AP® Calculus AB 2013 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 6,000 of the world’s leading educational institutions and is dedicated to promoting excellence and equityin 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.© 2013 The College Board. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks 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 onlineat: 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.org. www.mymathscloud.com
2013 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2013 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. On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by 29045cos,18tGtÈØ ÉÙÊÚ where t is measured in hours and 08.t
At the beginning of the workday 0,t the plant has 500 tons of unprocessed gravel. During the hours of operation, 08,t
the plant processes gravel at a constant rate of 100 tons per hour. (a) Find 5.G Using correct units, interpret your answer in the context of the problem. (b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday. (c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time 5t hours? Show the work that leads to your answer. (d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify your answer. www.mymathscloud.com
2013 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2013 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -3- 2. A particle moves along a straight line. For 05,t
the velocity of the particle is given by 652323 ,vtttt and the position of the particle is given by .st It is known that 010.s(a) Find all values of t in the interval 24t
for which the speed of the particle is 2. (b) Write an expression involving an integral that gives the position .st Use this expression to find the position of the particle at time 5.t(c) Find all times t in the interval 05t
at which the particle changes direction. Justify your answer. (d) Is the speed of the particle increasing or decreasing at time 4?t Give a reason for your answer. END OF PART A OF SECTION II www.mymathscloud.com
2013 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2013 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. t (minutes) 0 1 2 3 4 5 6 Ct(ounces) 0 5.3 8.8 11.2 12.8 13.8 14.5 3. Hot water is dripping through a coffeemaker, filling a large cup with coffee. The amount of coffee in the cup at time t, 06,t
is given by a differentiable function C, where t is measured in minutes. Selected values of ,Ct measured in ounces, are given in the table above. (a) Use the data in the table to approximate 3.5 .C Show the computations that lead to your answer, and indicate units of measure. (b) Is there a time t, 24,t
at which 2?Ct Justify your answer. (c) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate the value of 601.6Ct dtÔ Using correct units, explain the meaning of 6016Ct dtÔ in the context of the problem. (d) The amount of coffee in the cup, in ounces, is modeled by 0.41616.tBte Using this model, find the rate at which the amount of coffee in the cup is changing when 5.twww.mymathscloud.com
2013 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2013 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -5- 4. The figure above shows the graph of ,f the derivative of a twice-differentiable function f, on the closed interval 08.x
The graph of f has horizontal tangent lines at 1,x3,x and 5.x The areas of the regions between the graph of f and the x-axis are labeled in the figure. The function f is defined for all real numbers and satisfies 84.f(a) Find all values of x on the open interval 08x for which the function f has a local minimum. Justify your answer. (b) Determine the absolute minimum value of f on the closed interval 08.x
Justify your answer. (c) On what open intervals contained in 08x is the graph of f both concave down and increasing? Explain your reasoning. (d) The function g is defined by 3.gxf x If 53,2f find the slope of the line tangent to the graph of g at 3.xwww.mymathscloud.com
2013 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2012 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -6- 5. Let 2264fxxx and 14cos.4gxxp Let R be the region bounded by the graphs of f and g, as shown in the figure above. (a) Find the area of R. (b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line 4.y(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Write, but do not evaluate, an integral expression that gives the volume of the solid. www.mymathscloud.com
