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Unformatted text preview: AP® Calculus AB
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For the College Board’s online home for AP professionals, visit AP Central at apcentral.collegeboard.com. 2003 AP® CALCULUS AB FREERESPONSE QUESTIONS
CALCULUS AB
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 R be the shaded region bounded by the graphs of y =
as shown in the figure above. x and y = e −3x and the vertical line x = 1, (a) Find the area of R.
(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 xaxis is
a rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid. 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 2003 AP® CALCULUS AB FREERESPONSE QUESTIONS
2. A particle moves along the xaxis so that its velocity at time t is given by
v(t ) = (t + 1) sin t .
2
2 At time t = 0, the particle is at position x = 1.
(a) Find the acceleration of the particle at time t = 2. Is the speed of the particle increasing at t = 2 ? Why or
why not?
(b) Find all times t in the open interval 0 < t < 3 when the particle changes direction. Justify your answer.
(c) Find the total distance traveled by the particle from time t = 0 until time t = 3.
(d) During the time interval 0 ≤ t ≤ 3, what is the greatest distance between the particle and the origin? Show
the work that leads to your answer. 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.
3 2003 AP® CALCULUS AB FREERESPONSE QUESTIONS 3. The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twicedifferentiable and strictly increasing function R of time t. The graph of R and a table of selected values of
R t , for the time interval 0 ≤ t ≤ 90 minutes, are shown above. 05 05 (a) Use data from the table to find an approximation for R ′ 45 . Show the computations that lead to your
answer. Indicate units of measure. 05 (b) The rate of fuel consumption is increasing fastest at time t = 45 minutes. What is the value of R ′′ 45 ?
Explain your reasoning.
(c) Approximate the value of I 90 0 R(t ) dt using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation less than the value of
(d) For 0 < b 90 minutes, explain the meaning of
Explain the meaning of
both answers. 1
b I b 0 I b 0 I 90 0 R(t ) dt ? Explain your reasoning. R(t ) dt in terms of fuel consumption for the plane. R(t ) dt in terms of fuel consumption for the plane. Indicate units of measure in END OF PART A OF SECTION II 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. 4 2003 AP® CALCULUS AB FREERESPONSE QUESTIONS
CALCULUS AB
SECTION II, Part B
Time—45 minutes
Number of problems—3
No calculator is allowed for these problems. 05 4. Let f be a function defined on the closed interval 3 x 4 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 xcoordinate of each point of inflection of the graph of f on the open interval 3 < x < 4. Justify
your answer. 05 (c) Find an equation for the line tangent to the graph of f at the point 0, 3 . 05 (d) Find f ( 3) and f 4 . Show the work that leads to your answers. 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.
5 2003 AP® CALCULUS AB FREERESPONSE QUESTIONS 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 = pr 2 h. )
(a) Show that dh
h
=.
dt
5 (b) Given that h = 17 at time t = 0 , solve the differential equation dh
h
=−
for h as a function of t.
dt
5 (c) At what time t is the coffeepot empty? 6. Let f be the function defined by 0 5 %5 x +x 1
&
' fx= for 0 x 3
for 3 < x 5. (a) Is f continuous at x = 3 ? Explain why or why not. 05 (b) Find the average value of f x on the closed interval 0 x 5.
(c) Suppose the function g is defined by
g( x ) = %k x + 1
&mx + 2
' for 0 x 3
for 3 < x 5, where k and m are constants. If g is differentiable at x = 3, what are the values of k and m ? END OF EXAMINATION
Copyright © 2003 by College Entrance Examination Board. All rights reserved.
Available to AP professionals at apcentral.collegeboard.com and to
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 Spring '09
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