calculus_ab_frq_03 - AP® Calculus AB 2003 Free-Response...

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Unformatted text preview: AP® Calculus AB 2003 Free-Response Questions 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. These materials were produced by Educational Testing Service® (ETS®), which develops and administers the examinations of the Advanced Placement Program for the College Board. 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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 FREE-RESPONSE 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 x-axis 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 FREE-RESPONSE QUESTIONS 2. A particle moves along the x-axis 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 FREE-RESPONSE 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 ˆ 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 FREE-RESPONSE 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 ˆ x ˆ 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 x-coordinate 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 FREE-RESPONSE 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 ˆ x ˆ 3 for 3 < x ˆ 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 ˆ x ˆ 5. (c) Suppose the function g is defined by g( x ) = %k x + 1 &mx + 2 ' for 0 ˆ x ˆ 3 for 3 < x ˆ 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 students and parents at www.collegeboard.com/apstudents. 6 ...
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