2007 AP Calculus AB Free-Response Questions Section II, Part A (45 minutes) # of questions: 3 A graphing calculator may be used for this part

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1 2007 AP Calculus AB Free-Response Questions Section II, Part A (45 minutes) # of questions: 3 A graphing calculator may be used for this part 1. Let R be the region in the first and second quadrants bounded by the graph of and below by the horizontal line y = 2. (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the x-axis. (c) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are semicircles. Find the volume of this solid. The graph for this problem is shown to the right: In order to find the bounds, you can use your graphing calculator to get the points of intersection at x = ±3. (a) Area = or (b) To find the volume, we will use the washer method: or π (c) Base = 2 Cross Section: Semi-Circle Area: 2 Volume= By Hand Solution to finding the bounds:

2 2. The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval 0 t 7, where t is measured in hours. In this model, rates are given as follows: (i) The rate at which water enters the tank is 100 sin gallons per hour for 0 t 7. (ii) The rate at which water leaves the tank is 250 for for 3 7 gallons per hour. The graphs of f and g, which intersect at t = and t = 5.076, are shown in the figure above. At time t = 0, the amount of water in the tank is 5000 gallons. (a) How many gallons of water enter the tank during the time interval 0 t 7? Round your answer to the nearest gallon. (b) For 0 t 7, find the time intervals during which the amount of water in the tank is decreasing. Give a reason for each answer. (c) For 0 t 7, at what time t is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer. (a) 100 sin gallons (b) The amount of water in the tank will be decreasing over the intervals [0,1.617) and (3,5.076) because. (c) The amount of water in the tank will be at its greatest at t = 3. The only intervals in which the amount of water is increasing are (1.617,3) and (5.076,7]. Let A(t) = amount of water in the tank at any time t. A(0) = sin gal. A(3) = sin gal. A(5.076) = sin sin A(7) = sin sin gal. gal Therefore the maximum occurs at t = 3 with a maximum amount of 5127 gallons.

3 3. The functions f and g are differentiable for all real numbers, and g is strictly increasing. The table below gives values of the functions and their first derivatives at selected values of x. The function h is given by 6. (a) Explain why there must be a value r for 1 < r < 3 such that 5. (b) Explain why there must be a value c for 1 < c < 3 such that 5. (c) Let w be the function given by. Find the value of 3. (d) If is the inverse function of g, write an equation for the line tangent to the graph of at x = 2. x (a) Below is the table for h(x): x 6 h(x) 1 f(2) 6= f(3) 6= f(4) 6= Since f and g are differentiable, then f(g(x)) is also differentiable. Since h(2) = 4 and h(3) = 7, then, by the IVT there is a point c, 2 < c < 3, such that h(c) = 5 (b) By the Chain Rule: By the Mean Value Theorem, there exists a point c in the interval [a,b] such that h (c) = Therefore: (c) From the First Fundamental Theorem of Calculus: Therefore w (3) = f(4) g (3) = (d) At x = 2: 1 By Def of Inverse Functions: = f(g(x)) g (x) Taking the derivative of both sides: 1 Therefore: At x = 2: 2 Resulting in 1 2

4 2007 AP Calculus AB Free-Response Questions Section II, Part B (45 minutes) # of questions: 3 No calculator may be used for this part 4. A particle is moves along the x-axis with the position at time t given by sin for 0 t 2π. (a) Find the time t at which the particle is farthest to the left. Justify your answer. (b) Find the value of the constant A for which x(t) satisfies the equation " 0 for 0 t 2π. (a) cos sin cos sin = 0 cos sin 0 0 cos sin 0 cos sin 1 = tan t Q: I, III R:, Using the 1 st Derivative Test cos sin right left right Therefore the left most point will occur at (b) cos sin " sin cos cos sin " 2 cos " 0 2 cos cos sin sin 0 2 cos cos 0 cos

5 t (minutes) r'(t) (feet per minute) The volume of a spherical hot air balloon expands as the air inside the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0 < t < 12, the graph of r is concave down. The table above gives selected values of the rate of change, r (t), of the radius of the balloon over the time interval 0 t 12. The radius of the balloon is 30 feet when t = 5. (Note: The volume of a sphere of radius r is given by.) (a) Estimate the radius of the balloon when t = 5.4 using the tangent line approximation at t = 5. Is your estimate greater than or less than the true value? Given a reason for your answer. (b) Find the rate of change of the volume of the balloon with respect to time when t = 5. Indicate units of measure. (c) Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate. Using correct units, explain the meaning of in terms of the radius of the balloon. (d) Is your approximation in part (c) greater than or less than in? Give a reason for your answer. (a) At t = 5: r (t) = 2, r = 30 Therefore the tangent line is which simplifies to: 220 At t = 5.4: r = 2(5.4) + 20 = 30.8 feet Using the MVT, we can find r (5.4) =. 0.40concave down Since r is negative, then the estimate is greater than the true value. (b) 4 at t = 5, r = 30, 2: feet/min (c) is the amount the radius changes in first 12 minutes ft (d) Since r is a decreasing function, then a right Riemann sum will result in an approximation that is less that the true value.

6 6. Let f be the function defined by ln for x > 0, where k is a positive constant. (a) Find and ". (b) For what values of the constant k does f have a critical point at x = 1? For this value of k, determine whether f has a relative minimum, relative maximum, or neither at x = 1. Justify your answer. (c) For certain value of the constant k, the graph of f has a point of inflection on the x-axis. Find this value of k. (a) " (b) f has a critical point when f = at x = 1: k 2 = 0 k = 2. At x = 1 and k = 2: " 0 Therefore a relative minimum. (c) A point of inflection will occur when f = 0. The value of x for which the point of inflection will be on the x-axis will be when f(x) = 0: ln0 " At x = : k = ln4 0