AP® Calculus AB2015 Free-Response Questions© 2015 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. 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
2015 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2015 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. The rate at which rainwater flows into a drainpipe is modeled by the function R, where220sin35tRtÈØÉÙÊÚcubic feet per hour, t is measured in hours, and 0t
8. The pipe is partially blocked, allowing water to drain out the other end of the pipe at a rate modeled by 320.04 0.4 0.96D ttt tcubic feet per hour, for 0t
8. There are 30 cubic feet of water in the pipe at time 0.t(a) How many cubic feet of rainwater flow into the pipe during the 8-hour time interval 08?t
(b) Is the amount of water in the pipe increasing or decreasing at time 3t hours? Give a reason for your answer. (c) At what time t, 0t
8,is the amount of water in the pipe at a minimum? Justify your answer. (d) The pipe can hold 50 cubic feet of water before overflowing. For8t!, water continues to flow into and out of the pipe at the given rates until the pipe begins to overflow. Write, but do not solve, an equation involving one or more integrals that gives the time w when the pipe will begin to overflow. www.mymathscloud.com
2015 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -3- 2. Let f and g be the functions defined by 221xxfx x e and 426.5 6 2.gx x x x Let R and Sbe the two regions enclosed by the graphs of f and g shown in the figure above. (a) Find the sum of the areas of regions R and S. (b) Region S is the base of a solid whose cross sections perpendicular to the x-axis are squares. Find the volume of the solid. (c) Let h be the vertical distance between the graphs of f and g in region S. Find the rate at which h changes with respect to x when 1.8.xEND OF PART A OF SECTION II www.mymathscloud.com
CALCULUS AB 2015 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -4- SECTION II, Part B Time — 60 minutes Number of problems —4 No calculator is allowed for these problems. t (minutes) 0 12 20 24 40vt(meters per minute) 0 200 240 –220 1503. Johanna jogs along a straight path. For 040t
, Johanna’s velocity is given by a differentiable function v.Selected values of vt, where t is measured in minutes and vt is measured in meters per minute, are given inthe table above.(a) Use the data in the table to estimate the value of 16v(b) Using correct units, explain the meaning of the definite integral400vt dÔt in the context of the problem.Approximate the value of 400vt dÔt using a right Riemann sum with the four subintervals indicated in thetable. (c) Bob is riding his bicycle along the same path. For 010t
, Bob’s velocity is modeled by 326 300Bt t t , where t is measured in minutes and Bt is measured in meters per minute.Find Bob’s acceleration at time 5t(d) Based on the model B from part (c), find Bob’s average velocity during the interval .01t
0..www.mymathscloud.com
2015 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -5- 4. Consider the differential equation 2dyx ydx .(a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated. (b) Find22dydx in terms of x and y. Determine the concavity of all solution curves for the given differential equation in Quadrant II. Give a reason for your answer. (c) Let y fx be the particular solution to the differential equation with the initial condition 23.fDoes f have a relative minimum, a relative maximum, or neither at 2x? Justify your answer. (d) Find the values of the constants m and b for which y mx b is a solution to the differential equation. www.mymathscloud.com
2015 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -6- 5. The figure above shows the graph of ,f the derivative of a twice-differentiable function f, on the interval > @3, 4. The graph of fhas horizontal tangents at 1, 1,xx and 3.x The areas of the regions bounded by the x-axis and the graph of fon the intervals >@2,1 and > @1, 4 are 9 and 12, respectively. (a) Find all x-coordinates at which f has a relative maximum. Give a reason for your answer. (b) On what open intervals contained in 3x 4 is the graph of f both concave down and decreasing? Give a reason for your answer. (c) Find the x-coordinates of all points of inflection for the graph of f. Give a reason for your answer. (d) Given that 1f3, write an expression for f x that involves an integral. Find 4f and 2f.www.mymathscloud.com
