AP® Calculus AB 2006 Free-Response Questions Form B The College Board: Connecting Students to College Success 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 5,000 schools, colleges, universities, and other educational organizations. Each year, the College Board serves seven 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. © 2006 The College Board. All rights reserved. College Board, AP Central, APCD, Advanced Placement Program, AP, AP Vertical Teams, Pre-AP, SAT, and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation Service, CollegeEd, connect to college success, MyRoad, SAT Professional Development, SAT Readiness Program, and Setting the Cornerstones are trademarks owned by the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. Permission to use copyrighted College Board materials may be requested online at: www.collegeboard.com/inquiry/cbpermit.html. Visit the College Board on the Web: www.collegeboard.com. AP Central is the official online home for the AP Program: apcentral.collegeboard.com. www.mymathscloud.com
2006 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). GO ON TO THE NEXT PAGE. 2 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 f be the function given by ()323cos .432xxxfxx=--+ Let R be the shaded region in the second quadrant bounded by the graph of f, and let S be the shaded region bounded by the graph of f and line A, the line tangent to the graph of f at 0,x= as shown above. (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the horizontal line 2.y=-(c) Write, but do not evaluate, an integral expression that can be used to find the area of S. WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
2006 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). GO ON TO THE NEXT PAGE. 3 2. Let f be the function defined for 0x≥ with ()05f= and ,f¢ the first derivative of f, given by ()()()xfx ex42sin.-=¢ The graph of ()yfx=¢ is shown above. (a) Use the graph of f¢ to determine whether the graph of f is concave up, concave down, or neither on the interval 1.71.9.x<< Explain your reasoning. (b) On the interval 03,x££ find the value of x at which f has an absolute maximum. Justify your answer. (c) Write an equation for the line tangent to the graph of f at 2.x=WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
2006 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). 4 3. The figure above is the graph of a function of x, which models the height of a skateboard ramp. The function meets the following requirements. (i) At 0,x= the value of the function is 0, and the slope of the graph of the function is 0. (ii) At 4,x= the value of the function is 1, and the slope of the graph of the function is 1. (iii) Between 0x= and 4,x= the function is increasing. (a) Let ()2,fxax= where a is a nonzero constant. Show that it is not possible to find a value for a so that fmeets requirement (ii) above. (b) Let ()23,16xgxcx=- where c is a nonzero constant. Find the value of c so that g meets requirement (ii) above. Show the work that leads to your answer. (c) Using the function g and your value of c from part (b), show that g does not meet requirement (iii) above. (d) Let (),nxhxk= where k is a nonzero constant and n is a positive integer. Find the values of k and n so that h meets requirement (ii) above. Show that h also meets requirements (i) and (iii) above. WRITE ALL WORK IN THE EXAM BOOKLET. END OF PART A OF SECTION II www.mymathscloud.com
2006 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). GO ON TO THE NEXT PAGE. 5 CALCULUS AB SECTION II, Part B Time — 45 minutes Number of problems — 3 No calculator is allowed for these problems. 4. The rate, in calories per minute, at which a person using an exercise machine burns calories is modeled by the function f. In the figure above, ()3213142fttt=- + + for 04t££ and f is piecewise linear for 424.t££(a) Find ()22 .f¢ Indicate units of measure. (b) For the time interval 024,t££ at what time t is f increasing at its greatest rate? Show the reasoning that supports your answer. (c) Find the total number of calories burned over the time interval 618t££ minutes. (d) The setting on the machine is now changed so that the person burns ()ft c+ calories per minute. For this setting, find c so that an average of 15 calories per minute is burned during the time interval 618.t££WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
2006 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). GO ON TO THE NEXT PAGE. 6 5. Consider the differential equation ()()21cos.dyyxdxp=-(a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated. (Note: Use the axes provided in the exam booklet.) (b) There is a horizontal line with equation yc= that satisfies this differential equation. Find the value of c. (c) Find the particular solution ()yfx= to the differential equation with the initial condition ()10.f=www.mymathscloud.com
