APCalculus AB1998 Free-Response QuestionsThese materials are intended for non-commercial use by AP teachers for course andexam preparation; permission for any other use must be sought from the AdvancedPlacement Program. Teachers may reproduce them, in whole or in part, in limitedquantities, for face-to-face teaching purposes but may not mass distribute thematerials, electronically or otherwise. These materials and any copies made of themmay 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.These materials were produced by Educational Testing Service (ETS), which develops and administers the examinations of theAdvanced Placement Program for the College Board. The College Board and Educational Testing Service (ETS) are dedicatedto 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 studentsto 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, 22,000high 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 SAT, the PSAT/NMSQT, and theAdvanced Placement Program(AP). 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.For further information, contact www.collegeboard.com.Copyright2002 by College Entrance Examination Board. All rights reserved. College Board, Advanced PlacementProgram, AP, and the acorn logo are registered trademarks of the College Entrance Examination Board.www.mymathscloud.com
1998 AP Calculus AB Free-Response QuestionsCALCULUS ABSection IITime—1 hour and 30 minutesNumber of problems—6Percent of total grade—50A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THISSECTION OF THE EXAMINATION.REMEMBER TO SHOW YOUR SETUPS AS DESCRIBED IN THE GENERAL INSTRUCTIONS.1. LetRbe the region bounded by thex–axis, the graph ofy=px, and the linex= 4.(a) Find the area of the regionR.(b) Find the value ofhsuch that the vertical linex=hdivides the regionRinto two regionsof equal area.(c) Find the volume of the solid generated whenRis revolved about thex–axis.(d) The vertical linex=kdivides the regionRinto two regions such that when these tworegions are revolved about thex–axis, they generate solids with equal volumes. Find thevalue ofk.Copyright1998 College Entrance Examination Board. All rights reserved.www.mymathscloud.com
1998 AP Calculus AB Free-Response Questions2. Letfbe the function given byf(x)=2xe2x.(a) Find limx! 1f(x) and limx!1f(x).(b) Find the absolute minimum value off. Justify that your answer is an absolute minimum.(c) What is the range off?(d) Consider the family of functions defined byy=bxebx, wherebis a nonzero constant. Showthat the absolute minimum value ofbxebxis the same for all nonzero values ofb.Copyright1998 College Entrance Examination Board. All rights reserved.www.mymathscloud.com
1998 AP Calculus AB Free-Response QuestionsVelocity(feet per second)5101520253035404550102030405060708090tv(t)OTime (seconds)tv(t)(seconds)(feet per second)005121020153020552570307835814075456050723. The graph of the velocityv(t), in ft/sec, of a car traveling on a straight road, for 0t50,is shown above. A table of values forv(t), at 5 second intervals of timet, is shown to the rightof the graph.(a) During what intervals of time is the acceleration of the car positive? Give a reason foryour answer.(b) Find the average acceleration of the car, in ft/sec2, over the interval 0t50.(c) Find one approximation for the acceleration of the car, in ft/sec2, att= 40. Show thecomputations you used to arrive at your answer.(d) ApproximateZ500v(t)dtwith a Riemann sum, using the midpoints of five subintervals ofequal length. Using correct units, explain the meaning of this integral.Copyright1998 College Entrance Examination Board. All rights reserved.www.mymathscloud.com
1998 AP Calculus AB Free-Response Questions4. Letfbe a function withf(1) = 4 such that for all points (x,y) on the graph offthe slope isgiven by3x2+12y.(a) Find the slope of the graph offat the point wherex= 1.(b) Write an equation for the line tangent to the graph offatx= 1 and use it to approximatef(1.2).(c) Findf(x) by solving the separable di↵erential equationdydx=3x2+12ywith the initialconditionf(1) = 4.(d) Use your solution from part (c) to findf(1.2).Copyright1998 College Entrance Examination Board. All rights reserved.www.mymathscloud.com
1998 AP Calculus AB Free-Response Questions5. The temperature outside a house during a 24-hour period is given byF(t) = 80 10 cos✓⇡t12◆,0t24,whereF(t) is measured in degrees Fahrenheit andtis measured in hours.(a) Sketch the graph ofFon the grid below.TimeinHours06121824DegreesFahrenheit10090807060(b) Find the average temperature, to the nearest degree Fahrenheit, betweent= 6 andt= 14.(c) An air conditioner cooled the house whenever the outside temperature was at or above 78degrees Fahrenheit. For what values oftwas the air conditioner cooling the house?(d) The cost of cooling the house accumulates at the rate of0.05 per hour for each degreethe outside temperature exceeds 78 degrees Fahrenheit. What was the total cost, to thenearest cent, to cool the house for this 24–hour period?Copyright1998 College Entrance Examination Board. All rights reserved.www.mymathscloud.com
