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1 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB SECTION I, Part A Time 60 minutes Number of questions 0 NO CALCULATOR IS ALLOWED FOR THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After eamining the form of the choices, decide which is the best of the choices given and place the letter of your choice in the corresponding bo on the answer sheet. No credit will be given for anything written in this eam booklet. Do not spend too much time on any one problem. In this eam: () Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f()is a real number. () The inverse of a trigonometric function f may be indicated using the inverse function notation f prefi arc (e.g., sin = arcsin ). or with the GO ON TO THE NEXT PAGE. Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

2 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA. ( ) = (B) 4 (C) 8 (D). If f() = e ( ), then f ( ) = 6e 4 (B) e 4 (C) 4e 4 (D) 0e 4 GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam

3 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA. Let f be a differentiable function such that f ( ) = 4 and f ( ) =. What is the approimation for f (.) found by using the line tangent to the graph of f at =?.95 (B).95 (C) 4.05 (D) Let g be the function defined by g () = 4. How many relative etrema does g have? Zero (B) One (C) Two (D) Three GO ON TO THE NEXT PAGE. 4 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

4 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 5. The velocity of a particle moving along the -ais is given by vt () = t for time t 0. What is the average velocity of the particle from time t = to time t =? 4 (B) (C) 7 (D) 7 6. On a certain day, the rate at which material is deposited at a recycling center is modeled by the function R, where Rt ()is measured in tons per hour and t is the number of hours since the center opened. Using a trapezoidal sum with the three subintervals indicated by the data in the table, what is the approimate number of tons of material deposited in the first 9 hours since the center opened? 68 (B) 70.5 (C) 85 (D) 6 GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 5

5 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 7. What is the total area of the regions between the curves y = 6 8 and y = 6 from = to =? 4 (B) (C) 6 (D) 0 8. The function g is defined by g () = b, where b is a constant. If the line tangent to the graph of g at = is parallel to the line that contains the points ( 0, ) and (, 4), what is the value of b? (B) (C) 5 (D) 4 GO ON TO THE NEXT PAGE. 6 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

6 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA f() = for 0 0 for = 0 9. The function f is defined above. The value of f() is 5 (B) (C) 8 (D) noneistent 0. Let g be a continuous function. Using the substitution u =, the integral g ( ) is equal to which of the following? 5 5 gu ( ) du (B) gu ( ) du (C) gu ( ) du (D) gu ( ) du GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 7

7 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA. The graph of y = f()consists of a semicircle with endpoints at (, 6) and (, 6), as shown in the figure above. What is the value of f ()? 5p (B) 5p 5p (C) 60 (D) 60 5p GO ON TO THE NEXT PAGE. 8 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

8 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA. An object moves along a straight line so that at any time t its acceleration is given by at () = 6t. At time t = 0, the object s velocity is 0 and the object s position is 7. What is the object s position at time t =? (B) 7 (C) 8 (D) 5. If y = cos ln( ), then dy sin (B) (C) (D) sin sin sin = GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 9

9 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 4. The graph of the function f, shown above, consists of three line segments. If the function g is an antiderivative of f such that g( ) = 5, for how many values of c, where 0 c 6, does gc () =? Zero (B) One (C) Two (D) Three 6 c for f() = 9 ln for 5. Let f be the function defined above, where c is a constant. If f is continuous at =, what is the value of c? (B) (C) 5 (D) 9 GO ON TO THE NEXT PAGE. 0 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

10 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 6. Which of the following could be a slope field for the differential equation dy = y? (B) (C) (D) GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam

11 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 7. The function y = e 5 7 is a solution to which of the following differential equations? y y 5 = 0 (B) y y 5 = 0 (C) y y 5 = 0 (D) y y 5 = 0 8. If f() = sin, then f = p 6 (B) p (C) 4 7 (D) GO ON TO THE NEXT PAGE. Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

12 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 9. lim e 0 0 ( ) ( e e) e is 9 0 (B) 0e (C) e 0 e (D) noneistent 0. Let y = f()be a twice-differentiable function such that f ( ) = and dy at =? = y. What is the value of dy (B) 66 (C) (D) 65 GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam

13 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA. The table above gives values of f, f, g, and g for selected values of. If h ( ) = f( g ( )), what is the value of h( )? 9 (B) 4 (C) 7 (D) 9. Let y = f()be the particular solution to the differential equation dy f ( 0) =. Which of the following is an epression for f()? (B) (C) 4 (D) 4 = y with the initial condition GO ON TO THE NEXT PAGE. 4 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

14 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA. Let R be the shaded region bounded by the graph of y =, the graph of y =, and the -ais, as shown in the figure above. Which of the following gives the volume of the solid generated when R is revolved about the -ais? 4 p ( ( ) ) 0 (B) p ( ( )) (C) p p ( ( ) ) 0 (D) p p 0 4 ( ( )) GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 5

15 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 4. lim tan ( ) is e 0 (B) (C) (D) noneistent 5. Let f be a function with first derivative defined by f () = 6 for 0. It is known that f ( ) = 9 and f ( ) =. What value of in the open interval (, ) satisfies the conclusion of the Mean Value Theorem for f on the closed interval,? 6 (B) (C) (D) GO ON TO THE NEXT PAGE. 6 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

16 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 6. 5 = ln 4 (B) 5 (C) 5 ln 4 (D) A hemispherical water tank, shown above, has a radius of 6 meters and is losing water. The area of the surface of the water is A = ph ph square meters, where h is the depth, in meters, of the water in the tank. When h = meters, the depth of the water is decreasing at a rate of meter per minute. At that instant, what is the rate at which the area of the water s surface is decreasing with respect to time? (B) (C) (D) p square meters per minute 6p square meters per minute 9p square meters per minute 7p square meters per minute GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 7

17 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 8. Consider a triangle in the y-plane. Two vertices of the triangle are on the -ais at (, 0) and ( 5, 0), and a third verte is on the graph of y = ln( ) 5 for 8. What is the maimum area of such a triangle? 9 (B) ln 9 (C) ln4 8 (D) ln6 9. The function f is defined by f() = 4. If g is the inverse function of f and g( ) = 0, what is the value of g( )? 6 (B) 4 8 (C) 4 (D) 4 GO ON TO THE NEXT PAGE. 8 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

18 Calculus AB Practice Eam 9 0. Which of the following limits is equal to 5? k n n lim n k n = (B) k n n lim n k n = (C) k n n lim n k n = (D) k n n lim n k n = END OF PART A 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. AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Monday, May, 07 at 9:9: PM Central Daylight Time

19 B B B B B B B B B CALCULUS AB SECTION I, Part B Time 45 minutes Number of questions 5 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 eamining the form of the choices, decide which is the best of the choices given and place the letter of your choice in the corresponding bo on the answer sheet. No credit will be given for anything written in this eam booklet. Do not spend too much time on any one problem. In this eam: () The eact 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 approimates the eact numerical value. () Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f()is a real number. () The inverse of a trigonometric function f may be indicated using the inverse function notation f prefi arc (e.g., sin = arcsin ). or with the 0 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time GO ON TO THE NEXT PAGE.

20 B B B B B B B B B 76. To help restore a beach, sand is being added to the beach at a rate of st () = 65 4 sin( 0.t) tons per hour, where t is measured in hours since 5:00 A.M. How many tons of sand are added to the beach over the -hour period from 7:00 A.M. to 0:00 A.M.? (B) 5.7 (C) 85. (D) The graph of the function f is shown above. For what values of a does lim f() = 0? only (B) and 4 (C) 0 and only (D) 0,, and a GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam

21 B B B B B B B B B 78. The second derivative of a function f is given by f () = sin( ) cos( ). How many points of inflection does the graph of f have on the interval 0? One (B) Three (C) Four (D) Five 79. Over the time interval 0 t 5, a particle moves along the -ais. The graph of the particle s velocity, v, is shown above. Over the time interval 0 t 5, the particle s displacement is and the particle travels a 4 total distance of. What is the value of vt () dt? 0 (B) 5 (C) 5 (D) 0 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time GO ON TO THE NEXT PAGE.

22 B B B B B B B B B 80. The temperature in a room at midnight is 0 degrees Celsius. Over the net 4 hours, the temperature changes at a rate modeled by the differentiable function H, where Ht ()is measured in degrees Celsius per hour and time t is measured in hours since midnight. Which of the following is the best interpretation of The temperature of the room, in degrees Celsius, at 6:00 A.M. 6 0 Ht () dt? (B) (C) (D) The average temperature of the room, in degrees Celsius, between midnight and 6:00 A.M. The change in the temperature of the room, in degrees Celsius, between midnight and 6:00 A.M. The rate at which the temperature in the room is changing, in degrees Celsius per hour, at 6:00 A.M. GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam

23 B B B B B B B B B 8. The graph of f, the derivative of the function f, is shown above. Which of the following could be the graph of f? (B) (C) (D) GO ON TO THE NEXT PAGE. 4 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

24 B B B B B B B B B 8. Let f be the function with derivative given by f () = sin( ). At what values of in the interval does f have a relative maimum?.7 (B).478 and.478 only and.7 only (C).8, 0, and.8 (D).478,.7,.7, and The graph of the function f is shown above. At what value of does f have a jump discontinuity? (B) (C) 7 (D) 0 GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 5

25 B B B B B B B B B 84. Let f be a differentiable function such that f ( ) = p and f () = 6. What is the value of f ( 5)?.94 (B) (C) 4.67 (D) People are entering a building at a rate modeled by f()people t per hour and eiting the building at a rate modeled by gt ()people per hour, where t is measured in hours. The functions f and g are nonnegative and differentiable for all times t. Which of the following inequalities indicates that the rate of change of the number of people in the building is increasing at time t? f() t 0 (B) f () t 0 (C) f() t g() t 0 (D) f () t g() t 0 6 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time GO ON TO THE NEXT PAGE.

26 B B B B B B B B B 86. The velocity of a particle moving along the -ais is given by vt () = t cos( e)for t 0. Which of the following statements describes the motion of the particle at t =? t (B) (C) (D) The particle is moving to the left with positive acceleration. The particle is moving to the right with positive acceleration. The particle is moving to the left with negative acceleration. The particle is moving to the right with negative acceleration. 87. A tire that is leaking air has an initial air pressure of 0 pounds per square inch (psi). The function t = f( p) models the amount of time t, in hours, it takes for the air pressure of the tire to reach p psi. What are the units for f ( p)? hours (B) psi (C) psi per hour (D) hours per psi GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 7

27 B B B B B B B B B 88. The first derivative of the function f is defined by f () = only (B) 0.6 and 0.9 only (C) (D) There are no intervals on which f is increasing. e. On what intervals is f increasing? The table above shows selected values of a continuous function f. For 0, what is the fewest possible number of times f() = 4? One (B) Two (C) Three (D) Four 8 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time GO ON TO THE NEXT PAGE.

28 B B B B B B B B B 90. The function h is defined on the closed interval,. The graph of h, the derivative of h, is shown above. The graph consists of two semicircles with a common endpoint at =. Which of the following statements about h must be true? I. h( ) = h( ) II. h is continuous at =. III. The graph of h has a vertical asymptote at =. None (B) II only (C) I and II only (D) I and III only 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. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 9

29 AP Calculus AB Eam SECTION II: Free Response DO NOT OPEN THIS BOOKLET OR BEGIN PART B UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time hour, 0 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 0 minutes Electronic Device Graphing calculator required Percent of Section II Score.% Part B Number of Questions 4 Time 60 minutes Electronic Device None allowed Percent of Section II Score 66.6% Instructions ,.,,.,,,,. -,.,.,..,.,,,.,,,,...., 5 (,,, )., ( ).,.,., f f(). 40 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

30 CALCULUS AB SECTION II, Part A Time 0 minutes Number of problems A GRAPHING CALCULATOR IS REQUIRED FOR THESE PROBLEMS. GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 4

31 . The depth of water in tank A, in inches, is modeled by a differentiable and increasing function h for 0 t 0, where t is measured in minutes. Values of ht for selected values of t are given in the table above. (a) Use the data in the table to find an approimation for h6. Show the computations that lead to your answer. Indicate units of measure. (b) 0 Approimate the value of ht dt using a right Riemann sum with the four subintervals indicated by 0 0 the data in the table. Is this approimation greater than or less than ht dt? Give a reason for your 0 answer. 4 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

32 (c) The depth of water in tank B, in inches, is modeled by the function gt =. 7.5 sin0.6tfor 0 t 0, where t is measured in minutes. Find the average depth of the water in tank B over the interval 0 t 0. Is this value greater than or less than the average depth of the water in tank A over the interval 0 t 0? Give a reason for your answer. (d) According to the model given in part (c), is the depth of the water in tank B increasing or decreasing at time t = 6? Give a reason for your answer. GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 4

33 . Particle Q moves along the -ais so that its velocity at any time t is given by = t vq t cos, and its 5 6t t acceleration at any time t is given by aq= t 5 sin. The particle is at position = at time t 0 5 =. (a) In the interval 0 t 5, when is the velocity of particle Q increasing? Give a reason for your answer. (b) Find the position of particle Q at time t =. 44 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

34 (c) A second particle, R, moves along the -ais so that its position at any time t is given by a differentiable function R t, where R = 4 and R = 8. Eplain why there must be a time t, for t, at which the velocity of particle R is. (d) At time t =, the velocity of particle R described in part (c) is. Are particles Q and R moving toward each other or away from each other at time t =? Eplain your reasoning. GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 45

35 END OF PART A 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. 46 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

36 CALCULUS AB SECTION II, Part B Time 60 minutes Number of problems 4 NO CALCULATOR IS ALLOWED FOR THESE PROBLEMS. DO NOT BEGIN PART B UNTIL YOU ARE TOLD TO DO SO. GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 47

37 NO CALCULATOR ALLOWED. A continuous function g is defined on the closed interval 8 6. The graph of g, shown above, consists of three line segments and a quarter of a circle centered at the point 0,. Let f be the function given by f= g t dt. 8 (a) Find all values of in the interval 8 6 at which f has a critical point. Classify each critical point as the location of a local minimum, a local maimum, or neither. Justify your answers. 48 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

38 NO CALCULATOR ALLOWED (b) Find f 0. (c) Find lim 4 f. 4 (d) Let h be the function defined by h = g. Find h. Monday, May, 07 at 9:9: PM Central Daylight Time GO ON TO THE NEXT PAGE. Calculus AB Practice Eam 49

39 NO CALCULATOR ALLOWED 4. Consider the differential equation dy = y. (a) Find y = g, the particular solution to the given differential equation with initial condition g0 = 5. (b) For the particular solution y = g found in part (a), find lim g. 50 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

40 NO CALCULATOR ALLOWED (c) Let y = f be the particular solution to the given differential equation with initial condition f =. Find the value of dy at the point,. Is the graph of y = f concave up or concave down at the point,? Give a reason for your answer. GO ON TO THE NEXT PAGE. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 5

41 NO CALCULATOR ALLOWED 5. The function f is defined by (a) Is f continuous at = 0? Justify your answer. (b) Find f and f. = for 0 f e for 0. 5 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

42 NO CALCULATOR ALLOWED (c) Eplain why f 0 does not eist. (d) Let g be the function given by g = f t dt. Find g. Monday, May, 07 at 9:9: PM Central Daylight Time GO ON TO THE NEXT PAGE. Calculus AB Practice Eam 5

43 NO CALCULATOR ALLOWED 6. A hive contains 5 hundred bees at time t = 0. During the time interval 0 t 4 hours, bees enter the hive at a rate modeled by Et = 6t t, where Et is measured in hundreds of bees per hour. During the same time interval, bees leave the hive at a rate modeled by Lt = t5, where Lt is measured in hundreds of bees per hour. (a) How many bees leave the hive during the time interval 0 t? (b) Write an epression involving one or more integrals for the total number of bees, in hundreds, in the hive at time t for 0 t 4. Find the total number of bees in the hive at t = Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

44 NO CALCULATOR ALLOWED (c) Find the minimum number of bees in the hive for 0 t 4. Justify your answer. Monday, May, 07 at 9:9: PM Central Daylight Time GO ON TO THE NEXT PAGE. Calculus AB Practice Eam 55

45 STOP END OF EXAM 56 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

46 This section describes how the questions in the AP Practice Eam correspond to the components of the curriculum framework included in the AP Calculus AB and AP Calculus BC Course and Eam Description. For each of the questions in the AP Practice Eam, the targeted learning objectives, essential knowledge statements, and Mathematical Practices for AP Calculus from the curriculum framework are indicated. Eam questions assess the learning objectives detailed in the curriculum framework; as such, they require a strong conceptual understanding of calculus in conjunction with the application of one or more of the Mathematical Practices for AP Calculus (MPACs). Although topics in subject areas such as algebra, geometry, and precalculus are not eplicitly assessed, students must have mastered the relevant preparatory material in order to apply calculus techniques successfully and accurately. Students take either the AP Calculus AB Eam or the AP Calculus BC Eam. The eams, which are identical in format, consist of a multiple-choice section and a free-response section, as shown in the tables on the following page. In this publication, the multiple-choice and free-response questions include the following features: For multiple-choice questions, the correct response is indicated with a rationale for why it is correct. There are also eplanations for the incorrect responses. Note that in the cases where multiple learning objectives, essential knowledge statements, or MPACs are provided, the primary one is listed first. Free-response questions include scoring guidelines that eplain how students can use required knowledge learned in the AP Calculus course to answer the questions. Student performance on these two sections will be compiled and weighted to determine an AP Eam score. Each section of the eam counts toward 50 percent of the student s score. Points are not deducted for incorrect answers or unanswered questions. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 57

47 Section I: Multiple Choice Part Graphing Calculator Number of Questions Time Part A Not permitted 0 60 minutes Part B Required 5 45 minutes TOTAL 45 hour, 45 minutes Percentage of Total Eam Score 50% Section II: Free Response Part Graphing Calculator Number of Questions Time Part A Required 0 minutes Part B Not permitted 4 60 minutes TOTAL 6 hour, 0 minutes Percentage of Total Eam Score 50% Calculator Use on the Eams Both the multiple-choice and free-response sections of the AP Calculus Eams include problems that require the use of a graphing calculator. A graphing calculator appropriate for use on the eams is epected to have the built-in capability to do the following: Plot the graph of a function within an arbitrary viewing window Find the zeros of functions (solve equations numerically) Numerically calculate the derivative of a function Numerically calculate the value of a definite integral One or more of these capabilities should provide sufficient computational tools for successful development of a solution to any AP Calculus AB or Calculus BC Eam question that requires the use of a calculator. Care is taken to ensure that the eam questions do not favor students who use graphing calculators with more etensive built-in features. Students are epected to bring a graphing calculator with the capabilities listed above to the eams. AP teachers should check their own students calculators to ensure that the required conditions are met. Students and teachers should keep their calculators updated with the latest available operating systems. Information is available on calculator company websites. A list of acceptable calculators can be found at the AP student website ( apstudent.collegeboard.org/apcourse/ap-calculus-ab/calculator-policy). Note that requirements regarding calculator use help ensure that all students have sufficient computational tools for the AP Calculus Eams. Eam restrictions should not be interpreted as restrictions on classroom activities. 58 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

48 The multiple-choice section on each eam is designed for broad coverage of the course content. Multiple-choice questions are discrete, as opposed to appearing in question sets, and the questions do not appear in the order in which topics are addressed in the curriculum framework. There are 0 multiple-choice questions in Part A and 5 multiple-choice questions in Part B; students may use a graphing calculator only for Part B. Each part of the multiple-choice section is timed and students may not return to questions in Part A of the multiple-choice section once they have begun Part B. Curriculum Framework Alignment and Rationales for Answers Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.B(b) Evaluate definite integrals. (B) (C).B: If f is continuous on the interval [ a, b] and F is an antiderivative of f, then b f( ) = Fb ( ) F( a). a MPAC : Implementing algebraic/ computational processes MPAC 5: Building notational fluency This option is incorrect. Lack of parentheses is a common mistake, resulting in sign errors, when the antiderivative consists of more than one term: 7 = 9 0 ( ) = + =. This answer might also be obtained if the student uses parentheses correctly but makes an algebraic mistake by taking ( ) = to get: 7 = 9 0 ( ) ( ( ) ) ( = + ) =. This option is correct. This question involves using the basic power rule for antidifferentiation and then correctly substituting the endpoints and evaluating: 7 = 9 ( ) 4 ( ) =. This option is incorrect. If the process of antidifferentiation and differentiation are confused in evaluating the definite integral, the result is:. = 8 Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 59

49 (D) This option is incorrect. A possible error in applying the power rule for antiderivatives is to not divide by the new eponent. This error would give: =. Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.C Calculate derivatives..c: Sums, differences, products, and quotients of functions can be differentiated using derivative rules. (B) (C) MPAC : Implementing algebraic/ computational processes MPAC 5: Building notational fluency This option is incorrect. Subtraction was used in the product rule instead of addition. f ( ) = e + e ( ) ( ) f ( ) = 8e e = 6e This option is incorrect. The correct product rule was used but the chain rule is ignored in the derivative of e, thereby a factor of was lost. f ( ) = e + e ( ) + ( ) f ( ) = 9e + e = e This option is incorrect. The chain rule was correctly used but the product rule is incorrectly stated as being d ( f( ) g( ) ) = f ( ) g ( ). ( ) = ( ) 4 ( ) = ( ) = f e 4 f e 4e (D) This option is correct. Use a combination of the product rule and the chain rule to compute the derivative, then evaluate that derivative at =. ( ) + ( ) f ( ) = e + e f ( ) = 8e + e = 0e Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

50 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.B Solve problems involving the slope of a tangent line..b: The tangent line is the graph of a locally linear approimation of the function near the point of tangency. MPAC : Connecting concepts MPAC : Implementing algebraic/ computational processes This option is incorrect. An incorrect equation was used, namely y = f( a) + f ( a), in constructing the equation of a line given a slope and a point on the line. f (. ) f ( ) (. ) = = 95. (B) This option is correct. An equation of the tangent line at = a is y = f( a) + f ( a) ( a). In this question a = and f ( a) =. The value of y when =. would be an approimation to f (. ). f (. ) f ( ) (. ) = 4 ( 0. ) = = 95. (C) This option is incorrect. An error was made in constructing the equation of a line given a slope and a point on the line, using y = f( a) f ( a) ( a) for the equation of the tangent line. f (. ) f ( ) + (. ) = 4 + ( 0. ) = 405. The same answer could also be obtained with the correct tangent line equation by making a multiplication error or sign error in substituting the derivative value or performing the calculation. (D) This option is incorrect. This is a fundamental conceptual error in not recognizing the derivative as the slope of the tangent line. f(. ) f( ) + (. ) = = 4. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 6

51 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.A Use derivatives to analyze properties of a function. (B) (C) (D).A: First and second derivatives of a function can provide information about the function and its graph including intervals of increase or decrease, local (relative) and global (absolute) etrema, intervals of upward or downward concavity, and points of inflection. MPAC : Implementing algebraic/ computational processes MPAC : Connecting concepts This option is incorrect. The error comes from believing that g is always increasing because the derivative is the sum of two terms and is thus always positive. This option is correct. First find the critical values where the derivative is 0 or undefined. A relative etrema occurs at a critical value if the derivative changes sign there. g ( ) = 4 + = ( 4 + ) ( 4 + ) = 0 at = 0 or = There are two zeros, but g ( ) changes sign for = only. The sign of g ( ) at = 0 does not change because of the factor. This option is incorrect. The number of critical values has been confused with the number of relative etrema. There are two zeros for g ( ) = 0, but the derivative only changes sign at one of them. This option is incorrect. The error is the result of assuming that a quartic polynomial always has relative etrema or that a cubic polynomial always has zeros. 6 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

52 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.4B Apply definite integrals to problems involving the average value of a function..4b: The average value of a function f over an interval [ a, b] is b f b a ( ). a MPAC : Reasoning with definitions and theorems MPAC : Implementing algebraic/ computational processes This option is incorrect. The average velocity has been confused with the average rate of change of velocity over the interval: v( ) v() 7 = = 4. (B) (C) (D) This option is incorrect. The average velocity is not the arithmetic mean of the velocity values at the endpoints: v( ) + v() 7 + = =. This option is correct. The average value of vt () computed with a definite integral: vt () dt = ( ) = t dt t t ( ( )) = = ( ) over the interval [, ] is This option is incorrect. Speed and velocity have been confused and it was assumed that the average velocity must be positive: ( ) = 7 t dt. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 6

53 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.B Approimate a definite integral. (B) (C) (D).B: Definite integrals can be approimated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approimations can be computed using either uniform or nonuniform partitions. MPAC : Reasoning with definitions and theorems MPAC : Implementing algebraic/ computational processes This option is correct. The trapezoidal sum is the average of the left and right Riemann sums. The three intervals are of length, 5, and, respectively. Taking the average of the left and right endpoint values on each interval and multiplying by the length of the interval gives the following trapezoidal sum: ( 5 + 9) + 5 ( 9 + 5) + ( 5 + 4) = 68. This option is incorrect. Using the trapezoidal rule is not appropriate because the intervals are of different lengths. Ignoring the values of 9 0 t and believing that t = = as if the intervals were of equal length in the table would yield the following computation using the trapezoidal rule: [ ( ) + 5 ( ) + 4 ] = This option is incorrect. A trapezoidal sum has been confused with a left Riemann sum: = 85. This option is incorrect. The error is due to taking just the sum of the left and right Riemann sums rather than the average, thus not dividing each width by : 5 ( + 9) + 59 ( + 5) + ( 5 + 4) = Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

54 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.4D Apply definite integrals to problems involving area, volume, (BC) and length of a curve..b(b) Evaluate definite integrals. (B).4D: Areas of certain regions in the plane can be calculated with definite integrals. (BC) Areas bounded by polar curves can be calculated with definite integrals..b: If f is continuous on the interval [ a, b] and F is an antiderivative of f, then b f( ) = Fb ( ) F( a). a MPAC : Connecting concepts MPAC : Implementing algebraic/ computational processes This option is incorrect. If a sketch is not made or if it is not checked first whether the two graphs intersect on the interval, it is likely that the calculation for area will be done with just one integral: ( ) ( ) = ( f ( ) g( ) ) = 6 = 6 ( 54 54) ( 6). If the integration had been done as ( g ( ) f( ) ), the answer would be 4. But since area must be positive, the absolute value would give the same answer as the first integral. This option is correct. It is helpful to y sketch the graphs of the two functions and to check if the two graphs cross 0 inside the given interval. The first graph is a concave up parabola with zeros at = 0 and =, and the second graph is a line passing through O 0 the origin. The sketch indicates that there are two regions between the curves on the interval,. The line intersects the parabola where 6 8 = 6, or 0 = 6 = 6( ) ; therefore the top and bottom curves of the regions switch roles at =. Different integrals should be used to find the total area of the regions, one for the interval and one for the interval. Let f( ) = 6 8 and g( ) = 6. The total area is: f( ) g( ) = ( g( ) f( ) ) + ( f( ) g( ) ) = ( 6 ) + ( 6 ) = ( 6 ) + ( 6 ) = (( 4 6) ( 6 ) ) + (( 54 54) ( 6 4) ) = =. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 65

55 (C) This option is incorrect. The error comes from finding the area bounded by the curves and the -ais between = and =. Since the region is below the ais, the area is the opposite of the integral value: ( ( ) ) + g ( f ( ) ). y 0 O 0 (D) This option is incorrect. The error comes from finding the area bounded only by the parabola and -ais between = and =. Because the region is below the ais, the area is the opposite of the integral value: f ( ). y 0 O 0 66 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

56 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.B Solve problems involving the slope of a tangent line. (B).B: The derivative at a point is the slope of the line tangent to a graph at that point on the graph. MPAC : Implementing algebraic/ computational processes MPAC : Connecting concepts This option is incorrect. The slope of the tangent line has been incorrectly set equal to g( ) instead of g ( ) : This option is incorrect. Only the slope of the line was calculated through the two given points, but that was not connected to the calculation to find the value of b. (C) This option is incorrect. The slope of the line through (, 0 ) and (, 4 ) was mistakenly found using the wrong difference quotient, (D) g( ) = ( ) + b( ) = b =. y =. g ( ) = + b g ( ) = ( ) + b = b = 5 This option is correct. Parallel lines have the same slope. The slope of the tangent line at = is therefore equal to the slope of the line through (, 0 ) and (, 4 ), which is 4 ( ) =. Therefore find b so that 0 g ( ) =. g ( ) = + b g ( ) = ( ) + b = b = 4 Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 67

57 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.C Calculate a definite integral using areas and properties of definite integrals. (B) (C) (D).C: The definition of the definite integral may be etended to functions with removable or jump discontinuities. This option is correct. This function has a jump discontinuity at = 0. f( ) = for < 0 and f( ) = for > 0. The graph of f is shown to the right along with the regions A and B between the graph of f and the -ais on the intervals [ 50, ] and [ 0, ], respectively. MPAC : Implementing algebraic/ computational processes MPAC 5: Building notational fluency Computing the definite integral in terms of areas and taking into account which region is below the ais and which is above, we get: 5 The definite integral over the interval [ 5, ] can also be written as the sum of the definite integrals over [ 50, ] and [ 0, ], giving: This option is incorrect. Sign errors were made in the definition of f so that the graph of f is reflected across the -ais. This then yields: or 5 This option is incorrect. Rather than taking into account the signed areas, this is the total area: or f( ) = area ( A) + area( B ) = 5 + = f( ) = + = 5 + =. 5 f( ) = area ( A) area( B) = 5 = 5 0 f( ) = + = 5 =. 5 f( ) = area ( A) + area( B) = 5 + = f( ) = + = 5 + = This option is incorrect. The source of the error may be not recognizing that the definition of the definite integral can be etended to functions with removable or jump discontinuities. 5 A y B 68 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

58 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.B(b) Evaluate definite integrals. (B) (C).B5: Techniques for finding antiderivatives include algebraic manipulation such as long division and completing the square, substitution of variables, (BC) integration by parts, and nonrepeating linear partial fractions. MPAC : Implementing algebraic/ computational processes MPAC 5: Building notational fluency This option is incorrect. The technique of substitution of variables was not applied correctly because only the substitution u = was made. There is no substitution for or for the limits of integration. This option is incorrect. The technique of substitution of variables was not applied correctly because only the substitutions for and were made, but not for the limits of integration. This option is incorrect. The technique of substitution of variables was not applied correctly because only the substitutions for and the limits of integration were made, but not for. (D) This option is correct. Starting with the substitution u =, u = du = = du. Also change the limits of integration for to limits of integration for u: = u = = = u = = 5. Substituting for, for, and for the limits of integration gives: g ( 5 5 ) = gu ( ) du gu du = ( ). Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 69

59 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.C Calculate a definite integral using areas and properties of definite integrals. (B) (C) (D).C: In some cases, a definite integral can be evaluated by using geometry and the connection between the definite integral and area. MPAC : Implementing algebraic/ computational processes MPAC 4: Connecting multiple representations This option is incorrect. Only the area of the semicircle was considered. The opposite of that area was then taken, since the region is below the ais. This option is incorrect. Only the area of the semicircle was considered, ignoring both the area of the rectangle and the fact that the region is below the ais. This option is correct. The definite integral can be evaluated by using geometry and the connection of the definite integral with areas. In particular, the value of the integral here is the opposite of the area of the region between the semicircle and the -ais, i.e. the area of the rectangle with vertices at ( 0, ), (, 0), (, 6), and (, 6) minus the area of the semicircle. (, 6) (, 6) The area of the rectangle is 60 = 60. The area of the semicircle of radius 5 is 5π π ( 5) =. The definite integral is therefore equal to 5π ( 60 ). This option is incorrect. Only the area of the rectangle minus the area of the semicircle was computed; the need to take the opposite of this value, since the region is below the ais, was not recognized. 70 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

60 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.5A Analyze differential equations to obtain general and specific solutions. (B) (C) (D).5A: Antidifferentiation can be used to find specific solutions to differential equations with given initial conditions, including applications to motion along a line, eponential growth and decay, (BC) and logistic growth. MPAC : Implementing algebraic/ computational processes MPAC : Connecting concepts This option is incorrect. This is the velocity at t =, not the position: vt ()= t + 0 v( ) =. The velocity at t = can also be found directly using the Fundamental Theorem of Calculus: v( ) = v( 0) + v ( t) dt = 0 + at () dt = 0 + 6tdt =. 0 0 This option is incorrect. The assumption was made that the velocity is constant and the acceleration was ignored: p( ) = 7 + v( 0) = = 7. This option is incorrect. This is the total change in position without accounting for the initial position. The calculation uses the correct velocity function but takes pt ()= t + 0 t. This option is correct. Velocity is the antiderivative of acceleration and position is the antiderivative of velocity. In each case, the object s velocity and position at t = 0 can be used to find the appropriate + C after each antidifferentiation. The last step is to then evaluate the position function at t =. at ()= 6t vt ()= t + C v( 0) = ( 0) + C = 0 C = 0 vt ()= t + 0 pt ()= t + 0t + C p( 0) = ( 0) + 0( 0) + C = 7 C = 7 pt ()= t + 0t + 7 p( ) = + 0( ) + 7 = 5 0 Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 7

61 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.D Determine higher order derivatives. (B).D: Differentiating f produces the second derivative f, provided the derivative of f eists; repeating this process produces higher order derivatives of f. MPAC : Implementing algebraic/ computational processes MPAC 5: Building notational fluency This option is correct. This question requires repeated differentiation using knowledge of the derivatives of sine and cosine, the derivative of the natural logarithm, and the power rule for derivatives, as well as recognizing the need to use the chain rule for the first derivative. dy = sin = sin d y = cos + d y = sin This option is incorrect. A consistent sign error was made with the derivatives of the two trigonometric functions by using d ( sin ) = cos and d ( cos ) = sin. dy = sin = sin d y = cos + d y = sin 7 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

62 (C) (D) This option is incorrect. The only error was not using the chain rule when taking the first derivative. No chain rule is needed after that: dy = sin d y = cos + d y = sin = sin. This option is incorrect. A chain rule error was made on the first derivative, and there are consistent sign errors with the derivatives of the trigonometric functions. dy = sin d y = cos + d y = sin = sin Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 7

63 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.A Analyze functions defined by an integral..c Calculate a definite integral using areas and properties of definite integrals. (B) (C) (D).A: Graphical, numerical, analytical, and verbal representations of a function f provide information about the function g defined as g( ) = f() t dt..c: In some cases, a definite integral can be evaluated by using geometry and the connection between the definite integral and area. MPAC 4: Connecting multiple representations MPAC : Reasoning with definitions and theorems This option is incorrect. The Fundamental Theorem of Calculus was applied to calculate values of g( ), but all the areas were added to the value g( ). Therefore the conclusion was that g( ) 5 for all values of. This option is incorrect. It comes from confusing the graph of g with the graph of f and finding the number of times that f( ) =, which happens only between = 5 and = 6. This option is incorrect. The Fundamental Theorem of Calculus was applied correctly to find values of g( ) for >, thus finding the solutions c = 4 and 5 < c < 6. However, the integral f ( ) might 0 have been mishandled, thus missing the solution at c = 0. This option is correct. By the Fundamental Theorem of Calculus and using area and geometry to calculate the definite integral, 0 g( 0) = g( ) + g ( ) = 5 f( ) = 5 =. 0 So c = 0 is one solution to g( ) =. Since g = f, the graph of f indicates that g is increasing from = 0 to = since f is positive there. From = to = 4 the function g decreases by the same amount that it increased from = 0 to = by the symmetry of the regions. 4 g( 4) = 5 + f( ) = 5 + ( ) = a This gives a second solution at c = 4. According to the graph of the derivative f, the function g continues to decrease until = 5 and then begins increasing again. 5 g( 5) = 5 + f( ) = 5 + ( ) = 6 g( 6) = + f( ) = + = 4 5 Since g is continuous, the Intermediate Value Theorem guarantees a third solution somewhere on the interval (, 56 ). There are no other solutions possible. (The third solution is at c = 55..) 74 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time

64 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.A Analyze functions for intervals of continuity or points of discontinuity. (B) (C).A: A function f is continuous at = c provided that f( c) eists, lim f ( ) eists, and c lim f( ) = f( c). c MPAC : Implementing algebraic/ computational processes MPAC 5: Building notational fluency This option is incorrect. Rather than setting as equal the left- and rightsided limits at =, the derivatives of the two pieces were set equal at =. d d ( 6 + c) = ( 9 + ln ) c = c = This option is correct. Since the function f is continuous at =, then lim f( ) = lim f( ). Evaluate these two limits and set them equal, + then solve for c. lim f( ) = 6 + c = 6 + c lim f( ) = 9 + ln = c = 9 c = This option is incorrect. Rather than setting as equal the left- and rightsided limits at =, the derivatives of the two pieces were set equal at =, but a calculus error was made with respect to the derivative of a constant. d d ( 6 + c) = ( 9 + ln ) 6 + c = c = 9 + c = 5 OR The left- and right-sided limits were set equal at =, but an algebraic error was made in evaluating ln = : 6 + c()= 9 + ()= c = 5. (D) This option is incorrect. The value of the function at = was confused with c so that c = f()= 9 + ln = 9. Monday, May, 07 at 9:9: PM Central Daylight Time Calculus AB Practice Eam 75

65 Learning Objective Essential Knowledge Mathematical Practices for AP Calculus.F Estimate solutions to differential equations. (B) (C).F: Slope fields provide visual clues to the behavior of solutions to first order differential equations. MPAC 4: Connecting multiple representations MPAC : Connecting concepts This option is incorrect. For this differential equation, the slopes of the line segments in Quadrant II must be positive since y > 0 and > 0 in that quadrant. In this slope field that does not happen, as can be observed with the segments near the bottom left of Quadrant II. In addition, all line segments along the -ais should have positive slopes. This is not the case here. This slope field might be chosen if the squared term is not accounted for. [This is the slope field for dy = + y. ] This option is incorrect. For this differential equation, the slopes of the line segments in Quadrant I must be positive since y > 0 and > 0 in that quadrant. In this slope field, however, all the line segments in Quadrant I have negative slopes. In addition, all line segments along the -ais should have positive slopes. This is not the case here. This slope field might be chosen if one considers dy = 0 and thinks of the differential equation as relating to the parabola y =. [This is the slope field for dy = 8..] This option is incorrect. For this differential equation, the slopes of the line segments in Quadrant II must be positive since y > 0 and > 0 in that quadrant. In this slope field, however, that does not happen as can be observed with the segments near the bottom left of Quadrant II. In addition, all line segments along the -ais should have positive slopes. This is not the case here. (D) This option might be chosen if the and y variables are confused and one looks for the slope field for dy = + y. This option is correct. The line segments in the slope field have slopes given by dy = + y at the point (, y). In Quadrants I and II, all slopes must be positive or zero since y > 0 in those quadrants and 0. This is the only option in which that condition is true. 76 Calculus AB Practice Eam Monday, May, 07 at 9:9: PM Central Daylight Time