2000 Advanced Placement Program®Free-Response QuestionsThe materials included in these files are intended for use by AP® teachers for courseand exam preparation in the classroom; permission for any other use must besought from the Advanced Placement Program. Teachers may reproduce them, inwhole or in part, in limited quantities, for face-to-face teaching purposes but maynot mass distribute the materials, electronically or otherwise. These materials andany copies made of them may not be resold, and the copyright notices must beretained as they appear here. This permission does not apply to any third-partycopyrights contained herein.These materials were produced by Educational Testing Service (ETS), which develops and administers the examinations of the Advanced PlacementProgram for the College Board. The College Board and Educational Testing Service (ETS) are dedicated to the principle of equal opportunity, and theirprograms, 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 3,800 schools, colleges, universities, and other educational organizations. Each year, theCollege Board serves over three million students and their parents, 22,000 high schools, and 5,000 colleges, through major programs and services in collegeadmission, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT®,the Advanced Placement Program® (AP®), and Pacesetter®. The College Board is committed to the principles of equity and excellence, and thatcommitment is embodied in all of its programs, services, activities, and concerns.Copyright © 2000 by College Entrance Examination Board and Educational Testing Service. All rights reserved. College Board, Advanced PlacementProgram, AP, and the acorn logo are registered trademarks of the College Entrance Examination Board.www.mymathscloud.com
2000 AP®CALCULUS AB FREE-RESPONSE QUESTIONSCopyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved.AP is a registered trademark of the College Entrance Examination Board.GO ON TO THE NEXT PAGE.-2-CALCULUS ABSECTION II, Part ATime — 45 minutesNumber of problems — 3A graphing calculator is required for some problems or parts of problems.1. Let R be the shaded region in the first quadrant enclosed by the graphs of yex=−2,yxaE1cos, and they-axis, as shown in the figure above.(a) Find the area of the region R.(b) Find the volume of the solid generated when the region R is revolved about the x-axis.(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square.Find the volume of this solid.www.mymathscloud.com
2000 AP®CALCULUS AB FREE-RESPONSE QUESTIONSCopyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved.AP is a registered trademark of the College Entrance Examination Board.GO ON TO THE NEXT PAGE.-3-2. Two runners, A and B, run on a straight racetrack for 010≤≤t seconds. The graph above, which consists oftwo line segments, shows the velocity, in meters per second, of Runner A. The velocity, in meters per second, ofRunner B is given by the function v defined by vttt().aC2423(a) Find the velocity of Runner A and the velocity of Runner B at time t=2 seconds. Indicate units ofmeasure.(b) Find the acceleration of Runner A and the acceleration of Runner B at time t=2 seconds. Indicate unitsof measure.(c) Find the total distance run by Runner A and the total distance run by Runner B over the time interval010≤≤t seconds. Indicate units of measure.www.mymathscloud.com
2000 AP®CALCULUS AB FREE-RESPONSE QUESTIONSCopyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved.AP is a registered trademark of the College Entrance Examination Board.-4-3. The figure above shows the graph of f, the derivative of the function f, for E77x. The graph of f hashorizontal tangent lines at xxaEa32,, and xa5, and a vertical tangent line at xa3.(a) Find all values of x, for E``77x, at which fattains a relative minimum. Justify your answer.(b) Find all values of x, for E``77x, at which fattains a relative maximum. Justify your answer.(c) Find all values of x, for E``77x, at which @A`fx0.(d) At what value of x, for E77x, does f attain its absolute maximum? Justify your answer.END OF PART A OF SECTION IIwww.mymathscloud.com
2000 AP®CALCULUS AB FREE-RESPONSE QUESTIONSCopyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved.AP is a registered trademark of the College Entrance Examination Board.-5-CALCULUS ABSECTION II, Part BTime —45 minutesNumber of problems—3No calculator is allowed for these problems.4. Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tankat the rate of t+1 gallons per minute, for 0120≤≤t minutes. At time t=0, the tank contains 30 gallons ofwater.(a) How many gallons of water leak out of the tank from time t=0 to t=3 minutes?(b) How many gallons of water are in the tank at time ta3 minutes?(c) Write an expression for At(), the total number of gallons of water in the tank at time t.(d) At what time t, for 0120t, is the amount of water in the tank a maximum? Justify your answer.5. Consider the curve given by xyx y236Ea.(a) Show that dydxxy yxyx=−−32223.(b) Find all points on the curve whose x-coordinate is 1, and write an equation for the tangent line at each ofthese points.(c) Find the x-coordinate of each point on the curve where the tangent line is vertical.6. Consider the differential equation dydxxeya322.(a) Find a solution yfx=() to the differential equation satisfying f012()=.(b) Find the domain and range of the function f found in part (a).END OF EXAMINATIONwww.mymathscloud.com
