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Unformatted text preview: AP® Calculus AB
2003 FreeResponse Questions
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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 (Form B)
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 f be the function given by f ( x ) = 4 x 2  x 3 , and let l be the line y = 18  3 x, where l is tangent to the
graph of f. Let R be the region bounded by the graph of f and the xaxis, and let S be the region bounded by
the graph of f, the line l, and the xaxis, as shown above.
(a) Show that l 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 xaxis. 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 (Form B)
2. A tank contains 125 gallons of heating oil at time t = 0. During the time interval 0 t 12 hours, heating oil
is pumped into the tank at the rate
H (t ) = 2 + 10 01 + ln(t + 1)5 gallons per hour. During the same time interval, heating oil is removed from the tank at the rate
R(t ) = 12 sin t gallons per hour.
47
2 (a) How many gallons of heating oil are pumped into the tank during the time interval 0 t 12 hours?
(b) Is the level of heating oil in the tank rising or falling at time t = 6 hours? Give a reason for your answer.
(c) How many gallons of heating oil are in the tank at time t = 12 hours?
(d) At what time t, for 0 t 12, is the volume of heating oil in the tank the least? Show the analysis that
leads to your conclusion. 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 (Form B) 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 twicedifferentiable
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 p I B( x)
2 275 125 2 dx in terms of the blood vessel. (d) Explain why there must be at least one value x, for 0 < x < 360, such that B( x ) = 0. 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 (Form B)
CALCULUS AB
SECTION II, Part B
Time—45 minutes
Number of problems—3
No calculator is allowed for these problems. 4. A particle moves along the xaxis with velocity at time t 0 given by v (t ) = 1 + e1 t .
(a) Find the acceleration of the particle at time t = 3.
(b) Is the speed of the particle increasing at time t = 3 ? Give a reason for your answer.
(c) Find all values of t at which the particle changes direction. Justify your answer.
(d) Find the total distance traveled by the particle over the time interval 0 t 3. I 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 ) =
(a) Find g(3), g (3), and g (3). x 2 f (t ) dt. (b) Find the average rate of change of g on the interval 0 x 3.
(c) For how many values c, where 0 < c < 3, is g (c) equal to the average rate found in part (b) ? Explain
your reasoning.
(d) Find the xcoordinate of each point of inflection of the graph of g on the interval 0 < x < 7. Justify 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.
5 2003 AP® CALCULUS AB FREERESPONSE QUESTIONS (Form B) 05 05 6. Let f be the function satisfying f x = x f x for all real numbers x, where f (3) = 25.
(a) Find f (3).
(b) Write an expression for y = f ( x ) by solving the differential equation
condition f (3) = 25. dy
= x y with the initial
dx END OF EXAMINATION 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. 6 ...
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This note was uploaded on 03/12/2010 for the course CALCULUS 1561234586 taught by Professor Zalla during the Spring '10 term at Air Force Institute of Technology, Ohio.
 Spring '10
 Zalla
 Calculus

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