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2014 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2014 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. Grass clippings are placed in a bin, where they decompose. For 030t
, the amount of grass clippings remaining in the bin is modeled by 6.687 0.931tAt, where ()A t is measured in pounds and t is measured in days. (a) Find the average rate of change of A t over the interval 030t
. Indicate units of measure. (b) Find the value of 15A. Using correct units, interpret the meaning of the value in the context of the problem. (c) Find the time t for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval 030.t
(d) For 30t!,L t, the linear approximation to A at t30, is a better model for the amount of grass clippings remaining in the bin. Use L t to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer. www.mymathscloud.com
2014 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2014 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -3- 2. Let R be the region enclosed by the graph of 432.34fxxx and the horizontal line y4, as shown in the figure above. (a) Find the volume of the solid generated when R is rotated about the horizontal line 2y .(b) Region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in R. Find the volume of the solid. (c) The vertical line xk divides R into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value k. END OF PART A OF SECTION II www.mymathscloud.com
2014 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2014 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. The function f is defined on the closed interval >@5, 4. The graph of f consists of three line segments and is shown in the figure above. Let g be the function defined by 3xgxf t dÔ.t(a) Find 3g.(b) On what open intervals contained in5x 4 is the graph of g both increasing and concave down? Give a reason for your answer. (c) The function h is defined by 5gxhxx. Find 3.h(d) The function p is defined by 2p xfxx . Find the slope of the line tangent to the graph of p at the point where 1x .www.mymathscloud.com
2014 AP® CALCULUS AB FREE-RESPONSE QUESTIONS .© 2014 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -5- t minutes0 2 5 8 12vtAmeters minute0 100 40 –120 –150 4. Train A runs back and forth on an east-west section of railroad track. Train A’s velocity, measured in meters per minute, is given by a differentiable function ,Avt where time t is measured in minutes. Selected values for Avt are given in the table above. (a) Find the average acceleration of train A over the interval 2t
8.(b) Do the data in the table support the conclusion that train A’s velocity is 100 meters per minute at some time t with 5t8? Give a reason for your answer. (c) At time t2, train A’s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train A, in meters from the Origin Station, at time 12t Use a trapezoidal sum with three subintervals indicated by the table to approximate .the position of the train at time 12.t(d) A second train, train B, travels north from the Origin Station. At time t the velocity of train B is given by 256025,Bvtt t and at time 2t the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train A and train B is changing at time 2twww.mymathscloud.com
x22x111x11 1x33 fx12Positive8Positive2Positive7fx5Negative0Negative0Positive12gx1Negative0Positive3Positive1gx2Positive32Positive0Negative2 5. The twice-differentiable functions f and g are defined for all real numbers x. Values of f, f,g, and g for various values of x are given in the table above. (a) Find the x-coordinate of each relative minimum of f on the interval> @2, 3. Justify your answers. (b) Explain why there must be a value c, for 1c 1, such that 0.fc(c) The function h is defined by lnhxf x. Find 3h. Show the computations that lead to your answer. (d) Evaluate 32.fgx g x dxÔ2014 AP® CALCULUS AB FREE-RESPONSE QUESTIONS © 2014 The College Board. Visit the College Board on the Web: www.collegeboard.org. GO ON TO THE NEXT PAGE. -6- www.mymathscloud.com