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

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1 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 Instrument Pencil required Part A Number of Questions 8 Time 55 minutes Electronic Device None allowed Part B Number of Questions 17 Time 50 minutes Electronic Device Graphing calculator required Instructions Section I of this exam contains 45 multiple-choice questions and 4 survey questions. For Part A, fill in only the circles for numbers 1 through 8 on page of the answer sheet. For Part B, fill in only the circles for numbers 76 through 9 on page 3 of the answer sheet. The survey questions are numbers 93 through 96. Indicate all of your answers to the multiple-choice questions on the answer sheet. No credit will be given for anything written in this exam booklet, but you may use the booklet for notes or scratch work. After you have decided which of the suggested answers is best, completely fill in the corresponding circle on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely. Here is a sample question and answer. Use your time effectively, working as quickly as you can without losing accuracy. Do not spend too much time on any one question. Go on to other questions and come back to the ones you have not answered if you have time. It is not expected that everyone will know the answers to all of the multiple-choice questions. Your total score on the multiple-choice section is based only on the number of questions answered correctly. Points are not deducted for incorrect answers or unanswered questions. Form I Form Code 4LBP6-S 66

2 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB SECTION I, Part A Time 55 minutes Number of questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. In this exam: (1) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f( x) is a real number. () The inverse of a trigonometric function f may be indicated using the inverse function notation prefix arc (e.g., sin -1 x = arcsin x ). 1 f - or with the -3-

3 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA dy 1. If y = cos x, then = dx (A) - sin x (B) - sin x (C) sin x (D) sin x (E) sin x Ú x x - 1 dx = 3. ( ) 10 (A) 3 4 x Ê x 3 Á Ë 4 10 ˆ - x + C (B) (C) (D) (E) 3 ( x - 1) 11 x 11 + C ( x - 1) ( x - 1) 11 x 33 + C ( x - 1) C + C -4-

4 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 3. 4 lim 9x + 1 xæ 4 x + 3 is (A) 1 3 (B) 3 4 (C) 3 (D) 9 4 (E) infinite x x If ( ) 5 =, y dy then = dx (A) 51+ ( x ) 4 (B) ( x + 1 ) x x (C) ( x + 1) 4 4 5x (D) ( x + 1) 6 (E) 4 ( x ( x + 1) 5x ) -5-

5 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA t (minutes) rt ( ) (gallons per minute) Water is flowing into a tank at the rate rt, ( ) where ( ) rt is measured in gallons per minute and t is measured in minutes. The tank contains 15 gallons of water at time t = 0. Values of rt ( ) for selected values of t are given in the table above. Using a trapezoidal sum with the three intervals indicated by the table, what is the approximation of the number of gallons of water in the tank at time t = 9? (A) 5 (B) 57 (C) 67 (D) 77 (E) The slope of the line tangent to the graph of = ln( 1 - ) y x at x =-1 is (A) - 1 (B) 1 - (C) 1 (D) ln (E) 1-6-

6 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 7. For which of the following pairs of functions f and g is (A) f ( x) = x + x and ( ) gx = x + ln x (B) 3 4 f ( x) = 3x and gx ( ) = x (C) f( x ) = 3 x 3 and gx ( ) = x (D) 3 f ( x) 3e x = + x and gx ( ) = e x + x (E) f ( x) = ln( 3x) and gx ( ) = ln( x) ( ) ( ) f x lim Æ gx x infinite? -7-

7 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 8. Ú 4 x 0 x + 9 (A) - (B) dx = - (C) 1 (D) (E) Let f be the function with derivative given by f ( x) (A) [ 0, ) only (B) (-,0] only = ( 1 -x + x ). On what interval is f decreasing? (C) È - ÍÎ 1 1, 3 3 (D) (-, ) only (E) There is no such interval. -8-

8 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Ú e dx = 10. ( e x + ) (A) x e + C (B) e x + C (C) x e + e + C (D) x 1 e + + ex + C (E) x e + ex + C -9-

9 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 11. The graph of the function f is shown in the figure above. Which of the following could be the graph of f, the derivative of f? (A) (B) (C) (D) (E) -10-

10 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 1. If 0 < c < 1, what is the area of the region enclosed by the graphs of y = 0, (A) ln( 1 c) 1 - (B) ( ) ln c (C) ln c (D) 1 c y =, x (E) c x = c, and x = 1? d ( tan x x) dx 1 1 (A) - + sin x x + = (B) (C) (D) (E) x x x x x x x x -11-

11 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 14. If y = f( x) is a solution to the differential equation following is true? (A) f( x) (B) f( x) (C) ( ) (D) ( ) (E) ( ) = 1 + e = xe 1 x x x t f x = Ú e dt f x e dt = +Ú x t f x e dt 0 = +Ú x t dy dx x = e with the initial condition f ( 0 ) =, which of the -1-

12 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 15. A function f( t) gives the rate of evaporation of water, in liters per hour, from a pond, where t is measured in 10 hours since 1 noon. Which of the following gives the meaning of Ú f () t d t in the context described? 4 (A) The total volume of water, in liters, that evaporated from the pond during the first 10 hours after 1 noon (B) The total volume of water, in liters, that evaporated from the pond between 4 P.M. and 10 P.M. (C) The net change in the rate of evaporation, in liters per hour, from the pond between 4 P.M. and 10 P.M. (D) The average rate of evaporation, in liters per hour, from the pond between 4 P.M. and 10 P.M. (E) The average rate of change in the rate of evaporation, in liters per hour per hour, from the pond between 4 P.M. and 10 P.M. -13-

13 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 16. The first derivative of the function f is given by ( ) 4 of inflection of the graph of f? (A) x = 3 only (B) x = 4 only (C) x = 0 and x = (D) x = 0 and x = 3 (E) x = 0 and x = 4 3 f x = 3x -1x. What are the x-coordinates of the points 1 x 17. Let f be the function defined by f( x) =. What is the average value of f on the interval [ ] (A) 1 - (B) (C) 1 3 ln (D) 3 ln (E) 1 ln 4, 6? -14-

14 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA,0, Ê 1 x, ˆ Á, Ë x and Ê 1 3, ˆ Á Ë x are the vertices of a rectangle, where 3 x, as shown in the figure above. For what value of x does the rectangle have a maximum area? 18. The points ( 3, 0 ), ( x ) (A) 3 (B) 4 (C) 6 (D) 9 (E) There is no such value of x. 19. What are all values of x for which 3 Úx t dt is equal to 0? (A) - only (B) 0 only (C) only (D) - and only (E) -, 0, and -15-

15 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA hx = Ú sin tdt. Which of the following is an equation for the line tangent p 4 p to the graph of h at the point where x =? 4 0. Let h be the function defined by ( ) (A) y = 1 (B) y = x (C) p y = x p (D) y = ( x - 4) p (E) y ( x ) = - 4 x -16-

16 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA x f( x) The table above gives selected values for a twice-differentiable function f. Which of the following must be true? (A) f has no critical points in the interval - 1 < x < 5. (B) f ( x ) = 8 for some value of x in the interval - 1 < x < 5. (C) f ( x ) > 0 for all values of x in the interval - 1 < x < 5. (D) f ( x ) < 0 for all values of x in the interval - 1 < x < 5. (E) The graph of f has no points of inflection in the interval - 1 < x <

17 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA. A particle moves along the x-axis so that at time t 0, the acceleration of the particle is at () = 15 t. The position of the particle is 10 when t = 0, and the position of the particle is 0 when t = 1. What is the velocity of the particle at time t = 0? (A) -14 (B) 0 (C) 5 (D) 6 (E) Which of the following is the solution to the differential equation the point ( 0, 1 )? dy dx = x xy + 1 whose graph contains (A) y = e x (B) y = x + 1 ( ) (C) y = ln x + 1 (D) y = 1+ ln( x + 1) (E) y = 1+ ln( x + 1) -18-

18 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA r 4. Sand is deposited into a pile with a circular base. The volume V of the pile is given by V =, where r is the 3 radius of the base, in feet. The circumference of the base is increasing at a constant rate of 5p feet per hour. When the circumference of the base is 8p feet, what is the rate of change of the volume of the pile, in cubic feet per hour? (A) 8 p (B) 16 (C) 40 (D) 40p (E) 80p 3 5. e lim hæ h -1 - e h is (A) - 1 (B) -1 e (C) 0 (D) 1 e (E) nonexistent -19-

19 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 3 6. Let f be the function given by f ( x) = x + 5x. For what value of x in the closed interval [ ] instantaneous rate of change of f equal the average rate of change of f on that interval? (A) 7 3 (B) 13 3 (C) 5 (D) 6 (E) ,3 does the 7. If xy e - y = e - 4, then at 1 x = and y =, dy dx = (A) e 4 (B) e (C) 4e 8 - e (D) 4e 4 - e (E) e e -0-

20 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 3 8. Let f be the function defined by f ( x) = x + x + x. Let gx ( ) = f -1 ( x ), where g ( 3) of g ( 3 )? = 1. What is the value (A) 1 39 (B) 1 34 (C) 1 6 (D) 1 3 (E) 39 END OF PART A OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO. -1-

21 B B B B B B B B B CALCULUS AB SECTION I, Part B Time 50 minutes Number of questions 17 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO QUESTIONS NUMBERED YOU MAY NOT RETURN TO PAGE OF THE ANSWER SHEET. In this exam: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. () Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f( x) is a real number. (3) The inverse of a trigonometric function f may be indicated using the inverse function notation prefix arc (e.g., sin -1 x = arcsin x ). 1 f - or with the -4-

22 B B B B B B B B B 76. The graph of a function f is shown above. Which of the following limits does not exist? (A) lim f( x - xæ1 ) (B) lim f( x) (C) lim f( x) (D) lim f( x) (E) lim f( x) x Æ1 - xæ3 xæ3 xæ5-5-

23 B B B B B B B B B 77. Let f be a function that is continuous on the closed interval [ ] following statements must be true? (A) 10 f ( ) 18 (B) f is increasing on the interval [ 1, 3 ]. (C) f( x ) = 17 has at least one solution in the interval [ 1, 3 ]. (D) ( ) = 8 f x has at least one solution in the interval (1, 3). 3 (E) ( ) Ú f x dx > 0 1 1, 3 with f () 1 = 10 and f () 3 = 18. Which of the -6-

24 B B B B B B B B B 78. Let R be the region bounded by the graphs of y e, y = e and x = 0. Which of the following gives the volume of the solid formed by revolving R about the line y =-1? 3 3 x (A) pú ( - +1) 0 e e dx 3 3 x (B) p ( e - e -1) dx Ú x (C) Ú ( ) p È e - e + 1 dx 0 ÍÎ 3 p È e - e -1 dx 0 ÍÎ 3 x (D) Ú ( ) 3 3 (E) p È x Ú ( + 1) - ( + 1) 0 e e ÍÎ dx = x 3, 79. The number of people who have entered a museum on a certain day is modeled by a function f( t ), where t is measured in hours since the museum opened that day. The number of people who have left the museum since it opened that same day is modeled by a function ( ) f t = t and ( t 4) gt. If ( ) ( ) Ê p - g ( t) = sinÁ, at what time t, for 1 t 11, is the number of people in the Ë 1 ˆ museum at a maximum? (A) 1 (B) (C) (D) (E) 11-7-

25 B B B B B B B B B x f( x ) ( x) f The derivative of the function f is continuous on the closed interval [ 0, 4 ]. Values of f and f for selected values of x are given in the table above. If Ú f ( t) dt = 8, then ( 4) (A) 0 (B) 3 (C) 5 (D) 10 (E) f = -8-

26 B B B B B B B B B 81. A slope field for a differential equation is shown in the figure above. If y f( x) differential equation through the point (- 1, ) and hx ( ) = 3 x f( x), then ( 1) (A) - 6 (B) - (C) 0 (D) 1 (E) 1 = is the particular solution to the h - = 8. If f is a continuous function such that f ( ) (A) f( x) lim = 3 xæ1 (B) f( x) (C) lim = 1 xæ xæ ( ) - f( ) f x lim = 6 x - (D) f( x ) lim = 36 xæ (E) ( f( x) ) lim = 36 xæ = 6, which of the following statements must be true? -9-

27 B B B B B B B B B t A particle moves along a straight line with velocity given by vt ( ) 5 e = + for time t 0. What is the acceleration of the particle at time t = 4? (A) 0.4 (B) (C) 1.65 (D) (E) t 84. A home uses fuel oil at the rate rt = sin gallons per day, where t is the number of days from 60 the beginning of the heating season. To the nearest gallon, what is the total amount of fuel oil used from t = 0 to t = 60 days? () ( ) (A) 7 gal (B) 14 gal (C) 600 gal (D) 81 gal (E) 1004 gal -30-

28 B B B B B B B B B 85. The function f is defined on the open interval 0.4 < x <.4 and has first derivative f given by ( ) = sin( f x x ). Which of the following statements are true? I. f has a relative maximum on the interval 0.4 < x <.4. II. f has a relative minimum on the interval 0.4 < x <.4. III. The graph of f has two points of inflection on the interval 0.4 < x <.4. (A) I only (B) II only (C) III only (D) I and III only (E) II and III only -31-

29 B B B B B B B B B 86. The graph of the function f, which has a domain of [ 0, 7 ], is shown in the figure above. The graph consists of a quarter circle of radius 3 and a segment with slope 1 What is the value of b? (A) (B) (C) (D) (E) There is no such value of b. -. Let b be a positive number such that ( ) b Ú f x dx =

30 B B B B B B B B B 87. The first derivative of the function g is given by ( ) = cos( p ) following intervals is g decreasing? (A) < x < 0 (B) 0 < x < 1 (C) < x < 1.5 (D) 1.5 < x < (E) < x < 1.5 g x x for < x < On which of the -33-

31 B B B B B B B B B 88. The height above the ground of a passenger on a Ferris wheel t minutes after the ride begins is modeled by the Ht is measured in meters. Which of the following is an interpretation of the differentiable function H, where ( ) statement H ( 7.5) = ? (A) The Ferris wheel is turning at a rate of meters per minute when the passenger is 7.5 meters above the ground. (B) The Ferris wheel is turning at a rate of meters per minute 7.5 minutes after the ride begins. (C) The passenger s height above the ground is increasing by meters per minute when the passenger is 7.5 meters above the ground. (D) The passenger s height above the ground is increasing by meters per minute 7.5 minutes after the ride begins. (E) The passenger is meters above the ground 7.5 minutes after the ride begins. 89. A particle moves along a straight line for 6 seconds so that its velocity, in centimeters per second, is modeled by the graph shown. During the time interval 0 t 6, what is the total distance the particle travels? (A) cm (B) 3.5 cm (C) 4 cm (D) 6.5 cm (E) 8.5 cm -34-

32 B B B B B B B B B 90. Let f be a twice-differentiable function on the open interval (, ( ab),, which of the following could be the graph of f? ab). If ( ) > 0 f x on (, ab) and ( ) < 0 f x on (A) (B) (C) (D) (E) -35-

33 B B B B B B B B B 91. The graphs of f and g are shown above. If hx ( ) = f( xg ) ( x ), then ( 6) (A) - 9 (B) - 7 (C) 1 (D) 7 (E) 9 h = 9. In the xy-plane, the graph of the twice-differentiable function y = f( x) is concave up on the open interval (0, ) and is tangent to the line y = 3x - at x = 1. Which of the following statements must be true about the derivative of f? (A) f ( x ) 3 on the interval ( ) 0.9, 1. (B) f ( x ) 3 on the interval ( ) (C) ( ) < 0 0.9, 1. f x on the interval (0.9, 1.1). (D) ( ) > 0 f x on the interval (0.9, 1.1). (E) f ( x) is constant on the interval (0.9, 1.1). -36-

34 B B B B B B B B B END OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART B ONLY. DO NOT GO ON TO SECTION II UNTIL YOU ARE TOLD TO DO SO. MAKE SURE YOU HAVE DONE THE FOLLOWING. PLACED YOUR AP NUMBER LABEL ON YOUR ANSWER SHEET WRITTEN AND GRIDDED YOUR AP NUMBER CORRECTLY ON YOUR ANSWER SHEET TAKEN THE AP EXAM LABEL FROM THE FRONT OF THIS BOOKLET AND PLACED IT ON YOUR ANSWER SHEET AFTER TIME HAS BEEN CALLED, TURN TO PAGE 38 AND ANSWER QUESTIONS

35 Section II: Free-Response Questions This is the free-response section of the 016 AP exam. It includes cover material and other administrative instructions to help familiarize students with the mechanics of the exam. (Note that future exams may differ in look from the following content.)

36 AP Calculus AB Exam SECTION II: Free Response 016 DO NOT OPEN THIS BOOKLET OR BREAK THE SEALS ON PART B UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour, 30 minutes Number of Questions 6 Percent of Total Score 50% Writing Instrument Either pencil or pen with black or dark blue ink Weight The questions are weighted equally, but the parts of a question are not necessarily given equal weight. Part A Number of Questions Time 30 minutes Electronic Device Graphing calculator required Percent of Section II Score 33.3% Part B Number of Questions 4 Time 60 minutes Electronic Device None allowed Percent of Section II Score 66.6% Instructions The questions for Section II are printed in this booklet. Do not break the seals on Part B until you are told to do so. Write your solution to each part of each question in the space provided. Write clearly and legibly. Cross out any errors you make; erased or crossed-out work will not be scored. Manage your time carefully. During the timed portion for Part A, work only on the questions in Part A. You are permitted to use your calculator to solve an equation, find the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your question, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results. During the timed portion for Part B, you may continue to work on the questions in Part A without the use of a calculator. For each part of Section II, you may wish to look over the questions before starting to work on them. It is not expected that everyone will be able to complete all parts of all questions. Show all of your work. Clearly label any functions, graphs, tables, or other objects that you use. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit. Justifications require that you give mathematical (noncalculator) reasons. Your work must be expressed in standard mathematical notation rather than calculator 5 syntax. For example, x dx may not be written as fnint(x, X, 1, 5). 1 Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If you use decimal approximations in calculations, your work will be scored on accuracy. Unless otherwise specified, your final answers should be accurate to three places after the decimal point. Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f x is a real number. Form I Form Code 4LBP6-S 66

37 CALCULUS AB SECTION II, Part A Time 30 minutes Number of problems A graphing calculator is required for these problems. -3-

38 A company produces and sells chili powder. The company s weekly profit on the sale of x kilograms of chili Px = 48x + 1.4x x - 7 powder is modeled by the function P given by ( ) and 0 x , where ( ) Px is in dollars (a) Find the rate, in dollars per kilogram, at which the company s weekly profit is changing when it sells 3 kilograms of chili powder. Is the company s weekly profit increasing or decreasing when it sells 3 kilograms of chili powder? Give a reason for your answer. Do not write beyond this border. (b) How many kilograms of chili powder must the company sell to maximize its weekly profit? Justify your answer. Do not write beyond this border. -4- Continue problem 1 on page 5.

39 (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 p St () = + cos( t 10 ) 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. Do not write beyond this border. (d) Using the function S from part (c), find the value of S (3). Interpret the meaning of this value in the context of the problem. Do not write beyond this border. -5-

40 t (weeks) Gt ( ) (games per week) A store tracks the sales of one of its popular board games over a 1-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 0 t Gt are given in the table above for selected values of t. 1. Values of ( ) (a) Approximate the value of G ( 8) using the data in the table. Show the computations that lead to your answer. Do not write beyond this border. 1 (b) Approximate the value of ( ) table. Explain the meaning of ( ) Ú Gt d t using a right Riemann sum with the four subintervals indicated by the 0 Ú 1 Gt d t in the context of this problem. 0 Do not write beyond this border. -6- Continue problem on page 7.

41 (c) One salesperson believes that, starting with 400 games per week at time t = 1, the rate at which games will be sold will increase at a constant rate of 100 games per week per week. Based on this model, how many total games will be sold in the 8 weeks between time t = 1 and t = 0? Do not write beyond this border. (d) Another salesperson believes the best model for the rate at which games will be sold in the 8 weeks between 0.01 t 1 time t = 1 and = 0 Mt = 400e - - games per week. Based on this model, how many total games, to the nearest whole number, will be sold during this period? t is ( ) ( ) Do not write beyond this border. -7-

42 END OF PART A OF SECTION II IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO. -8-

43 CALCULUS AB SECTION II, Part B Time 60 minutes Number of problems 4 No calculator is allowed for these problems. DO NOT BREAK THE SEALS UNTIL YOU ARE TOLD TO DO SO. -13-

44 NO CALCULATOR ALLOWED Do not write beyond this border. 3. The function f is defined on the interval -5 x c, where 0 c > and f( c ) = 0. The graph of f, which consists of three line segments and a quarter of a circle with center (- 3, 0) and radius, is shown in the figure above. (a) Find the average rate of change of f over the interval [- 5, 0]. Show the computations that lead to your answer. Do not write beyond this border. (b) For -5 x c, let g be the function defined by gx ( ) = Ú ft ( ) dt. Find the x-coordinate of each point of inflection of the graph of g. Justify your answer. x Continue problem 3 on page 15.

45 NO CALCULATOR ALLOWED (c) Find the value of c for which the average value of f over the interval -5 x c is 1. Do not write beyond this border. (d) Assume 3 c >. The function h is defined by ( ) ( x hx = f ). Find h (6) in terms of c. Do not write beyond this border. -15-

46 NO CALCULATOR ALLOWED Do not write beyond this border. 4. Let S be the shaded region in the first quadrant bounded above by the horizontal line y = 3, below by the graph of p y = 3sin x, and on the left by the vertical line x = k, where 0 < k <, as shown in the figure above. p (a) Find the area of S when k =. 3 Do not write beyond this border Continue problem 4 on page 17.

47 NO CALCULATOR ALLOWED (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 p =. 6 Do not write beyond this border. (c) Region S is revolved about the horizontal line y = 5 to form a solid. Write, but do not evaluate, p an expression involving one or more integrals that gives the volume of the solid when k =. 4 Do not write beyond this border. -17-

48 NO CALCULATOR ALLOWED 5. For 0 t 4 hours, the temperature inside a refrigerator in a kitchen is given by the function W that satisfies dw the differential equation = 3cos t dt W. Wt ( ) is measured in degrees Celsius ( C ), and t is measured in hours. At time t = 0 hours, the temperature inside the refrigerator is 3 C. (a) Write an equation for the line tangent to the graph of = ( ) y Wt at the point where t = 0. Use the equation to approximate the temperature inside the refrigerator at t = 0.4 hour. Do not write beyond this border. (b) Find y = Wt, ( ) the particular solution to the differential equation with initial condition W(0) = 3. Do not write beyond this border Continue problem 5 on page 19.

49 NO CALCULATOR ALLOWED (c) The temperature in the kitchen remains constant at 0 C for 0 t 4. The cost of operating the refrigerator accumulates at the rate of $0.001 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. Do not write beyond this border. Do not write beyond this border. -19-

50 NO CALCULATOR ALLOWED 6. Let f be the function defined above. ( ) f x (a) Is f continuous at x = 1? Why or why not? 10 - x - x for x 1 = e x- for x > 1 Do not write beyond this border. (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. Do not write beyond this border. -0- Continue problem 6 on page 1.

51 NO CALCULATOR ALLOWED (c) Find the value of Ú f ( x ) 0 dx. Do not write beyond this border. Do not write beyond this border. -1-

52 STOP END OF EXAM THE FOLLOWING INSTRUCTIONS APPLY TO THE COVERS OF THE SECTION II BOOKLET. MAKE SURE YOU HAVE COMPLETED THE IDENTIFICATION INFORMATION AS REQUESTED ON THE FRONT AND BACK COVERS OF THE SECTION II BOOKLET. CHECK TO SEE THAT YOUR AP NUMBER LABEL APPEARS IN THE BOX ON THE COVER. MAKE SURE YOU HAVE USED THE SAME SET OF AP NUMBER LABELS ON ALL AP EXAMS YOU HAVE TAKEN THIS YEAR. --

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