dx Let y f x equation with initial condition f 0 1. If f 3, find the slope of the line tangent to the graph of g at xĦ 5. (d) The function g is defined by 3 5 gx f x. (c) On what open intervals contained in 0 x 8 is the graph of f both concave down and increasing? Explain your reasoning. (b) Determine the absolute minimum value of f on the closed interval 0 x 8. (a) Find all values of x on the open interval 0 x 8 for which the function f has a local minimum. The function f is defined for all real numbers and satisfies f 8 4. The areas of the regions between the graph of f and the x-axis are labeled in the figure. The graph of f has horizontal tangent lines at x 1, x 3, and x 5. The figure above shows the graph of f, the derivative of a twice-differentiable function f, on the closed interval 0 x 8. Using this model, find the rate at which the amount of coffee in the cup is changing when tĥ 4. 0.4t (d) The amount of coffee in the cup, in ounces, is modeled by Bt e. 6 Using correct units, explain the meaning of 0 6 Ct dtin the context of the 0 problem. Show the computations that lead to your answer, and (c) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate the value of Ct dt. (b) Is there a time t, t 4, at which C t? Justify your answer. Selected values of Ct measured in ounces, are given in the table above., (a) Use the data in the table to approximate indicate units of measure. The amount of coffee in the cup at time t, 0 t 6, is given by a differentiable function C, where t is measured in minutes. t (minutes) Ct (ounces) Hot water is dripping through a coffeemaker, filling a large cup with coffee. END OF PART A OF SECTION II -3-Ĥ CALCULUS BC SECTION II, Part B Time 60 minutes Number of problems 4 No calculator is allowed for these problems. Find the time t in (c) For the particle described in part (b), find the position vector in terms of t. (b) A particle moves along the polar curve r 4 sinq so that at time t seconds, the interval 1 t for which the x-coordinate of the particle s position is 1. 6 (a) Let S be the shaded region that is inside the graph of r 3 and also inside the graph of r 4 sin q. The curves intersect when p q and 6 q 5 p. The graphs of the polar curves r 3 and r 4 sinq are shown in the figure above. (d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify your answer. (c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time t 5 hours? Show the work that leads to your answer. (b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday. Using correct units, interpret your answer in the context of the problem. During the hours of operation, 0 t 8, the plant processes gravel at a constant rate of 100 tons per hour. At the beginning of the workday t 0, the plant has 500 tons of unprocessed gravel. On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant t is modeled by Gt 90 45cos, 18 where t is measured in hours and 0 t 8. Visit the College Board on the Web: Permission to use copyrighted College Board materials may be requested online at: AP Central is the official online home for the AP Program: .Ģ CALCULUS BC SECTION II, Part A Time 30 minutes Number of problems A graphing calculator is required for these problems. All other products and services may be trademarks of their respective owners. Admitted Class Evaluation Service and inspiring minds are trademarks owned by the College Board. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. The organization also serves the education community through research and advocacy on behalf of students, educators, and schools. 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 Advanced Placement Program. 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 equity in education. Founded in 1900, the College Board was created to expand access to higher education. 1 AP Calculus BC 013 Free-Response Questions About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity.
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