Answer Key for AP Calculus AB Practice Exam, Section I. Question 23: B

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1 Answer Key for AP Calculus AB Practice Exam, Section I Question : A Question : D Question : B Question 4: D Question 5: C Question 6: B Question 7: C Question 8: D Question 9: A Question : E Question : B Question : B Question : E Question 4: D Question 5: B Question 6: A Question 7: C Question 8: C Question 9: D Question : D Question : B Question : D Question 4: C Question 5: B Question 6: B Question 7: C Question 8: C Question 76: D Question 77: C Question 78: E Question 79: B Question 8: E Question 8: E Question 8: E Question 8: C Question 84: D Question 85: D Question 86: D Question 87: C Question 88: D Question 89: D Question 9: C Question 9: A Question 9: A Question : B

2 6 SCORING GUIDELINES Question A company produces and sells chili powder. The company s weekly profit on the sale of x kilograms of chili.8 powder is modeled by the function P given by Px ( ) = 48x +.4x.5x 7, where Px ( ) is in dollars and x 8. (a) Find the rate, in dollars per kilogram, at which the company s weekly profit is changing when it sells kilograms of chili powder. Is the company s weekly profit increasing or decreasing when it sells kilograms of chili powder? Give a reason for your answer. (b) How many kilograms of chili powder must the company sell to maximize its weekly profit? Justify your answer. (c) The company plans to have a one-day sale on chili powder. Management estimates that t hours after the company store opens, chili powder will sell at a rate modeled by the function S given by St ( ) = + cos( t ) kilograms per hour. Based on this model, estimate the amount of chili powder, in kilograms, that will be sold during the first 5 hours of the sale. (d) Using the function S from part (c), find the value of S ( ). Interpret the meaning of this value in the context of the problem. (a) P ( ) = 65.9 Since P ( ) >, the company s profit is increasing when it sells kilograms of chili powder..8 (b) P ( x) = x.4x = when x = x Px ( ) : P ( ) : : increasing with reason : sets P ( x) = : : answer : justification The company must sell kilograms of chili powder to maximize its profit. (c) Based on this model, the estimate, in kilograms, is 5 : integral : S( t) dt =.4 (or.4). : answer (d) S ( ) =.58 The rate of sale of chili powder is decreasing at a rate of.58 kilogram per hour per hour at time t = hours. : S ( ) : : interpretation 6 The College Board.

3 6 SCORING GUIDELINES Question t (weeks) Gt ( ) (games per week) A store tracks the sales of one of its popular board games over a -week period. The rate at which games are being sold is modeled by the differentiable function G, where Gt ( ) is measured in games per week and t is measured in weeks for t. Values of Gt ( ) are given in the table above for selected values of t. (a) Approximate the value of G ( 8) using the data in the table. Show the computations that lead to your answer. (b) Approximate the value of G ( t ) dt using a right Riemann sum with the four subintervals indicated by the table. Explain the meaning of G ( t ) dt in the context of this problem. (c) One salesperson believes that, starting with 4 games per week at time t =, the rate at which games will be sold will increase at a constant rate of games per week per week. Based on this model, how many total games will be sold in the 8 weeks between time t = and t =? (d) Another salesperson believes the best model for the rate at which games will be sold in the 8 weeks between.( t ) time t = and t = is M( t) = 4e games per week. Based on this model, how many total games, to the nearest whole number, will be sold during this period? G( ) G( 6) 9 (a) G ( 8) = = 6 6 games per week per week : approximation (b) G( t) dt G( ) + G( 6) + 4 G( ) + G( ) = = 75 Gt ( ) dt represents the total number of games sold over the -week period t. (c) The rate of sales of the game for t is Rt ( ) = 4 + ( t ). Based on this model, the total number of games that will be sold between time t = and t = is R( t) dt = 4. (d) M ( t) dt = Based on this model, the total number of games that will be sold between time t = and t = is : right Riemann sum : : approximation : explanation : rate function : : integral : integral : : answer 6 The College Board.

4 6 SCORING GUIDELINES Question The function f is defined on the interval 5 x c, where c > and f( c ) =. The graph of f, which consists of three line segments and a quarter of a circle with center (, ) and radius, is shown in the figure above. (a) Find the average rate of change of f over the interval [ 5, ]. Show the computations that lead to your answer. (b) For 5 x c, let g be the function defined by x g( x) = f ( t) dt. Find the x-coordinate of each point of inflection of the graph of g. Justify your answer. (c) Find the value of c for which the average value of f over the interval 5 x c is. (d) Assume x c >. The function h is defined by hx ( ) f( ) =. Find h ( 6) in terms of c. (a) The average rate of change of f over the interval [ 5, ] is f( ) f( 5) =. ( 5) 5 (b) g ( x) = f( x) The graph of g has a point of inflection at x = because g = f changes from decreasing to increasing at this point. The graph of g has a point of inflection at x = because g = f changes from increasing to decreasing at this point. : answer : g ( x) = f( x) : : identifies x = and : justification x = c (c) f( x) = c ( + ( ) + + c) = c + 5 c = + (d) h x ( x ) = f ( ) h ( 6) = f ( ) = = c c : integral : : equation : h ( x) : 6 The College Board.

5 6 SCORING GUIDELINES Question 4 Let S be the shaded region in the first quadrant bounded above by the horizontal line y =, below by the graph of y = sin x, and on the left by the vertical line x = k, where < k <, as shown in the figure above. (a) Find the area of S when k =. (b) The area of S is a function of k. Find the rate of change of the area of S with respect to k when k =. 6 (c) Region S is revolved about the horizontal line y = 5 to form a solid. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid when k =. 4 (a) Area = ( sin x) x cos x = + : integrand : : antiderivative = ( + ) ( + ) = (b) Let Ak ( ) be the area of S. Ak ( ) = ( sin x) k A ( k) = + sin k A ( ) = + sin 6 ( 6) = : expression for area : : expression for A ( k) (c) Volume = ( 5 sinx) ( 5 ) 4 = ( 5 sin x) 4 4 : integrand : : limits and constant 6 The College Board.

6 6 SCORING GUIDELINES Question 5 For t 4 hours, the temperature inside a refrigerator in a kitchen is given by the function W that satisfies dw cos t the differential equation =. W( t ) is measured in degrees Celsius dt W At time t = hours, the temperature inside the refrigerator is C. ( C), and t is measured in hours. (a) Write an equation for the line tangent to the graph of y = W( t) at the point where t =. Use the equation to approximate the temperature inside the refrigerator at t =.4 hour. (b) Find y = W( t), the particular solution to the differential equation with initial condition W ( ) =. (c) The temperature in the kitchen remains constant at C for t 4. The cost of operating the refrigerator accumulates at the rate of $. per hour for each degree that the temperature in the kitchen exceeds the temperature inside the refrigerator. Write, but do not evaluate, an expression involving an integral that can be used to find the cost of operating the refrigerator for the 4-hour interval. (a) dw cos = = dt ( t, W) = (, ) ( ) An equation for the tangent line is y = t +. : tangent line equation : : approximation W (. 4 ) (.4 ) + =. C (b) W dw = cos t dt W dw = cos t dt W = sin t + C = sin + C C = 9 W = sin t + 9 Since W( ) =, W = sin t + 9 for t 4. : separation of variables : antiderivatives 5: : constant of integration and uses initial condition : solves for W Note: max 5 [---] if no constant of integration Note: 5 if no separation of variables 4 dollars : : integrand (c). ( W ( t) ) dt : limits and constant 6 The College Board.

7 Let f be the function defined above. (a) Is f continuous at x =? Why or why not? AP CALCULUS AB 6 SCORING GUIDELINES Question 6 x x for x f( x) = + 4e x for x > (b) Find the absolute minimum value and the absolute maximum value of f on the closed interval x. Show the analysis that leads to your conclusion. (c) Find the value of f ( x ). (a) f ( x ) ( x x ) lim = lim = 7 x x x ( ) ( e ) lim f x = lim + 4 = x x Therefore, lim f( x) = 7. x Since lim f( x) = f( ), f is continuous at x =. x (b) For x <, f ( x) =. x f ( x) = x = x For x >, f ( x) = 4e =/. At x =, f ( x) is not defined. : considers one-sided limits : : answer with explanation d x ( x x ) ( e ) d : and + 4 : identifies x = and x = as 4: critical points : evaluates f at endpoints s with analysis x f( x ) 7 + 4e The absolute minimum value of f on the interval x is 7. The absolute maximum value of f on the interval x is + 4. e (c) ( ) ( ) ( 4 x f x = + ) x x + e : sum of integrals : : antiderivatives = x x x + x + 4e x = ( ) + ( e) ( + 4 ) = + 4e 6 The College Board.

DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO.

AP Calculus AB Exam SECTION I: Multiple Choice 016 DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Score 50% Writing