APÆCalculus AB2002 Free-Response Questions These materials were produced by Educational Testing ServiceÆ (ETSÆ), which develops and administers the examinations of the Advanced Placement Program for the College Board. The College Board and Educational Testing Service (ETS) are dedicated to the principle of equal opportunity, and their programs, services, and employment policies are guided by that principle. The College Board is a national nonprofit membership association dedicated to preparing, inspiring, and connecting students to college and opportunity. Founded in 1900, the association is composed of more than 4,200 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 22,000 high schools, and 3,500 colleges, through major programs and services in college admission, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SATÆ, the PSAT/NMSQTÆ, and the Advanced Placement ProgramÆ (APÆ). The College Board is committed to the principles of equity and excellence, and that commitment is embodied in all of its programs, services, activities, and concerns. Copyright © 2002 by College Entrance Examination Board. All rights reserved. College Board, Advanced Placement Program, AP, SAT, and the acorn logo are registered trademarks of the College Entrance Examination Board. APIEL is a trademark owned by 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. The materials included in these files are intended for use by AP teachers for course and exam preparation in the classroom; permission for any other use must be sought from the Advanced Placement ProgramÆ. 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. These materials and any copies made of them may not be resold, and the copyright notices must be retained as they appear here. This permission does not apply to any third-party copyrights contained herein. www.mymathscloud.com
2002 AP® CALCULUS AB FREE-RESPONSE QUESTIONS Copyright © 2002 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. GO ON TO THE NEXT PAGE. 2CALCULUS 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 and g be the functions given by fxexHSa and gxxHSaln .(a) Find the area of the region enclosed by the graphs of f and g between xa12 and xa1.(b) Find the volume of the solid generated when the region enclosed by the graphs of f and g between xa12and xa1 is revolved about the line ya4.(c) Let h be the function given by hxf xgxHSHSHSaE. Find the absolute minimum value of hx@A on the closed interval 121x, and find the absolute maximum value of hx@A on the closed interval 121x.Show the analysis that leads to your answers. www.mymathscloud.com
2002 AP® CALCULUS AB FREE-RESPONSE QUESTIONS Copyright © 2002 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. GO ON TO THE NEXT PAGE. 3 2. The rate at which people enter an amusement park on a given day is modeled by the function E defined by Ettt@AaEC15600241602QV. The rate at which people leave the same amusement park on the same day is modeled by the function Ldefined by Lttt@AaEC9890383702QV. Both EtHS and LtHS are measured in people per hour and time t is measured in hours after midnight. These functions are valid for 923t, the hours during which the park is open. At time ta9, there are no people in the park. (a) How many people have entered the park by 5:00 P.M. (ta17 )? Round your answer to the nearest whole number. (b) The price of admission to the park is $15 until 5:00 P.M. (ta17). After 5:00 P.M., the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number. (c) Let HtE xLx dxt@Aa@AE@AsHS9 for 923t. The value of H17@A to the nearest whole number is 3725. Find the value of @AH17 , and explain the meaning of H17@A and @AH17 in the context of the amusement park. (d) At what time t, for 923t, does the model predict that the number of people in the park is a maximum? www.mymathscloud.com
2002 AP® CALCULUS AB FREE-RESPONSE QUESTIONS Copyright © 2002 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. 4 3. An object moves along the x-axis with initial position x02@Aa. The velocity of the object at time t0 is given by vtt@Aa%'(0sin.p3(a) What is the acceleration of the object at time ta4? (b) Consider the following two statements. Statement I: For 345``t., the velocity of the object is decreasing. Statement II: For 345``t., the speed of the object is increasing. Are either or both of these statements correct? For each statement provide a reason why it is correct or not correct. (c) What is the total distance traveled by the object over the time interval 04t? (d) What is the position of the object at time ta4? END OF PART A OF SECTION II www.mymathscloud.com
2002 AP®CALCULUS AB FREE-RESPONSE QUESTIONSCopyright © 2002 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. GO ON TO THE NEXT PAGE. 5CALCULUS AB SECTION II, Part B Time —45 minutes Number of problems —3 No calculator is allowed for these problems. 4. The graph of the function f shown above consists of two line segments. Let g be the function given by gxf t dtxHSHSas0.(a) Find gE@A1,E@Ag1, and E@Ag1.(b) For what values of x in the open interval E22,HS is g increasing? Explain your reasoning. (c) For what values of x in the open interval E22,HS is the graph of g concave down? Explain your reasoning. (d) On the axes provided, sketch the graph of g on the closed interval E22,.(Note: The axes are provided in the pink test booklet only.) www.mymathscloud.com
2002 AP®CALCULUS AB FREE-RESPONSE QUESTIONSCopyright © 2002 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board. GO ON TO THE NEXT PAGE. 6 5. A container has the shape of an open right circular cone, as shown in the figure above. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that its depth h is changing at the constant rate of E310 cm/hr. (Note: The volume of a cone of height h and radius r is given by Vrha132p.) (a) Find the volume V of water in the container when ha5 cm. Indicate units of measure. (b) Find the rate of change of the volume of water in the container, with respect to time, when ha5 cm. Indicate units of measure. (c) Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportionality? www.mymathscloud.com
