# AP Calculus AB 2012 Free-Response Questions

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2 CALCULUS AB SECTION II, Part A Time 3 minutes Number of problems A graphing calculator is required for these problems. t (minutes) Wt (degrees Fahrenheit) The temperature of water in a tub at time t is modeled by a strictly increasing, twice-differentiable function W, where Wt is measured in degrees Fahrenheit and t is measured in minutes. At time t, the temperature of the water is 55 F. The water is heated for 3 minutes, beginning at time t. Values of Wt at selected times t for the first minutes are given in the table above. W 1. Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem. (a) Use the data in the table to estimate (b) Use the data in the table to evaluate W t dt. Using correct units, interpret the meaning of W t dt in the context of this problem. 1 (c) For t, the average temperature of the water in the tub is. Wt dt Use a left Riemann sum 1 with the four subintervals indicated by the data in the table to approximate. Wt dt Does this approximation overestimate or underestimate the average temperature of the water over these minutes? Explain your reasoning. (d) For t 5, the function W that models the water temperature has first derivative given by W t.4 t cos.6 t. Based on the model, what is the temperature of the water at time t 5? --

3 . Let R be the region in the first quadrant bounded by the x-axis and the graphs of y ln x and y 5 x, as shown in the figure above. (a) Find the area of R. (b) Region R is the base of a solid. For the solid, each cross section perpendicular to the x-axis is a square. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid. (c) The horizontal line y k divides R into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of k. END OF PART A OF SECTION II -3-

4 CALCULUS AB SECTION II, Part B Time 6 minutes Number of problems 4 No calculator is allowed for these problems. 3. Let f be the continuous function defined on 4, 3 whose graph, consisting of three line segments and a x semicircle centered at the origin, is given above. Let g be the function given by gx ft dt. (a) Find the values of g and g (b) For each of g 3 and g 3,. find the value or state that it does not exist. (c) Find the x-coordinate of each point at which the graph of g has a horizontal tangent line. For each of these points, determine whether g has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers. (d) For 4 x 3, find all values of x for which the graph of g has a point of inflection. Explain your reasoning. 1-4-

5 f x 4. The function f is defined by (a) Find f x. 5 x for 5 x 5. (b) Write an equation for the line tangent to the graph of f at x 3. (c) Let g be the function defined by gx for 5 x 3 f x x 7 for 3 x 5. Is g continuous at x 3? Use the definition of continuity to explain your answer. (d) Find the value of 5 x 5 x dx. -5-

6 5. The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time t, when the bird is first weighed, its weight is grams. If Bt is the weight of the bird, in grams, at time t days after it is first weighed, then Let y db 1 1 B. dt 5 Bt be the solution to the differential equation above with initial condition B. (a) Is the bird gaining weight faster when it weighs 4 grams or when it weighs 7 grams? Explain your reasoning. d (b) Find B dt in terms of B. Use d B dt to explain why the graph of B cannot resemble the following graph. (c) Use separation of variables to find condition B. y, Bt the particular solution to the differential equation with initial -6-

7 6. For t 1, a particle moves along the x-axis. The velocity of the particle at time t is given by p vt cos t. The particle is at position x at time t. 6 (a) For t 1, when is the particle moving to the left? (b) Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from time t to time t 6. (c) Find the acceleration of the particle at time t. Is the speed of the particle increasing, decreasing, or neither at time t 4? Explain your reasoning. (d) Find the position of the particle at time t 4. STOP END OF EXAM -7-

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