AP® Calculus BC
2011 Free-Response Questions
Form B
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2011 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
CALCULUS BC
SECTION II, Part A
Time— 30 minutes
Number of problems— 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 £ t £ 60. The rate at which the height
of the water is rising in the can is given by S ¢(t ) = 2sin (0.03t ) + 1.5.
(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 t = 7 ?
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
1
3t 3 - 30t 2 + 330t . The height M (t ) is measured in millimeters, and t is measured in days
M (t ) =
400
for 0 £ t £ 60. Let D(t ) = M ¢(t ) - S ¢(t ) . Apply the Intermediate Value Theorem to the function D on
the interval 0 £ t £ 60 to justify that there exists a time t, 0 < t < 60, at which the heights of water in the
two cans are changing at the same rate.
(
)
2. The polar curve r is given by r (q ) = 3q + sin q , where 0 £ q £ 2 p .
(a) Find the area in the second quadrant enclosed by the coordinate axes and the graph of r.
p
£ q £ p , there is one point P on the polar curve r with x-coordinate -3. Find the angle q that
2
corresponds to point P. Find the y-coordinate of point P. Show the work that leads to your answers.
(b) For
(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
dy
dq
2p
, and interpret the meaning of your answer in the context of
= 2. Find
at the instant that q =
3
dt
dt
the problem.
WRITE ALL WORK IN THE EXAM BOOKLET.
END OF PART A OF SECTION II
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2011 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
CALCULUS BC
SECTION II, Part B
Time— 60 minutes
Number of problems— 4
No calculator is allowed for these problems.
3. The functions f and g are given by f ( x ) = x and g( x ) = 6 - x. Let R be the region bounded by the x-axis
and the graphs of f and g, as shown in the figure above.
(a) Find the area of R.
(b) The region R is the base of a solid. For each y, where 0 £ y £ 2, 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.
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2011 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
4. The graph of the differentiable function y = f ( x ) with domain 0 £ x £ 10 is shown in the figure above. The
area of the region enclosed between the graph of f and the x-axis for 0 £ x £ 5 is 10, and the area of the region
enclosed between the graph of f and the x-axis for 5 £ x £ 10 is 27. The arc length for the portion of the graph
of f between x = 0 and x = 5 is 11, and the arc length for the portion of the graph of f between x = 5 and
x = 10 is 18. The function f has exactly two critical points that are located at x = 3 and x = 8.
(a) Find the average value of f on the interval 0 £ x £ 5.
(b) Evaluate
10
Ú0
(c) Let g( x ) =
(3 f ( x ) + 2) dx. Show the computations that lead to your answer.
x
Ú5 f (t ) dt. 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 h( x ) = 2 f
( 2x ). The derivative of h is h¢(x) = f ¢( 2x ). Find the arc length of
the graph of y = h( x ) from x = 0 to x = 20.
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2011 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
t
(seconds)
0
B(t )
(meters)
v (t )
(meters per second)
10
40
60
100 136
9
49
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 t = 5 seconds. Indicate units of measure.
(b) Using correct units, interpret the meaning of
60
Ú0
60
Ú0
v(t ) dt in the context of this problem. Approximate
v(t ) dt using a left Riemann sum with the subintervals indicated by the data in the table.
(c) For 40 £ t £ 60, 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 ( L (t )) = 122 + ( B(t )) . At what rate is the distance between Ben and the light
changing at time t = 40 ?
2
2
WRITE ALL WORK IN THE EXAM BOOKLET.
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2011 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
(
)
6. Let f ( x ) = ln 1 + x 3 .
n
x2
x3 x 4
n +1 x
+
+ " + ( -1) ⴢ
+ ". Use the series to write
n
2
3
4
the first four nonzero terms and the general term of the Maclaurin series for f.
(a) The Maclaurin series for ln (1 + x ) is x -
(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 f ¢ t 2 . If g( x ) =
two nonzero terms of the Maclaurin series for g to approximate g(1) .
2
Ú0 f ¢(t ) dt, use the first
x
(d) The Maclaurin series for g, evaluated at x = 1, 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 g(1) by
1
less than .
5
WRITE ALL WORK IN THE EXAM BOOKLET.
END OF EXAM
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AP® Calculus BC
2009 Free-Response Questions
Form B
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.
© 2009 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.
2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
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. A baker is creating a birthday cake. The base of the cake is the region R in the first quadrant under the graph of
px
y = f ( x ) for 0 £ x £ 30, where f ( x ) = 20sin
. Both x and y are measured in centimeters. The region R
30
2p
px
is shown in the figure above. The derivative of f is f ¢( x ) =
cos
.
3
30
( )
( )
(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.
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2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
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
f (t ) = t + cos t - 3 meters per hour, t hours after the storm began. The edge of the water was 35 meters from
1
- sin t.
the road when the storm began, and the storm lasted 5 hours. The derivative of f (t ) is f ¢(t ) =
2 t
(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 f ¢( 4 ) = 1.007 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.
WRITE ALL WORK IN THE EXAM BOOKLET.
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2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
3. A continuous function f is defined on the closed interval - 4 £ x £ 6. The graph of f consists of a line segment
and a curve that is tangent to the x-axis at x = 3, as shown in the figure above. On the interval 0 < x < 6, the
function f is twice differentiable, with f ¢¢( x ) > 0.
(a) Is f differentiable at x = 0 ? Use the definition of the derivative with one-sided limits to justify your
answer.
(b) For how many values of a, - 4 £ a < 6, is the average rate of change of f on the interval [a, 6] equal to 0 ?
Give a reason for your answer.
(c) Is there a value of a, - 4 £ a < 6, for which the Mean Value Theorem, applied to the interval [a, 6],
1
guarantees a value c, a < c < 6, at which f ¢(c ) = ? Justify your answer.
3
(d) The function g is defined by g( x ) =
x
Ú0 f (t ) dt for - 4 £ x £ 6. On what intervals contained in [- 4, 6]
is the graph of g concave up? Explain your reasoning.
WRITE ALL WORK IN THE EXAM BOOKLET.
END OF PART A OF SECTION II
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2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
CALCULUS BC
SECTION II, Part B
Time— 45 minutes
Number of problems— 3
No calculator is allowed for these problems.
4. The graph of the polar curve r = 1 - 2 cos 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
dy
dx
and
in terms of q.
dq
dq
(c) Write an equation in terms of x and y for the line tangent to the graph of the polar curve at the point
p
where q = . Show the computations that lead to your answer.
2
WRITE ALL WORK IN THE EXAM BOOKLET.
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2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
5. Let f be a twice-differentiable function defined on the interval -1.2 < x < 3.2 with f (1) = 2. The graph of f ¢,
the derivative of f, is shown above. The graph of f ¢ crosses the x-axis at x = -1 and x = 3 and has
a horizontal tangent at x = 2. Let g be the function given by g( x ) = e f ( x ) .
(a) Write an equation for the line tangent to the graph of g at x = 1.
(b) For -1.2 < x < 3.2, find all values of x at which g has a local maximum. Justify your answer.
2
(c) The second derivative of g is g ¢¢( x ) = e f ( x ) ÈÎ( f ¢( x )) + f ¢¢( x )˘˚ . Is g ¢¢( -1) 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].
WRITE ALL WORK IN THE EXAM BOOKLET.
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2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
6. The function f is defined by the power series
f ( x ) = 1 + ( x + 1) + ( x + 1) + " + ( x + 1) + " =
2
n
•
 ( x + 1)n
n =0
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 x = -1. Find the sum of the series for f.
(c) Let g be the function defined by g( x ) =
( 21 ) cannot be determined.
Ú-1 f (t ) dt. Find the value of g(- 2 ) , if it exists, or explain why
x
1
g -
(
)
(d) Let h be the function defined by h( x ) = f x 2 - 1 . Find the first three nonzero terms and the general term
of the Taylor series for h about x = 0, and find the value of h
( 12 ).
WRITE ALL WORK IN THE EXAM BOOKLET.
END OF EXAM
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AP® Calculus BC
2009 Free-Response Questions
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.
© 2009 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.
2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
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. Caren rides her bicycle along a straight road from home to school, starting at home at time t = 0 minutes and
arriving at school at time t = 12 minutes. During the time interval 0 £ t £ 12 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 t = 7.5 minutes. Indicate units of measure.
(b) Using correct units, explain the meaning of
12
Ú0
12
Ú0
v(t ) dt in terms of Caren’s trip. Find the value of
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
p
p
sin
t , where w(t ) is in miles per minute for 0 £ t £ 12 minutes.
by the function w given by w(t ) =
15
12
Who lives closer to school: Caren or Larry? Show the work that leads to your answer.
( )
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2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
2. The rate at which people enter an auditorium for a rock concert is modeled by the function R given by
R(t ) = 1380t 2 - 675t 3 for 0 £ t £ 2 hours; R(t ) is measured in people per hour. No one is in the auditorium
at time t = 0, when the doors open. The doors close and the concert begins at time t = 2.
(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 w ¢(t ) = (2 - t ) R(t ) . Find w(2 ) - w(1) , the total wait time for those who enter the auditorium
after time t = 1.
(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).
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
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2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
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
dy
dx
= 0.8 and
= 3.6 - 9.8t ,
dt
dt
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 < q <
p
, between the path of the diver and the water at the instant the diver’s
2
shoulders enter the water.
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
END OF PART A OF SECTION II
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2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
CALCULUS BC
SECTION II, Part B
Time— 45 minutes
Number of problems— 3
No calculator is allowed for these problems.
dy
= 6 x 2 - x 2 y. Let y = f ( x ) be a particular solution to this differential
dx
equation with the initial condition f ( -1) = 2.
4. Consider the differential equation
(a) Use Euler’s method with two steps of equal size, starting at x = -1, to approximate f (0 ) . Show the work
that leads to your answer.
(b) At the point ( -1, 2 ) , the value of
f about x = -1.
d2y
is -12. Find the second-degree Taylor polynomial for
dx 2
(c) Find the particular solution y = f ( x ) to the given differential equation with the initial condition f ( -1) = 2.
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2009 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
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 £ x £ 13.
(a) Estimate f ¢( 4 ) . Show the work that leads to your answer.
(b) Evaluate
13
Ú2
(3 - 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
Ú2
f ( x ) dx.
Show the work that leads to your answer.
(d) Suppose f ¢(5) = 3 and f ¢¢( x ) < 0 for all x in the closed interval 5 £ x £ 8. Use the line tangent to the
graph of f at x = 5 to show that f (7) £ 4. Use the secant line for the graph of f on 5 £ x £ 8 to show
4
that f (7) ≥ .
3
6. The Maclaurin series for e x is e x = 1 + x +
x2
x3
xn
+
+"+
+ ". The continuous function f is defined by
n!
2
6
e( x -1) - 1
for x π 1 and f (1) = 1. The function f has derivatives of all orders at x = 1.
( x - 1)2
2
f ( x) =
2
(a) Write the first four nonzero terms and the general term of the Taylor series for e( x -1) about x = 1.
(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 x = 1.
(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 x = 1 to determine whether the graph of f has any points of inflection.
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
END OF EXAM
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-6-
AP® Calculus BC
2010 Free-Response Questions
Form B
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.
© 2010 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.com.
AP Central is the official online home for the AP Program: apcentral.collegeboard.com.
2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
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. In the figure above, R is the shaded region in the first quadrant bounded by the graph of y = 4 ln(3 - x ) , the
horizontal line y = 6, and the vertical line x = 2.
(a) Find the area of R.
(b) Find the volume of the solid generated when R is revolved about the horizontal line y = 8.
(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.
WRITE ALL WORK IN THE EXAM BOOKLET.
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-2-
2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
2. The velocity vector of a particle moving in the xy-plane has components given by
( ) ( )
dx
= 14 cos t 2 sin et
dt
and
( )
dy
= 1 + 2sin t 2 , for 0 £ t £ 1.5.
dt
At time t = 0, the position of the particle is ( -2, 3) .
(a) For 0 < t < 1.5, 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 t = 1.
(c) Find the speed of the particle at t = 1.
(d) Find the acceleration vector of the particle at t = 1.
t
0
2
P(t)
0
46 53 57 60 62 63
4
6
8
10 12
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 t = 0. During the time interval 0 £ t £ 12
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 R(t ) = 25e - 0.05t . (Note: The volume V of a cylinder with radius r and height h is given by
V = p r 2 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 £ t £ 12 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 £ t £ 12 hours.
(c) Use the results from parts (a) and (b) to approximate the volume of water in the pool at time t = 12 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 t = 8 hours. How fast is the
water level in the pool rising at t = 8 hours? Indicate units of measure in both answers.
WRITE ALL WORK IN THE EXAM BOOKLET.
END OF PART A OF SECTION II
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2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
CALCULUS BC
SECTION II, Part B
Time— 45 minutes
Number of problems— 3
No calculator is allowed for these problems.
4. A squirrel starts at building A at time t = 0 and travels along a straight, horizontal wire connected to
building B. For 0 £ t £ 18, the squirrel’s velocity is modeled by the piecewise-linear function defined
by the graph above.
(a) At what times in the interval 0 < t < 18, if any, does the squirrel change direction? Give a reason
for your answer.
(b) At what time in the interval 0 £ t £ 18 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 £ t £ 18.
(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 < t < 10.
WRITE ALL WORK IN THE EXAM BOOKLET.
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2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
5. Let f and g be the functions defined by f ( x ) =
1
4x
, for all x > 0.
and g ( x ) =
x
1 + 4 x2
(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 x = 1, below
the graph of f, and above the graph of g.
6. The Maclaurin series for the function f is given by f ( x ) =
• ( -1)n (2 x )n
Â
n=2
n -1
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 x y ¢ - y =
4 x2
for x < R, where R is the
1 + 2x
radius of convergence from part (a).
WRITE ALL WORK IN THE EXAM BOOKLET.
END OF EXAM
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AP® Calculus BC
2010 Free-Response Questions
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2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
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. 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 f (t ) = 7tecos t cubic feet per hour, where t is measured
in hours since midnight. Janet starts removing snow at 6 A.M. (t = 6 ) . 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
for 0 £ t < 6
Ï0
Ô
g (t ) = Ì125 for 6 £ t < 7
ÔÓ108 for 7 £ t £ 9 .
(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 £ t £ 9.
(d) How many cubic feet of snow are on the driveway at 9 A.M.?
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
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2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
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 (t = 0 ) and 8 P.M. (t = 8). The number of entries in the box t hours after noon is modeled by a
differentiable function E for 0 £ t £ 8. 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 t = 6. 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
Using correct units, explain the meaning of
1 8
E (t ) dt.
8 Ú0
1 8
E (t ) dt in terms of the number of entries.
8 Ú0
(c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the
function P, where P (t ) = t 3 - 30t 2 + 298t - 976 hundreds of entries per hour for 8 £ t £ 12. According
to the model, how many entries had not yet been processed by midnight (t = 12 ) ?
(d) According to the model from part (c), at what time were the entries being processed most quickly? Justify
your answer.
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
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2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
3. A particle is moving along a curve so that its position at time t is ( x (t ) , y(t )) , where x (t ) = t 2 - 4t + 8 and
y(t ) is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known
dy
that
= tet - 3 - 1.
dt
(a) Find the speed of the particle at time t = 3 seconds.
(b) Find the total distance traveled by the particle for 0 £ t £ 4 seconds.
(c) Find the time t, 0 £ t £ 4, 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 y(2 ) = 3 +
1
e
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
END OF PART A OF SECTION II
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2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
CALCULUS BC
SECTION II, Part B
Time— 45 minutes
Number of problems— 3
No calculator is allowed for these problems.
4. Let R be the region in the first quadrant bounded by the graph of y = 2 x , the horizontal line y = 6, 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 y = 7.
(c) Region R is the base of a solid. For each y, where 0 £ y £ 6, 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.
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
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2010 AP® CALCULUS BC FREE-RESPONSE QUESTIONS
dy
= 1 - y. Let y = f ( x ) be the particular solution to this differential
dx
equation with the initial condition f (1) = 0. For this particular solution, f ( x ) < 1 for all values of x.
5. Consider the differential equation
(a) Use Euler’s method, starting at x = 1 with two steps of equal size, to approximate f (0 ) . Show the work
that leads to your answer.
f ( x)
(b) Find lim
x Æ1 x 3
-1
. Show the work that leads to your answer.
(c) Find the particular solution y = f ( x ) to the differential equation
f (1) = 0.
dy
= 1 - y with the initial condition
dx
Ï cos x - 1 for x π 0
2
Ô
f (x) = Ì x
Ô- 1
for x = 0
Ó 2
6. The function f, defined above, has derivatives of all orders. Let g be the function defined by
g( x ) = 1 +
x
Ú0 f (t ) dt.
(a) Write the first three nonzero terms and the general term of the Taylor series for cos x about x = 0. Use this
series to write the first three nonzero terms and the general term of the Taylor series for f about x = 0.
(b) Use the Taylor series for f about x = 0 found in part (a) to determine whether f has a relative maximum,
relative minimum, or neither at x = 0. Give a reason for your answer.
(c) Write the fifth-degree Taylor polynomial for g about x = 0.
(d) The Taylor series for g about x = 0, evaluated at x = 1, is an alternating series with individual terms that
decrease in absolute value to 0. Use the third-degree Taylor polynomial for g about x = 0 to estimate the
1
value of g(1) . Explain why this estimate differs from the actual value of g(1) by less than .
6!
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
END OF EXAM
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