AP® Calculus AB2011 Free-Response QuestionsAbout 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 © 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. For 06,t££ a particle is moving along the x-axis. The particle’s position, (),xt is not explicitly given. The velocity of the particle is given by ()()42sin1.tvte=+ The acceleration of the particle is given by ()()441cos2ttatee= and ()02.x=(a) Is the speed of the particle increasing or decreasing at time 5.5 ?t= Give a reason for your answer. (b) Find the average velocity of the particle for the time period 06.t££(c) Find the total distance traveled by the particle from time 0t= to 6.t=(d) For 06,t££ the particle changes direction exactly once. Find the position of the particle at that time. WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
2011 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2011 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -3- t(minutes) 0 2 5 9 10 ()Ht(degrees Celsius)66 60 52 44 43 2. As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for 010,t££ where time t is measured in minutes and temperature ()Ht is measured in degrees Celsius. Values of ()Ht at selected values of time t are shown in the table above. (a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time 3.5.t= Show the computations that lead to your answer. (b) Using correct units, explain the meaning of ()100110Ht dtÚ in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate ()1001.10Ht dtÚ(c) Evaluate ()100.Htdt¢Ú Using correct units, explain the meaning of the expression in the context of this problem. (d) At time 0,t= biscuits with temperature 100 C∞ were removed from an oven. The temperature of the biscuits at time t is modeled by a differentiable function B for which it is known that ()0.17313.84.tBte-=-¢ Using the given models, at time 10,t= how much cooler are the biscuits than the tea? WRITE ALL WORK IN THE EXAM BOOKLET. END OF PART A OF SECTION II www.mymathscloud.com
2011 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2011 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -4- CALCULUS AB SECTION II, Part B Time — 60 minutes Number of problems — 4 No calculator is allowed for these problems. 3. Let R be the region in the first quadrant enclosed by the graphs of ()38fxx= and () ( )sin,gxxp= as shown in the figure above. (a) Write an equation for the line tangent to the graph of f at 1.2x=(b) Find the area of R. (c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated about the horizontal line 1.y=WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
2011 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2011 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -5- 4. The continuous function f is defined on the interval 43.x-£ £ The graph of f consists of two quarter circles and one line segment, as shown in the figure above. Let ()()02.xgxxf t dt=+Ú(a) Find ()3.g- Find ()gx¢ and evaluate ()3.g-¢(b) Determine the x-coordinate of the point at which g has an absolute maximum on the interval 43.x-£ £Justify your answer. (c) Find all values of x on the interval 43x-< < for which the graph of g has a point of inflection. Give a reason for your answer. (d) Find the average rate of change of f on the interval 43.x-£ £ There is no point c, 43,c-< < for which ()fc¢ is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem. WRITE ALL WORK IN THE EXAM BOOKLET. www.mymathscloud.com
2011 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2011 The College Board. Visit the College Board on the Web: www.collegeboard.org. -6- 5. At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners estimate that W will satisfy the differential equation ()130025dWWdt=- for the next 20 years. W is measured in tons, and t is measured in years from the start of 2010. (a) Use the line tangent to the graph of W at 0t= to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of 2010 (time 14t=). (b) Find 22dWdt in terms of W. Use 22dWdt to determine whether your answer in part (a) is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time 1.4t=(c) Find the particular solution ()WWt= to the differential equation ()130025dWWdt=- with initial condition ()01400.W= 6. Let f be a function defined by ()412sin for 0for 0.xxxfxex--£Ï=Ì>Ó(a) Show that f is continuous at 0.x=(b) For 0,xπ express ()fx¢ as a piecewise-defined function. Find the value of x for which ()3.fx=-¢(c) Find the average value of f on the interval []1, 1 .-WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM www.mymathscloud.com
