 : 11 : equation

t

dydxxy

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(c) Speed ( )( ) ( )( )2 21 1 4.105x y′ ′= + =

1 : answer

(d) Acceleration vector: ( ) ( )1 , 1 28.425, 2.161x y′′ ′′ = −

2 : ( )( )

1 : 11 : 1

xy′′⎧

⎨ ′′⎩


Question 3


t 0 2 4 6 8 10 12

P(t) 0 46 53 57 60 62 63

The figure above shows an aboveground swimming pool in the shape of a cylinder with a radius of 12 feet and a height of 4 feet. The pool contains 1000 cubic feet of water at time 0.t = During the time interval 0 12t≤ ≤ hours, water is pumped into the pool at the rate ( )P t cubic feet per hour. The table above gives values of ( )P t for selected values of t. During the same time interval, water is leaking from the pool at the rate ( )R t cubic feet

per hour, where ( ) 0.0525 .tR t e−= (Note: The volume V of a cylinder with radius r and height h is given by 2 .V r hπ= )

(a) Use a midpoint Riemann sum with three subintervals of equal length to approximate the total amount of water that was pumped into the pool during the time interval 0 12t≤ ≤ hours. Show the computations that lead to your answer.

(b) Calculate the total amount of water that leaked out of the pool during the time interval 0 12t≤ ≤ hours. (c) Use the results from parts (a) and (b) to approximate the volume of water in the pool at time 12t = hours.

Round your answer to the nearest cubic foot. (d) Find the rate at which the volume of water in the pool is increasing at time 8t = hours. How fast is the water

level in the pool rising at 8t = hours? Indicate units of measure in both answers.

(a) ( )12 30

46 4 57 4 62 4 660 ftP t dt ≈ ⋅ + ⋅ + ⋅ =∫

2 : { 1 : midpoint sum 1 : answer

(b) ( )12 30

225.594 ftR t dt =∫


(c) ( ) ( )12 12

0 01000 1434.406P t dt R t dt+ − =∫ ∫

At time 12t = hours, the volume of water in the pool is approximately 31434 ft .

1 : answer

(d) ( ) ( ) ( )V t P t R t′ = −

( ) ( ) ( ) 0.4 38 8 8 60 25 43.241 or 43.242 ft hrV P R e−′ = − = − =

( )212V hπ=

144dV dhdt dtπ=

8 8

1 0.095144t t

dh dVdt dtπ= =

= ⋅ = or 0.096 ft hr

4 :

( )

83

1 : 8

1 : equation relating and

1 :

1 : units of ft hr and ft hrt

VdV dhdt dt

dhdt =

′⎧⎪⎪⎪⎨⎪⎪⎪⎩


Question 4


A squirrel starts at building A at time 0t = and travels along a straight wire connected to building B. For 0 18,t≤ ≤ the squirrel’s velocity is modeled by the piecewise-linear function defined by the graph above. (a) At what times in the interval 0 18,t< < if any, does the

squirrel change direction? Give a reason for your answer. (b) At what time in the interval 0 18t≤ ≤ is the squirrel

farthest from building A ? How far from building A is the squirrel at this time?

(c) Find the total distance the squirrel travels during the time interval 0 18.t≤ ≤ (d) Write expressions for the squirrel’s acceleration ( ) ,a t velocity ( ) ,v t and distance ( )x t from building A that

are valid for the time interval 7 10.t< <

(a) The squirrel changes direction whenever its velocity changes sign. This occurs at 9t = and 15.t = 2 : { 1 : -values

1 : explanationt

(b) Velocity is 0 at 0,t = 9,t = and 15.t =

t position at time t 0 0

9 9 5 20 1402+ ⋅ =

15 6 4140 10 902+− ⋅ =

18 3 290 10 1152++ ⋅ =

The squirrel is farthest from building A at time 9;t = its greatest distance from the building is 140.

2 : { 1 : identifies candidates 1 : answers

(c) The total distance traveled is ( )18

0140 50 25 215.v t dt = + + =∫

1 : answer

(d) For 7 10,t< < ( ) ( )20 10 107 10a t − −= = −

−

( ) ( )20 10 7 10 90v t t t= − − = − +

( ) 7 57 20 1202x += ⋅ =

( ) ( ) ( )

( )7

27

2

7 10 90

120 5 90

5 90 265

t

u t

u

x t x u du

u u

t t

=

=

= + − +

= + − +

= − + −

∫

4 : ( )( )( )

1 : 1 :

2 :

a tv tx t

⎧⎪⎨⎪⎩


Question 5


Let f and g be the functions defined by ( ) 1f x x= and ( ) 24 ,

1 4xg xx

=+

for all 0.x >

(a) Find the absolute maximum value of g on the open interval ( )0, ∞ if the maximum exists. Find the absolute minimum value of g on the open interval ( )0, ∞ if the minimum exists. Justify your answers.

(b) Find the area of the unbounded region in the first quadrant to the right of the vertical line 1,x = below the graph of f, and above the graph of g.

(a) ( )( ) ( )

( )( )

( )2 2

2 22 2

4 1 4 4 8 4 1 4

1 4 1 4

x x x xg x

x x

+ − −′ = =

+ +

For 0,x > ( ) 0g x′ = for 1 .2x =

( ) 0g x′ > for 10 2x< <

( ) 0g x′ < for 12x >

( )1 12g =

Therefore g has a maximum value of 1 at 1 ,2x = and

g has no minimum value on the open interval ( )0, .∞

5 :

( ) 2 : 1 : critical point

1 : answers1 : justification

g x′⎧⎪⎪⎨⎪⎪⎩

(b) ( ) ( )( ) ( ) ( )( )1 1

limb

bf x g x dx f x g x dx

→

∞∞

− = −∫ ∫

( ) ( )( )2

1

1lim ln ln 1 42

x b

b xx x

=

→ =∞= − +

( ) ( ) ( )( )21 1lim ln ln 1 4 ln 52 2bb b

→∞= − + +

2

5lim ln1 4b

bb→∞

⎛ ⎞= ⎜ ⎟⎝ ⎠+

2

25lim ln

1 4bb

b→∞⎛ ⎞

= ⎜ ⎟+⎝ ⎠

2

21 5lim ln2 1 4b

bb→∞

⎛ ⎞= ⎜ ⎟+⎝ ⎠

1 5ln2 4=


1 : answer

⎧⎪⎨⎪⎩


Question 6


The Maclaurin series for the function f is given by ( ) ( ) ( )2

1 21

n n

n

xf x n=

∞ −= −∑ on its interval of convergence.

(a) Find the interval of convergence for the Maclaurin series of f. Justify your answer.

(b) Show that ( )y f x= is a solution to the differential equation 24

1 2xx y y x

′ − = + for ,x R< where R is the

radius of convergence from part (a).

(a)

( )( )

( )

121 1 1 1lim lim 2 lim 2 2

21

n

nn n n

xn n nx x xn nx

n

+

→ → →∞ ∞ ∞+ − − −= ⋅ = ⋅ =

−

2 1x < for 12x <

Therefore the radius of convergence is 1 .2

When 1 ,2x = − the series is ( ) ( )2 2

1 1 1 .1 1

n n

n nn n= =

∞ ∞− −=− −∑ ∑

This is the harmonic series, which diverges.

When 1 ,2x = the series is ( ) ( )2 2

1 1 1 .1 1

n nn

n nn n= =

∞ ∞− −=− −∑ ∑

This is the alternating harmonic series, which converges.

The interval of convergence for the Maclaurin series of f is ( 1 1, .2 2⎤− ⎥⎦

5 :

1 : sets up ratio 1 : limit evaluation 1 : radius of convergence1 : considers both endpoints 1 : analysis and interval of

convergence

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(b) ( ) ( ) ( ) ( ) ( )

( ) ( )

2 3 4

2 3 4

2 2 2 1 21 2 3 1

1 2164 4 3 1

n n

n n

x x x xy nxx x x n

−= − + − + +−

−= − + − + +−

( ) ( ) 12 3 1 2 2648 12 3 1

n nn xy x x x n

−− ⋅′ = − + − + +−

( ) ( )2 3 4 1 2648 12 3 1

n nn xxy x x x n−′ = − + − + +

−

( ) ( )( ) ( )( )

2 3 4

22 2

4 8 16 1 2

4 1 2 4 1 2

n n

n n

xy y x x x x

x x x x −

′ − = − + − + − +

= − + − + − +

The series ( ) ( ) ( )22

01 2 4 1 2 2n n n

nx x x x−

=

∞− + − + − + = −∑ is a

geometric series that converges to 11 2x+

for 1 .2x < Therefore

2 14 1 2xy y x x′ − = ⋅

+ for 1 .2x <

4 :

1 : series for 1 : series for 1 : series for 1 : analysis with geometric series

yxyxy y

′⎧⎪ ′⎪⎨ ′ −⎪⎪⎩


About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the College Board was created to expand access to higher education. Today, the membership association is made up of more than 5,900 of the world’s leading educational institutions and is dedicated to promoting excellence and equity in education. Each year, the College Board helps more than seven million students prepare for a successful transition to college through programs and services in college readiness and college success — including the SAT® and the Advanced Placement Program®. The organization also serves the education community through research and advocacy on behalf of students, educators and schools. © 2011 The College Board. College Board, Advanced Placement Program, AP, AP Central, SAT and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation Service and inspiring minds are trademarks owned by the College Board. All other products and services may be trademarks of their respective owners. Visit the College Board on the Web: www.collegeboard.org. Permission to use copyrighted College Board materials may be requested online at: www.collegeboard.org/inquiry/cbpermit.html. Visit the College Board on the Web: www.collegeboard.org. AP Central is the official online home for the AP Program: apcentral.collegeboard.com.


© 2011 The College Board. Visit the College Board on the Web: www.collegeboard.org.




A graphing calculator is required for these problems.

1. At time t, a particle moving in the xy-plane is at position ( ) ( )( ), ,x t y t where ( )x t and ( )y t are not explicitly

given. For 0,t ≥ 4 1dx tdt

= + and ( )2sin .dy

tdt

= At time 0,t = ( )0 0x = and ( )0 4.y = -

(a) Find the speed of the particle at time 3,t = and find the acceleration vector of the particle at time 3.t =

(b) Find the slope of the line tangent to the path of the particle at time 3.t =

(c) Find the position of the particle at time 3.t =

(d) Find the total distance traveled by the particle over the time interval 0 3.t£ £





t

(minutes) 0 2 5 9 10

( )H t (degrees Celsius)

66 60 52 44 43

2. As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for 0 10,t£ £ where

time t is measured in minutes and temperature ( )H t is measured in degrees Celsius. Values of ( )H t at selected values of time t are shown in the table above.

(a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time 3.5.t = Show the computations that lead to your answer.


0

110

H t dtÚ in the context of this problem. Use a trapezoidal

sum with the four subintervals indicated by the table to estimate ( )10

0

1 .10

H t dtÚ

(c) Evaluate ( )10

0.H t dt¢Ú Using correct units, explain the meaning of the expression in the context of this

problem.

(d) At time 0,t = biscuits with temperature 100 C∞ were removed from an oven. The temperature of the biscuits at time t is modeled by a differentiable function B for which it is known that

( ) 0.17313.84 .tB t e-= -¢ Using the given models, at time 10,t = how much cooler are the biscuits than the tea?









3. Let ( ) 2 .xf x e= Let R be the region in the first quadrant bounded by the graph of f, the coordinate axes, and the vertical line ,x k= where 0.k > The region R is shown in the figure above.

(a) Write, but do not evaluate, an expression involving an integral that gives the perimeter of R in terms of k.

(b) The region R is rotated about the x-axis to form a solid. Find the volume, V, of the solid in terms of k.

(c) The volume V, found in part (b), changes as k changes. If 1 ,3

dkdt

= determine dVdt

when 1 .2

k =





4. The continuous function f is defined on the interval 4 3.x- £ £ The graph of f consists of two quarter circles

and one line segment, as shown in the figure above. Let ( ) ( )0

2 .x

g x x f t dt= + Ú

(a) Find ( )3 .g - Find ( )g x¢ and evaluate ( )3 .g -¢

(b) Determine the x-coordinate of the point at which g has an absolute maximum on the interval 4 3.x- £ £ Justify your answer.

(c) Find all values of x on the interval 4 3x- < < for which the graph of g has a point of inflection. Give a reason for your answer.

(d) Find the average rate of change of f on the interval 4 3.x- £ £ There is no point c, 4 3,c- < < for

which ( )f c¢ is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem.





5. At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners estimate that W will satisfy the differential

equation ( )1 30025

dW Wdt

= - for the next 20 years. W is measured in tons, and t is measured in years from

the start of 2010.

(a) Use the line tangent to the graph of W at 0t = to approximate the amount of solid waste that the landfill

contains at the end of the first 3 months of 2010 (time 14

t = ).

(b) Find 2

2d W

dt in terms of W. Use

2

2d W

dt to determine whether your answer in part (a) is an underestimate or

an overestimate of the amount of solid waste that the landfill contains at time 1 .4

t =

(c) Find the particular solution ( )W W t= to the differential equation ( )1 30025

dW Wdt

= - with initial

condition ( )0 1400.W =




-7-

6. Let ( ) ( )2sin cos .f x x x= + The graph of ( ) ( )5y f x= is shown above.

(a) Write the first four nonzero terms of the Taylor series for sin x about 0,x = and write the first four

nonzero terms of the Taylor series for ( )2sin x about 0.x =

(b) Write the first four nonzero terms of the Taylor series for cos x about 0.x = Use this series and the series

for ( )2sin ,x found in part (a), to write the first four nonzero terms of the Taylor series for f about 0.x =

(c) Find the value of ( ) ( )6 0 .f

(d) Let ( )4P x be the fourth-degree Taylor polynomial for f about 0.x = Using information from the graph of

( ) ( )5y f x= shown above, show that ( ) ( )41 1 1 .4 4 3000

P f- <


END OF EXAM


The College Board 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 College Board is composed of more than 5,700 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,800 colleges through major programs and services in college readiness, college admission, guidance, assessment, financial aid and enrollment. Among its widely recognized programs are the SAT®, the PSAT/NMSQT®, the Advanced Placement Program®

(AP®), SpringBoard® and ACCUPLACER®. 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. © 2011 The College Board. College Board, ACCUPLACER, Advanced Placement Program, AP, AP Central, SAT, SpringBoard and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation Service is a trademark 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.org. AP Central is the official online home for the AP Program: apcentral.collegeboard.com.


Question 1


At time t, a particle moving in the xy-plane is at position ( ) ( )( ), ,x t y t where ( )x t and ( )y t are not explicitly

given. For 0,t ≥ 4 1dx tdt = + and ( )2sin .dy tdt = At time 0,t = ( )0 0x = and ( )0 4.y = −

(a) Find the speed of the particle at time 3,t = and find the acceleration vector of the particle at time 3.t =

(b) Find the slope of the line tangent to the path of the particle at time 3.t = (c) Find the position of the particle at time 3.t = (d) Find the total distance traveled by the particle over the time interval 0 3.t≤ ≤

(a) Speed ( )( ) ( )( )2 23 3 13.006x y′ ′= + = or 13.007 Acceleration ( ) ( )3 , 3

4, 5.466 or 4, 5.467x y′′ ′′=

= − −

2 : { 1 : speed1 : acceleration

(b) Slope ( )( )3 0.0313

yx′

= =′ or 0.032

1 : answer

(c) ( )3

03 0 21dxx dtdt= + =⌠

⌡

( )3

03 4 3.226dyy dtdt= − + = −⌠

⌡

At time 3,t = the particle is at position ( )21, 3.226 .−

4 :

2 : -coordinate 1 : integral 1 : answer2 : -coordinate 1 : integral 1 : answer

x

y

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(d) Distance ( ) ( )223

021.091dydx dtdt dt= + =⌠

⎮⌡



Question 2


t (minutes) 0 2 5 9 10

( )H t (degrees Celsius)

66 60 52 44 43

As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for 0 10,t≤ ≤ where time t is measured in minutes and temperature ( )H t is measured in degrees Celsius. Values of ( )H t at selected values of time t are shown in the table above. (a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time

3.5.t = Show the computations that lead to your answer.


01

10 H t dt∫ in the context of this problem. Use a trapezoidal

sum with the four subintervals indicated by the table to estimate ( )10

01 .10 H t dt∫

(c) Evaluate ( )10

0.H t dt′∫ Using correct units, explain the meaning of the expression in the context of this

problem. (d) At time 0,t = biscuits with temperature 100 C° were removed from an oven. The temperature of the

biscuits at time t is modeled by a differentiable function B for which it is known that ( ) 0.17313.84 .tB t e−′ = − Using the given models, at time 10,t = how much cooler are the biscuits than

the tea?

(a) ( ) ( ) ( )5 23.5 5 252 60 2.666 or 2.6673

H HH −′ ≈−

−= = − − degrees Celsius per minute

1 : answer

(b) ( )10

01

10 H t dt∫ is the average temperature of the tea, in degrees Celsius,

over the 10 minutes.

( ) ( )10

01 1 66 60 60 52 52 44 44 432 3 4 110 10 2 2 2 2

52.95

H t dt + + + +≈ ⋅ + ⋅ + ⋅ + ⋅

=

∫

3 : 1 : meaning of expression 1 : trapezoidal sum 1 : estimate

⎧⎪⎨⎪⎩

(c) ( ) ( ) ( )10

010 0 43 66 23H t dt H H′ = − = − = −∫

The temperature of the tea drops 23 degrees Celsius from time 0t = to time 10t = minutes.

2 : { 1 : value of integral 1 : meaning of expression

(d) ( ) ( )10

010 100 34.18275;B B t dt′= + =∫ ( ) ( )10 10 8.817H B− =

The biscuits are 8.817 degrees Celsius cooler than the tea.

3 : ( ) 1 : integrand 1 : uses 0 100 1 : answer

B⎧⎪ =⎨⎪⎩


Question 3


Let ( ) 2 .xf x e= Let R be the region in the first quadrant bounded by the graph of f, the coordinate axes, and the vertical line ,x k= where

0.k > The region R is shown in the figure above.

(a) Write, but do not evaluate, an expression involving an integral that gives the perimeter of R in terms of k.

(b) The region R is rotated about the x-axis to form a solid. Find the volume, V, of the solid in terms of k.

(c) The volume V, found in part (b), changes as k changes. If 1 ,3dkdt =

determine dVdt when 1 .2k =

(a) ( ) 22 xf x e′ = Perimeter ( )2 2

0

21 1 2

kk xk e e dx= + + + +∫

3 : ( ) 1 :

1 : integral1 : answer

f x′⎧⎪⎨⎪⎩

(b)

Volume ( )2 4 4 40 0

2

04 4 4x kk kx x x k

xe dx e dx e eπ π ππ π

=

== = = = −∫ ∫ 4 :

1 : integrand 1 : limits1 : antiderivative

1 : answer

⎧⎪⎪⎨⎪⎪⎩

(c) 4kdV dkedt dtπ=

When 1 ,2k = 2 1 .3dV edt π= ⋅

2 : { 1 : applies chain rule 1 : answer


Question 4


The continuous function f is defined on the interval 4 3.x− ≤ ≤ The graph of f consists of two quarter circles and one line segment, as shown in the figure above.

Let ( ) ( )0

2 .x

g x x f t dt= + ∫

(a) Find ( )3 .g − Find ( )g x′ and evaluate ( )3 .g′ −

(b) Determine the x-coordinate of the point at which g has an absolute maximum on the interval 4 3.x− ≤ ≤ Justify your answer.

(c) Find all values of x on the interval 4 3x− < < for which the graph of g has a point of inflection. Give a reason for your answer.

(d) Find the average rate of change of f on the interval 4 3.x− ≤ ≤ There is no point c, 4 3,c− < < for which ( )f c′ is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem.

(a) ( ) ( ) ( )3

093 63 2 4f t dtg π−

− + = − −− = ∫

( ) ( )2g x f x′ = +

( ) ( )3 2 3 2g f′ − = + − =

3 : ( )( )( )

1 : 3 1 : 1 : 3

gg xg

−⎧⎪ ′⎨⎪ ′ −⎩

(b) ( ) 0g x′ = when ( ) 2.f x = − This occurs at 5 .2x =

( ) 0g x′ > for 54 2x− < < and ( ) 0g x′ < for 5 3.2 x< <

Therefore g has an absolute maximum at 5 .2x =

3 : ( ) 1 : considers 0

1 : identifies interior candidate1 : answer with justification

g x′ =⎧⎪⎨⎪⎩

(c) ( ) ( )g x f x′′ ′= changes sign only at 0.x = Thus the graph of g has a point of inflection at 0.x =


(d) The average rate of change of f on the interval 4 3x− ≤ ≤ is ( )3 ( 4) 2 .3 ( 4) 7

f f− −= −

− −

To apply the Mean Value Theorem, f must be differentiable at each point in the interval 4 3.x− < < However, f is not differentiable at 3x = − and 0.x =

2 : { 1 : average rate of change 1 : explanation


Question 5


At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners estimate that W will satisfy the differential

equation ( )1 30025dW Wdt = − for the next 20 years. W is measured in tons, and t is measured in years from

the start of 2010. (a) Use the line tangent to the graph of W at 0t = to approximate the amount of solid waste that the landfill

contains at the end of the first 3 months of 2010 (time 14t = ).

(b) Find 2

2d Wdt

in terms of W. Use 2

2d Wdt

to determine whether your answer in part (a) is an underestimate or

an overestimate of the amount of solid waste that the landfill contains at time 1 .4t =

(c) Find the particular solution ( )W W t= to the differential equation ( )1 30025dW Wdt = − with initial

condition ( )0 1400.W =

(a) ( )( ) ( )0

1 10 300 1400 300 4425 25t

dW Wdt == − = − =

The tangent line is 1400 44 .y t= +

( ) ( )1 11400 44 14114 4W ≈ + = tons

2 : 1 : at 0

1 : answer

dW tdt⎧ =⎪⎨⎪⎩

(b) ( )2

21 1 30025 625

d W dW Wdtdt= = − and 1400W ≥

Therefore 2

2 0d Wdt

> on the interval 10 .4t≤ ≤

The answer in part (a) is an underestimate.

2 : 2

2 1 :


d Wdt

⎧⎪⎨⎪⎩

(c) ( )1 30025dW Wdt = −

1 1

300 25dW dtW =−

⌠ ⌠⌡ ⌡

1ln 300 25W t C− = +

( ) ( ) ( )1ln 1400 300 0 ln 110025 C C− = + ⇒ =

125300 1100

tW e− =

( )125300 1100 ,

tW t e= + 0 20t≤ ≤

5 :


1 : solves for W

⎧⎪⎪⎨⎪⎪⎩

Note: max 2 5 [1-1-0-0-0] if no constant of

integration Note: 0 5 if no separation of variables


Question 6


Let ( ) ( )2sin cos .f x x x= + The graph of ( ) ( )5y f x= is

shown above.

(a) Write the first four nonzero terms of the Taylor series for sin x about 0,x = and write the first four nonzero terms

of the Taylor series for ( )2sin x about 0.x =

(b) Write the first four nonzero terms of the Taylor series for cos x about 0.x = Use this series and the series for

( )2sin ,x found in part (a), to write the first four nonzero

terms of the Taylor series for f about 0.x =

(c) Find the value of ( ) ( )6 0 .f

(d) Let ( )4P x be the fourth-degree Taylor polynomial for f about 0.x = Using information from the graph of

( ) ( )5y f x= shown above, show that ( ) ( )41 1 1 .4 4 3000P f− <

(a) 3 5 7

sin 3! 5! 7!x x xx x= − + − +

( )6 10 1

24

2sin 3! 5! 7!x x xx x= − + − +

3 : ( )2 1 : series for sin 2 : series for sin

xx

⎧⎪⎨⎪⎩

(b) 2 4 6

cos 1 2! 4! 6!x x xx = − + − +

( )2 4 61211 2 4! 6!

x x xf x = + + − +

3 : ( )

1 : series for cos 2 : series for

xf x

⎧⎨⎩

(c) ( ) ( )6 06!

f is the coefficient of 6x in the Taylor series for f about

0.x = Therefore ( )(6) 0 121.f = −

1 : answer

(d) The graph of ( ) ( )5y f x= indicates that ( )14

(5)

0max 40.

xf x

≤ ≤<

Therefore

( ) ( )( )

( )14

(5)5

4 50max

1 40 1 1 .5! 4 3071 14 4 2 3000120 4

xf x

P f≤ ≤

≤ ⋅ < = <⋅

−

2 : { 1 : form of the error bound 1 : analysis


Form B

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-2-



A graphing calculator is required for these problems.

1. A cylindrical can of radius 10 millimeters is used to measure rainfall in Stormville. The can is initially empty,

and rain enters the can during a 60-day period. The height of water in the can is modeled by the function S, where ( )S t is measured in millimeters and t is measured in days for 0 60.t£ £ The rate at which the height

of the water is rising in the can is given by ( ) ( )2sin 0.03 1.5.S t t= +¢

(a) According to the model, what is the height of the water in the can at the end of the 60-day period?

(b) According to the model, what is the average rate of change in the height of water in the can over the 60-day period? Show the computations that lead to your answer. Indicate units of measure.

(c) Assuming no evaporation occurs, at what rate is the volume of water in the can changing at time 7 ?t = Indicate units of measure.

(d) During the same 60-day period, rain on Monsoon Mountain accumulates in a can identical to the one in Stormville. The height of the water in the can on Monsoon Mountain is modeled by the function M, where

( ) ( )3 21 3 30 330 .400

M t t t t= - + The height ( )M t is measured in millimeters, and t is measured in days

for 0 60.t£ £ Let ( ) ( ) ( ).D t M t S t= -¢ ¢ Apply the Intermediate Value Theorem to the function D on the interval 0 60t£ £ to justify that there exists a time t, 0 60,t< < at which the heights of water in the two cans are changing at the same rate.

2. The polar curve r is given by ( ) 3 sin ,r q q q= + where 0 2 .q p£ £

(a) Find the area in the second quadrant enclosed by the coordinate axes and the graph of r.

(b) For ,2p

q p£ £ there is one point P on the polar curve r with x-coordinate 3.- Find the angle q that

corresponds to point P. Find the y-coordinate of point P. Show the work that leads to your answers.

(c) A particle is traveling along the polar curve r so that its position at time t is ( ) ( )( ),x t y t and such that

2.ddtq = Find

dydt

at the instant that 2 ,3p

q = and interpret the meaning of your answer in the context of

the problem.






-3-




3. The functions f and g are given by ( )f x x= and ( ) 6 .g x x= - Let R be the region bounded by the x-axis and the graphs of f and g, as shown in the figure above.


(b) The region R is the base of a solid. For each y, where 0 2,y£ £ the cross section of the solid taken perpendicular to the y-axis is a rectangle whose base lies in R and whose height is 2y. Write, but do not evaluate, an integral expression that gives the volume of the solid.

(c) There is a point P on the graph of f at which the line tangent to the graph of f is perpendicular to the graph of g. Find the coordinates of point P.





-4-

4. The graph of the differentiable function ( )y f x= with domain 0 10x£ £ is shown in the figure above. The area of the region enclosed between the graph of f and the x-axis for 0 5x£ £ is 10, and the area of the region enclosed between the graph of f and the x-axis for 5 10x£ £ is 27. The arc length for the portion of the graph of f between 0x = and 5x = is 11, and the arc length for the portion of the graph of f between 5x = and

10x = is 18. The function f has exactly two critical points that are located at 3x = and 8.x =

(a) Find the average value of f on the interval 0 5.x£ £


03 2 .f x dx+Ú Show the computations that lead to your answer.

(c) Let ( ) ( )5

.x

g x f t dt= Ú On what intervals, if any, is the graph of g both concave up and decreasing? Explain

your reasoning.

(d) The function h is defined by ( ) ( )2 .2xh x f= The derivative of h is ( ) ( ).2

xh x f=¢ ¢ Find the arc length of

the graph of ( )y h x= from 0x = to 20.x =





-5-

t

(seconds) 0 10 40 60

( )B t (meters)

100 136 9 49

( )v t (meters per second)

2.0 2.3 2.5 4.6

5. Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiable function B models Ben’s position on the track, measured in meters from the western end of the track, at time t, measured in seconds from the start of the ride. The table above gives values for ( )B t and Ben’s velocity, ( ),v t measured in meters per second, at selected times t.

(a) Use the data in the table to approximate Ben’s acceleration at time 5t = seconds. Indicate units of measure.

(b) Using correct units, interpret the meaning of ( )60

0v t dtÚ in the context of this problem. Approximate

( )60

0v t dtÚ using a left Riemann sum with the subintervals indicated by the data in the table.

(c) For 40 60,t£ £ must there be a time t when Ben’s velocity is 2 meters per second? Justify your answer.

(d) A light is directly above the western end of the track. Ben rides so that at time t, the distance ( )L t between

Ben and the light satisfies ( )( ) ( )( )2 2212 .L t B t= + At what rate is the distance between Ben and the light changing at time 40 ?t =




-6-

6. Let ( ) ( )3ln 1 .f x x= +

(a) The Maclaurin series for ( )ln 1 x+ is ( )2 3 4

11 .2 3 4

nnx x x xx

n+- + - + + - +� Use the series to write

the first four nonzero terms and the general term of the Maclaurin series for f.

(b) The radius of convergence of the Maclaurin series for f is 1. Determine the interval of convergence. Show the work that leads to your answer.

(c) Write the first four nonzero terms of the Maclaurin series for ( )2 .f t¢ If ( ) ( )2

0,

xg x f t dt= ¢Ú use the first

two nonzero terms of the Maclaurin series for g to approximate ( )1 .g

(d) The Maclaurin series for g, evaluated at 1,x = is a convergent alternating series with individual terms that decrease in absolute value to 0. Show that your approximation in part (c) must differ from ( )1g by

less than 1 .5


END OF EXAM

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APfi Calculus BC 2002 Free-Response Questions Calculus BC... · APfi Calculus BC 2002 Free-Response Questions These materials were produced by Educational Testing Service fi (ETS