ap04_frq_calculus_ab_b - AP® Calculus AB 2004...

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Unformatted text preview: AP® Calculus AB 2004 Free-Response Questions Form B The materials included in these files are intended for noncommercial 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 face-to-face teaching purposes but may not mass distribute the materials, electronically or otherwise. This permission does not apply to any third-party copyrights contained herein. These materials and any copies made of them may not be resold, and the copyright notices must be retained as they appear here. 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 4,500 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three 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. For further information, visit www.collegeboard.com Copyright © 2004 College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, AP Vertical Teams, APCD, Pacesetter, Pre-AP, SAT, Student Search Service, and the acorn logo are registered trademarks of the College Entrance Examination Board. PSAT/NMSQT is a registered trademark jointly owned by the College Entrance Examination Board and the National Merit Scholarship Corporation. Educational Testing Service and ETS are registered trademarks of Educational Testing Service. Other products and services may be trademarks of their respective owners. For the College Board’s online home for AP professionals, visit AP Central at apcentral.collegeboard.com. 2004 AP® CALCULUS AB FREE-RESPONSE 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. - x 1, the vertical line x = = 1. Let R be the region enclosed by the graph of y 10, and the x-axis. (a) Find the area of R. (b) Find the volume of the solid generated when R is revolved about the horizontal line y = (c) Find the volume of the solid generated when R is revolved about the vertical line x 10. 31, the rate of change of the number of mosquitoes on Tropical Island at time t days is modeled t 5 t cos mosquitoes per day. There are 1000 mosquitoes on Tropical Island at time t 0. 5 £ KH IF = =)( (a) Show that the number of mosquitoes is increasing at time t = £ by R t t = 2. For 0 3. 6. (b) At time t 6, is the number of mosquitoes increasing at an increasing rate, or is the number of mosquitoes increasing at a decreasing rate? Give a reason for your answer. = £ (d) To the nearest whole number, what is the maximum number of mosquitoes for 0 analysis that leads to your conclusion. 31 ? Round your answer t £ = (c) According to the model, how many mosquitoes will be on the island at time t to the nearest whole number. 31 ? Show the Copyright © 2004 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). GO ON TO THE NEXT PAGE. 2 2004 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) t (minutes) 0 v (t ) (miles per minute) 5 10 15 20 25 30 35 40 7.0 9.2 9.5 7.0 4.5 2.4 2.4 4.3 7.3 3. A test plane flies in a straight line with positive velocity v t , in miles per minute at time t minutes, where fa t £ £ v is a differentiable function of t. Selected values of v t for 0 40 are shown in the table above. fa (a) Use a midpoint Riemann sum with four subintervals of equal length and values from the table to 40 z 0 v t dt . Show the computations that lead to your answer. Using correct units, explain the v t dt in terms of the plane’s flight. )( z meaning of 0 40 )( approximate (b) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval 0 t 40 ? Justify your answer. < + K I H F cos H F + = fa £ £ = t t t 7t 3 sin , is used to model the velocity of the plane, 10 40 40. According to this model, what is the acceleration of the plane at 6 K I in miles per minute, for 0 < (c) The function f, defined by f t 23 ? Indicate units of measure. (d) According to the model f, given in part (c), what is the average velocity of the plane, in miles per minute, over the time interval 0 t 40 ? £ £ END OF PART A OF SECTION II Copyright © 2004 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 3 2004 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) CALCULUS AB SECTION II, Part B Time—45 minutes Number of problems—3 No calculator is allowed for these problems. 4. The figure above shows the graph of f , the derivative of the function f, on the closed interval 1 x 5. The graph of f has horizontal tangent lines at x 1 and x 3. The function f is twice differentiable with f2 6. £ £- ¢ = = ¢ =) ( (a) Find the x-coordinate of each of the points of inflection of the graph of f. Give a reason for your answer. (b) At what value of x does f attain its absolute minimum value on the closed interval 1 x 5 ? At what value of x does f attain its absolute maximum value on the closed interval 1 x 5 ? Show the analysis that leads to your answers. £ £ £- £- x f x . Find an equation for the line tangent to the graph of g )( (c) Let g be the function defined by g x at x 2. =) ( = Copyright © 2004 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). GO ON TO THE NEXT PAGE. 4 2004 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) x4 y - ( = dy dx 2. ) 5. Consider the differential equation (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated. (Note: Use the axes provided in the test booklet.) (b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the xy-plane. Describe all points in the xy-plane for which the slopes are negative. f x to the given differential equation with the initial condition f 0 = )( )( = (c) Find the particular solution y 0. Copyright © 2004 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). GO ON TO THE NEXT PAGE. 5 2004 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) a = 1, as shown above. 1n 0 x dx in terms of n. (b) Let T be the triangular region bounded by , the x-axis, and the line x = z (a) Find x n at the point 1, 1 , where n f be the line tangent to the graph of y > 6. Let 1. Show that the area of T is 1 . 2n x n , the line , and the x-axis. Express the area of S in (c) Let S be the region bounded by the graph of y terms of n and determine the value of n that maximizes the area of S. = END OF EXAMINATION Copyright © 2004 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents). 6 ...
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