2024年4月1日发(作者：任浩思)
满分网——AP真题
AP
®
Calculus BC
2006 Free-Response Questions
Form B
The College Board: Connecting Students to College Success
The College Board is a not-for-profit membership association whose mission is to connect students to college success and
opportunity. Founded in 1900, the association is composed of more than 5,000 schools, colleges, universities, and other
educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and
3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and
teaching and learning. Among its best-known programs are the SAT
®
, the PSAT/NMSQT
®
, and the Advanced Placement
Program
®
(AP
®
). The College Board is committed to the principles of excellence and equity, and that commitment is embodied
in all of its programs, services, activities, and concerns.
© 2006 The College Board. All rights reserved. College Board, AP Central, APCD, Advanced Placement Program, AP, AP
Vertical Teams, Pre-AP, SAT, and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation
Service, CollegeEd, connect to college success, MyRoad, SAT Professional Development, SAT Readiness Program, and Setting the
Cornerstones are trademarks owned by the College Board. PSAT/NMSQT is a registered trademark of the College Board and
National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners.
Permission to use copyrighted College Board materials may be requested online at:
/inquiry/.
Visit the College Board on the Web: .
AP Central is the official online home for the AP Program: .
2006 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.
x
3
x
2
x
1. Let f be the function given by
f
(
x
)
=--+
3cos
x
.
Let R be the shaded region in the second
432
quadrant bounded by the graph of f, and let S be the shaded region bounded by the graph of f and line
A,
the
line tangent to the graph of f at
x=0,
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
y=-2.
(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.
© 2006 The College Board. All rights reserved.
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2
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2006 AP
®
CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
2. An object moving along a curve in the xy-plane is at position
(
x
(
t
)
,y
(
t
)
)
at time t, where
dy
dx
=tan
e
-t
and
=sec
e
-t
dt
dt
()
()
for
t≥0.
At time
t=1,
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
t
=
1.
(c) Find the total distance traveled by the object over the time interval
1
£
t
£
2.
(d) Is there a time
t
≥
0
at which the object is on the y-axis? Explain why or why not.
WRITE ALL WORK IN THE EXAM BOOKLET.
© 2006 The College Board. All rights reserved.
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3
GO ON TO THE NEXT PAGE.
2006 AP
®
CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
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
x=0,
the value of the function is 0, and the slope of the graph of the function is 0.
(ii) At
x=4,
the value of the function is 1, and the slope of the graph of the function is 1.
(iii) Between
x
=
0
and
x=4,
the function is increasing.
(a) Let
f
(
x
)
=ax
2
,
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.
x
2
,
where c is a nonzero constant. Find the value of c so that g meets requirement (ii)
(b) Let
g
(
x
)
=cx-
16
above. Show the work that leads to your answer.
3
(c) Using the function g and your value of c from part (b), show that g does not meet requirement (iii) above.
x
n
(d) Let
h
(
x
)
=
,
where k is a nonzero constant and n is a positive integer. Find the values of k and n so
k
that h meets requirement (ii) above. Show that h also meets requirements (i) and (iii) above.
WRITE ALL WORK IN THE EXAM BOOKLET.
END OF PART A OF SECTION II
© 2006 The College Board. All rights reserved.
Visit (for AP professionals) and /apstudents (for students and parents).
4
2006 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 rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the
13
function f. In the figure above,
f
(
t
)
=-t
3
+t
2
+1
for
0£t£4
and f is piecewise linear for
4£t£24.
42
(a) Find
f¢
(
22
)
.
Indicate units of measure.
(b) For the time interval
0£t£24,
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£t£18
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£t£18.
WRITE ALL WORK IN THE EXAM BOOKLET.
© 2006 The College Board. All rights reserved.
Visit (for AP professionals) and /apstudents (for students and parents).
5
GO ON TO THE NEXT PAGE.
2006 AP
®
CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
5. Let f be a function with
f
(
4
)
=1
such that all points
(
x,y
)
on the graph of f satisfy the differential equation
dy
=2y
(
3-x
)
.
dx
Let g be a function with
g
(
4
)
=1
such that all points
(
x,y
)
on the graph of g satisfy the logistic differential
equation
dy
=2y
(
3-y
)
.
dx
(a) Find
y=f
(
x
)
.
(b) Given that
g
(
4
)
=1,
find
limg
(
x
)
and
limg
¢
(
x
)
.
(It is not necessary to solve for
g
(
x
)
or to show how
you arrived at your answers.)
xÆ
•
xÆ
•
(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
f
(
x
)
=
1
.
The Maclaurin series for f is given by
1+x
3
n
1-x
3
+x
6
-x
9
+"+
(
-1
)
x
3n
+",
which converges to
f
(
x
)
for
-1 (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 - 369 n 3n 1 +-+"+-+" . () 2 2 2 5 2 8 2 3n-1 (c) Find the first four nonzero terms and the general term for the Maclaurin series representing (d) Use the first three nonzero terms of the infinite series found in part (c) to approximate are the properties of the terms of the series representing is within WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM Ú 0 x f ( t ) dt. Ú 0 12 f ( t ) dt. What Ú 0 12 f ( t ) dt that guarantee that this approximation 1 of the exact value of the integral? 10,000 © 2006 The College Board. All rights reserved. Visit (for AP professionals) and /apstudents (for students and parents). 6
2024年4月1日发(作者：任浩思)
满分网——AP真题
AP
®
Calculus BC
2006 Free-Response Questions
Form B
The College Board: Connecting Students to College Success
The College Board is a not-for-profit membership association whose mission is to connect students to college success and
opportunity. Founded in 1900, the association is composed of more than 5,000 schools, colleges, universities, and other
educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and
3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and
teaching and learning. Among its best-known programs are the SAT
®
, the PSAT/NMSQT
®
, and the Advanced Placement
Program
®
(AP
®
). The College Board is committed to the principles of excellence and equity, and that commitment is embodied
in all of its programs, services, activities, and concerns.
© 2006 The College Board. All rights reserved. College Board, AP Central, APCD, Advanced Placement Program, AP, AP
Vertical Teams, Pre-AP, SAT, and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation
Service, CollegeEd, connect to college success, MyRoad, SAT Professional Development, SAT Readiness Program, and Setting the
Cornerstones are trademarks owned by the College Board. PSAT/NMSQT is a registered trademark of the College Board and
National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners.
Permission to use copyrighted College Board materials may be requested online at:
/inquiry/.
Visit the College Board on the Web: .
AP Central is the official online home for the AP Program: .
2006 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.
x
3
x
2
x
1. Let f be the function given by
f
(
x
)
=--+
3cos
x
.
Let R be the shaded region in the second
432
quadrant bounded by the graph of f, and let S be the shaded region bounded by the graph of f and line
A,
the
line tangent to the graph of f at
x=0,
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
y=-2.
(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.
© 2006 The College Board. All rights reserved.
Visit (for AP professionals) and /apstudents (for students and parents).
2
GO ON TO THE NEXT PAGE.
2006 AP
®
CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
2. An object moving along a curve in the xy-plane is at position
(
x
(
t
)
,y
(
t
)
)
at time t, where
dy
dx
=tan
e
-t
and
=sec
e
-t
dt
dt
()
()
for
t≥0.
At time
t=1,
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
t
=
1.
(c) Find the total distance traveled by the object over the time interval
1
£
t
£
2.
(d) Is there a time
t
≥
0
at which the object is on the y-axis? Explain why or why not.
WRITE ALL WORK IN THE EXAM BOOKLET.
© 2006 The College Board. All rights reserved.
Visit (for AP professionals) and /apstudents (for students and parents).
3
GO ON TO THE NEXT PAGE.
2006 AP
®
CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
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
x=0,
the value of the function is 0, and the slope of the graph of the function is 0.
(ii) At
x=4,
the value of the function is 1, and the slope of the graph of the function is 1.
(iii) Between
x
=
0
and
x=4,
the function is increasing.
(a) Let
f
(
x
)
=ax
2
,
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.
x
2
,
where c is a nonzero constant. Find the value of c so that g meets requirement (ii)
(b) Let
g
(
x
)
=cx-
16
above. Show the work that leads to your answer.
3
(c) Using the function g and your value of c from part (b), show that g does not meet requirement (iii) above.
x
n
(d) Let
h
(
x
)
=
,
where k is a nonzero constant and n is a positive integer. Find the values of k and n so
k
that h meets requirement (ii) above. Show that h also meets requirements (i) and (iii) above.
WRITE ALL WORK IN THE EXAM BOOKLET.
END OF PART A OF SECTION II
© 2006 The College Board. All rights reserved.
Visit (for AP professionals) and /apstudents (for students and parents).
4
2006 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 rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the
13
function f. In the figure above,
f
(
t
)
=-t
3
+t
2
+1
for
0£t£4
and f is piecewise linear for
4£t£24.
42
(a) Find
f¢
(
22
)
.
Indicate units of measure.
(b) For the time interval
0£t£24,
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£t£18
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£t£18.
WRITE ALL WORK IN THE EXAM BOOKLET.
© 2006 The College Board. All rights reserved.
Visit (for AP professionals) and /apstudents (for students and parents).
5
GO ON TO THE NEXT PAGE.
2006 AP
®
CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
5. Let f be a function with
f
(
4
)
=1
such that all points
(
x,y
)
on the graph of f satisfy the differential equation
dy
=2y
(
3-x
)
.
dx
Let g be a function with
g
(
4
)
=1
such that all points
(
x,y
)
on the graph of g satisfy the logistic differential
equation
dy
=2y
(
3-y
)
.
dx
(a) Find
y=f
(
x
)
.
(b) Given that
g
(
4
)
=1,
find
limg
(
x
)
and
limg
¢
(
x
)
.
(It is not necessary to solve for
g
(
x
)
or to show how
you arrived at your answers.)
xÆ
•
xÆ
•
(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
f
(
x
)
=
1
.
The Maclaurin series for f is given by
1+x
3
n
1-x
3
+x
6
-x
9
+"+
(
-1
)
x
3n
+",
which converges to
f
(
x
)
for
-1 (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 - 369 n 3n 1 +-+"+-+" . () 2 2 2 5 2 8 2 3n-1 (c) Find the first four nonzero terms and the general term for the Maclaurin series representing (d) Use the first three nonzero terms of the infinite series found in part (c) to approximate are the properties of the terms of the series representing is within WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM Ú 0 x f ( t ) dt. Ú 0 12 f ( t ) dt. What Ú 0 12 f ( t ) dt that guarantee that this approximation 1 of the exact value of the integral? 10,000 © 2006 The College Board. All rights reserved. Visit (for AP professionals) and /apstudents (for students and parents). 6