AP Calculus AB 2011 Free-Response Questions

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2 11 AP CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part A Time 3 minutes Number of problems A graphing calculator is required for these problems. 1. For t 6, a particle is moving along the x-axis. The particle s position, xt (), is not explicitly given. t 4 vt = sin e + 1. The acceleration of the particle is given by The velocity of the particle is given by () ( ) 1 t () 4 t at = e cos( e 4 ) and x ( ) =. (a) Is the speed of the particle increasing or decreasing at time t = 5.5? Give a reason for your answer. (b) Find the average velocity of the particle for the time period t 6. (c) Find the total distance traveled by the particle from time t = to t = 6. (d) For t 6, the particle changes direction exactly once. Find the position of the particle at that time. --

3 11 AP CALCULUS AB FREE-RESPONSE QUESTIONS t (minutes) H() t (degrees Celsius) As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for t 1, where time t is measured in minutes and temperature H() t is measured in degrees Celsius. Values of H() t 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 t = 3.5. Show the computations that lead to your answer. 1 1 (b) Using correct units, explain the meaning of () 1 Ú Ht dtin the context of this problem. Use a trapezoidal 1 1 sum with the four subintervals indicated by the table to estimate (). 1 Ú Ht dt 1 (c) Evaluate Ú H () t dt. Using correct units, explain the meaning of the expression in the context of this problem. (d) At time t =, biscuits with temperature 1 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.173t B () t = e -. Using the given models, at time t = 1, how much cooler are the biscuits than the tea? END OF PART A OF SECTION II -3-

4 11 AP CALCULUS AB FREE-RESPONSE QUESTIONS CALCULUS AB SECTION II, Part B Time 6 minutes Number of problems 4 No calculator is allowed for these problems Let R be the region in the first quadrant enclosed by the graphs of f( x) = 8x and gx ( ) ( px) = sin, as shown in the figure above. 1 (a) Write an equation for the line tangent to the graph of f at x =. (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 y =

5 11 AP CALCULUS AB FREE-RESPONSE QUESTIONS 4. The continuous function f is defined on the interval - 4 x 3. The graph of f consists of two quarter circles gx x ft dt. = +Ú and one line segment, as shown in the figure above. Let ( ) ( ) (a) Find g( - 3. ) Find g ( x) and evaluate ( ) g -3. (b) Determine the x-coordinate of the point at which g has an absolute maximum on the interval - 4 x 3. Justify your answer. (c) Find all values of x on the interval - 4 < x < 3 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 - 4 x 3. There is no point c, - 4 < c < 3, for which f () c is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem. x -5-

6 11 AP CALCULUS AB FREE-RESPONSE QUESTIONS 5. At the beginning of 1, a landfill contained 14 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 dw 1 equation = ( W - 3) for the next years. W is measured in tons, and t is measured in years from dt 5 the start of 1. (a) Use the line tangent to the graph of W at t = to approximate the amount of solid waste that the landfill 1 contains at the end of the first 3 months of 1 (time t = ). 4 dw (b) Find dw in terms of W. Use dt dt an overestimate of the amount of solid waste that the landfill contains at time (c) Find the particular solution W W() t condition W ( ) = 14. to determine whether your answer in part (a) is an underestimate or t = = to the differential equation ( W 3) 1. 4 dw 1 = - with initial dt 5 f x 6. Let f be a function defined by ( ) -4 (a) Show that f is continuous at x =. Ï1- sin x for x = Ì x Óe for x >. (b) For x π, express f ( x) as a piecewise-defined function. Find the value of x for which f ( x ) =-3. (c) Find the average value of f on the interval [- 1, 1 ]. END OF EXAM -6-

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