AP® Calculus AB2011 Free-Response QuestionsForm BAbout the College BoardThe College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the College Board was created to expand access to higher education. Today, the membership association is made up of more than 5,900 of the world’s leading educational institutions and is dedicated to promoting excellence and equity in education. Each year, the College Board helps more than seven million students prepare for a successful transition to college through programs and services in college readiness and college success — including the SAT® and the AdvancedPlacement Program®. The organization also serves the education community through research and advocacy on behalf of students, educators and schools.© 2011 The College Board. College Board, Advanced Placement Program, AP, AP Central, SAT and the acorn logo are registeredtrademarks of the College Board. Admitted Class Evaluation Service and inspiring minds are trademarks owned by the College Board. All other products and services may be trademarks of their respective owners. Visit the College Board on the Web:www.collegeboard.org. Permission to use copyrighted College Board materials may be requested online at:www.collegeboard.org/inquiry/cbpermit.html. Visit the College Board on the Web: www.collegeboard.org. AP Central is the official online home for the AP Program: apcentral.collegeboard.com.www.mymathscloud.com
2011 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2011 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -2- CALCULUS AB SECTION II, Part A Time — 30 minutes Number of problems — 2 A graphing calculator is required for these problems. 1. A cylindrical can of radius 10 millimeters is used to measure rainfall in Stormville. The can is initially empty, and rain enters the can during a 60-day period. The height of water in the can is modeled by the function S, where ()St is measured in millimeters and t is measured in days for 060.t££ The rate at which the height of the water is rising in the can is given by ()( )2 sin 0.031.5.Stt=+¢(a) According to the model, what is the height of the water in the can at the end of the 60-day period? (b) According to the model, what is the average rate of change in the height of water in the can over the 60-day period? Show the computations that lead to your answer. Indicate units of measure. (c) Assuming no evaporation occurs, at what rate is the volume of water in the can changing at time 7?t=Indicate units of measure. (d) During the same 60-day period, rain on Monsoon Mountain accumulates in a can identical to the one in Stormville. The height of the water in the can on Monsoon Mountain is modeled by the function M, where ()()321330330.400Mtttt=-+ The height ()Mt is measured in millimeters, and t is measured in days for 060.t££ Let () () ().DtMtSt=-¢¢ Apply the Intermediate Value Theorem to the function D on the interval 060t££ to justify that there exists a time t, 060,t<< at which the heights of water in the two cans are changing at the same rate. 2. A 12,000-liter tank of water is filled to capacity. At time 0,t= water begins to drain out of the tank at a rate modeled by (),rt measured in liters per hour, where r is given by the piecewise-defined function ()0.2600f or 0531000for 5ttttrtet-Ï££Ô+=ÌÔ>Ó(a) Is r continuous at 5?t= Show the work that leads to your answer. (b) Find the average rate at which water is draining from the tank between time 0t= and time 8t= hours. (c) Find ()3.r¢ Using correct units, explain the meaning of that value in the context of this problem. (d) Write, but do not solve, an equation involving an integral to find the time A when the amount of water in the tank is 9000 liters. WRITE ALL WORK IN THE EXAM BOOKLET. END OF PART A OF SECTION II www.mymathscloud.com
2011 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2011 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -3- CALCULUS AB SECTION II, Part B Time —60 minutes Number of problems — 4 No calculator is allowed for these problems. 3. The functions f and g are given by ()fxx= and ()6.gxx=- Let R be the region bounded by the x-axis and the graphs of f and g, as shown in the figure above. (a) Find the area of R. (b) The region R is the base of a solid. For each y, where 02,y££ the cross section of the solid taken perpendicular to the y-axis is a rectangle whose base lies in R and whose height is 2y. Write, but do not evaluate, an integral expression that gives the volume of the solid. (c) There is a point P on the graph of f at which the line tangent to the graph of f is perpendicular to the graph of g. Find the coordinates of point P. 4. Consider a differentiable function f having domain all positive real numbers, and for which it is known that () ( )34fxxx-=-¢ for 0.x>(a) Find the x-coordinate of the critical point of f. Determine whether the point is a relative maximum, a relative minimum, or neither for the function f. Justify your answer. (b) Find all intervals on which the graph of f is concave down. Justify your answer. (c) Given that ()12,f= determine the function f. WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
2011 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2011 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -4- t(seconds) 0 10 40 60 ()Bt(meters) 100 1369 49 ()vt(meters per second)2.0 2.3 2.5 4.6 5. Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiable function B models Ben’s position on the track, measured in meters from the western end of the track, at time t, measured in seconds from the start of the ride. The table above gives values for ()Bt and Ben’s velocity, (),vt measured in meters per second, at selected times t. (a) Use the data in the table to approximate Ben’s acceleration at time 5t= seconds. Indicate units of measure. (b) Using correct units, interpret the meaning of ()600vt dtÚ in the context of this problem. Approximate ()600vt dtÚ using a left Riemann sum with the subintervals indicated by the data in the table. (c) For 4060,t££ must there be a time t when Ben’s velocity is 2 meters per second? Justify your answer. (d) A light is directly above the western end of the track. Ben rides so that at time t, the distance ()Lt between Ben and the light satisfies ()()()()22212.LtBt=+ At what rate is the distance between Ben and the light changing at time 40 ?t=WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
2011 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B) © 2011 The College Board. Visit the College Board on the Web: www.collegeboard.org. -5- 6. Let g be the piecewise-linear function defined on []2,4pp- whose graph is given above, and let () ()()cos.2xfxgx=-(a) Find ()42.fxdxpp-Ú Show the computations that lead to your answer. (b) Find all x-values in the open interval ()2,4pp- for which f has a critical point. (c) Let ()()30.xhxgt dt=Ú Find ().3hp-¢WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM www.mymathscloud.com
