The Princeton Review AP Calculus BC Practice Test 2

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1 0 The Princeton Review AP Calculus BC Practice Test

2 CALCULUS BC SECTION I, Part A Time 55 Minutes Number of questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem. In this test: 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.. What is the slope of the line tangent to the curve x + xy + y = when y =? (A) (B) 8 (C) (D) 0 (E) 8 x. xe dx = (A) e (B) e (C) 0 (D) e (E) e T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 7 9 5

3 . If, for t > 0, x = t and y = cos(t ), then dy dx = (A) cos(t ) (B) sin(t ) (C) sin(t) (D) sin(t ) (E) cos(t) 4. The function f(x) = 4x 8x + on the interval [, ] has an absolute minimum at x = (A) (B) (C) 0 (D) (E) 4 5. x dx x + 5x + 6 (A) ln ( x + ) ( x + ) + C (B) ln ( x + ) ( x + ) + C (C) ln ( x + ) ( x + ) + C (D) ln ( x + ) ( x + ) + C (E) ln ( x + )( x + ) + C n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

4 6. d dx ( x sin x) = (A) x sin x (B) x cos x (C) x sin x (D) x sin x + x cos x (E) x sin x + x sin x x 7. The line normal to the curve y = x at x = has slope + (A) 8 5 (B) 5 8 (C) (D) 8 5 (E) T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 7 9 7

5 8. If f and g are differentiable functions and h(x) = f(x)e g(x), then h9(x) = (A) f9(x)e g9(x) (B) f9(x)e g(x) + f(x)e g9(x) (C) e g(x) [f9(x) + f(x)g9(x)] (D) e g(x) [f9(x) + ] (E) e g9(x) [f9(x) + g9(x)] n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

6 9. y 0 x The graph of y = f(x) is shown above. Which of the following could be the graph of y = f9(x)? (A) y (D) y 0 x 0 x (B) y (E) y 0 x 0 x (C) y 0 x T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 7 9 9

7 e 0. x + e dx = x (A) 4 e + e + e (B) + e + (C) e 4 + e + e (D) (E) e e + e e e e + +. dx x (A) π 6 (B) π 4 (C) π (D) p (E) Divergent n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

8 . What is the equation of the line tangent to the graph of y = sin x at x = π 4? (A) y x π = 4 (B) y x π = 4 (C) y x π = 4 (D) y = x π 4 (E) y = x π 4. If f(x) = ax + ax + 5; x, find the value of a that makes f(x) continuous for all real values of x. 4ax 6ax + 9; x < (A) (B) 0 (C) (D) (E) 4 T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 8 0

9 4. xsin( x) dx = (A) x cos( x) + C (B) x cos( x) + C 4 (C) x cos( x) + sin( x) + C 4 (D) x cos( x) + cos( x) + C (E) cos( x) + sin( x) + C 4 x + 5x 4 5. If f ( x) =, then lim f ( x) x + 0x + 6 x 8 is (A) 0 (B) (C) (D) 6 (E) Nonexistent 8 0 n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

10 6. What is the approximation of the value of e obtained by using the fourth-degree Taylor polynomial about x = 0 for e x? 4 (A) (B) !! 4! 4 (C) + +!! 4! 4 (D) (E) +! 4! 7. A rock is thrown straight upward with an initial velocity of 50 m/s from a point 00 m above the ground. If the acceleration of the rock at any time t is a = 0 m/s, what is the maximum height of the rock (in meters)? (A) 5 (B) 50 (C) 75 (D) 00 (E) 5 T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 8 0

11 8. The sum of the infinite geometric series is (A) 6 (B) (C) 0 (D) 7 (E) 9. What are all values of x for which the series (A) x 8 (B) < x 8 (C) x < 8 (D) 5 x 5 (E) 5 x < 5 n ( x ) n n ( 5 ) converges? n = 0. Find the area inside one loop of the curve r = sin q. (A) π 6 (B) π 8 (C) π 4 (D) π (E) p n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

12 . The average value of sec x on the interval (A) 8 π π π 6, 4 is (B) (C) (D) (E) π 4 π 6 6 π 6 6 π. Find the length of the arc of the curve defined by x = t and y = t ( ), from t = 0 to t =. (A) 8 (B) 0 (C) (D) 4 (E) 6 T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 8 0 5

13 . The function f is given by f(x) = x 4 + 4x. On which of the following intervals is f decreasing? (A) (, 0) (B) (0, ) (C) (, ) (D) (, ) (E) (, 0) 4. Which of the following series converge(s)? I. II. n = III. ( ) n n n = n n = n (A) I only (B) II only (C) I and II (D) I and III (E) I, II, and III 5. Given the differential equation dz z = z dt 4 00, where z(0) = 50, what is lim z ( t )? t (A) 400 (B) 00 (C) 00 (D) 50 (E) n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

14 6. 0 The slope field shown above corresponds to which of the following differential equations? (A) (B) (C) (D) (E) dy x = dx y dy y = dx x dy = xy dx dy = x y dx dy = x + y dx T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 8 0 7

15 7. The value of c that satisfies the mean value theorem for derivatives on the interval [0, 5] for the function f(x) = x 6x is (A) 5 (B) 0 (C) (D) 5 (E) The graph of f is shown in the figure above. If g( x) = f ( t) dt, for what positive value of x does g(x) have a minimum? 0 x (A) 0 (B) (C) (D) (E) 4 STOP END OF PART A 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 n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

16 CALCULUS BC SECTION I, Part B Time 50 Minutes Number of questions 7 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem. In this test: () 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. 9. If f(x) is the function given by f(x) = e x +, at what value of x is the slope of the tangent line to f(x) equal to? (A).5 (B) 0 (C). (D).66 (E).69 T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 8 0 9

17 0. If y = (sin x) ex, then, when defined, y = (A) (sin x) ex (cos x) (B) (cos x) ex (C) e x (cot x + ln(sin x)) (D) e x (sin x) ex (cot x + ln(sin x)) (E) e x (sin x) ex (cot x). The side of a square is increasing at a constant rate of 0.4 cm/sec. In terms of the perimeter, P, what is the rate of change of the area of the square, in cm /sec? (A) 0.05P (B) 0.P (C) 0.4P (D) 6.4P (E) 5.P 8 0 n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

18 . If f is a vector-valued function defined by f(t) = (sin t, sin t), then f99(t) = (A) ( 4 sin t, cos t) (B) ( sin t, cos t) (C) (4 sin t, cos t) (D) (4 sin t, cos t) (E) ( cos t, 4 sin t). The height of a mass hanging from a spring at time t seconds, where t > 0, is given by h(t) = 4 cos(t). In the first two seconds, how many times is the velocity of the mass equal to 0? (A) 0 (B) (C) (D) (E) 4 T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 8

19 tan ( + h) π 4. lim 4 h 0 h (A) (B) π (C) π (D) (E) Nonexistent x 5. What is the trapezoidal approximation of e dx using n = 4 subintervals? 0 (A) (B).565 (C) 9.97 (D) (E) Given x y + x = y +, find d y at (,). dx (A) 6 (B) (C) 4 (D) (E) 6 8 n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

20 4 7. If f ( x) dx = a and f ( x) dx = b, then f ( x) dx = 4 (A) a + b (B) a b (C) a b (D) b a (E) b a 8. 5x d cos t dt dx = x (A) 5 cos 5x cos x (B) 5 sin 5x sin x (C) cos 5x cos x (D) sin 5x sin x (E) cos 5x sin x 5 T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 8

21 9. Using the Taylor series about x = 0 for sin x, approximate sin(0.4) to four decimal places. ( 0. 4) ( 0. 4) (A) ! 5! 5 ( 0. 4) ( 0. 4) (B) ! 5! 5 ( 0. 4) ( 0. 4) (C) ( 0. 4) ( 0. 4) ( 0. 4) ( 0. 4) (D) !! 4! 5! ( 0. 4) (E) 0. 4! 4 5 ( 0. 4) ( 0. 4) ( 0. 4) + +! 4! 5! Let R be the region in the first quadrant between the graphs of y = e x, y = sin x, and the y-axis. The volume of the solid that results when R is revolved about the x-axis is (A) (B) (C) 0.77 (D) (E) n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

22 4. Use Euler s Method, with h = 0. to estimate y(), if dy dx = y 4x and y() = 6. (A) (B) 0.44 (C).78 (D).756 (E) sec 4 x dx = (A) tan 4 x + C (B) tan x + tan x + C (C) tan x + C (D) sec 5 x + C 5 (E) sec x tan x + C 4. Let f(x) = cot x dx; 0 < x < p. If f π =, then f() = 6 (A).86 (B) (C) 0.4 (D) 0.54 (E).5 T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 8 5

23 44. 4 x dx = (A) 4 ( x ) + C (B) sin (C) sin x x 4 x C + + x x 4 x C + + (D) 4 ( x ) + C (E) sin x 4x 4 x C A force of 50 N is required to stretch a spring 5 m from rest. Using Hooke s law, F = kx, how much work, in Joules, is required to stretch the spring 7 m from rest? (A) 4.86 (B) 7.49 (C) 45 (D) 490 (E) 5 STOP 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. 8 6 n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

24 SECTION II GENERAL INSTRUCTIONS You may wish to look over the problems before starting to work on them, since it is not expected that everyone will be able to complete all parts of all problems. All problems are given equal weight, but the parts of a particular problem are not necessarily given equal weight. A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THIS SECTION OF THE EXAMINATION. You should write all work for each part of each problem in the space provided for that part in the booklet. Be sure to write clearly and legibly. If you make an error, you may save time by crossing it out rather than trying to erase it. Erased or crossed-out work will not be graded. Show all your work. You will be graded on the correctness and completeness of your methods as well as your answers. Correct answers without supporting work may not receive credit. Justifications require that you give mathematical (noncalculator) reasons and that you clearly identify functions, graphs, tables, or other objects you use. 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 problem, 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. Your work must be expressed in standard mathematical notation rather than calculator syntax. For 5 example, x dx may not be written as fnint (X, X,, 5). Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct 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. SECTION II, PART A Time 45 minutes Number of problems A graphing calculator is required for some problems or parts of problems. During the timed portion for Part A, you may work only on the problems in Part A. On 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 problem, 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. T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 8 7

25 . An object moving along a curve in the xy-plane has its position given by (x(t), y(t)) at time t seconds, 0 t, with dx dt = 8t cos t units/sec and dy = 8t sin t units/sec. dt At time t = 0, the object is located at (5, ). (a) Find the speed of the object at time t =. (b) Find the length of the arc described by the curve s position from time t = 0 to time t =. (c) Find the location of the object at time t = π. 8 8 n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

26 . Second Third D A α First C B Home A baseball diamond is a square with each side 90 feet in length. A player runs from second base to third base at a rate of 8 ft/sec. (a) At what rate is the player s distance from first base, A, changing when his distance from third base, D, is.5 feet? (b) At what rate is angle a increasing when D is.5 feet? (c) At what rate is the area of the trapezoidal region, formed by line segments A, B, C, and D, changing when D is.5 feet? T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 8 9

27 . A body is coasting to a stop and the only force acting on it is a resistance proportional to its speed, according to the equation ds k m = v f = v e t 0 ; s(0) = 0, where v dt 0 is the body s initial velocity (in m/s), v f is its final velocity, m is its mass, k is a constant, and t is time. (a) If a body with mass m = 50 kg and k =.5 kg/sec initially has a velocity of 0 m/s, how long, to the nearest second, will it take to slow to m/s? (b) How far, to the nearest 0 meters, will the body coast during the time it takes to slow from 0 m/s to m/s? (c) If the body coasts from 0 m/s to a stop, how far will it coast? 8 0 n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

28 No calculator is allowed for these problems. PART B Time 45 minutes Number of problems During the timed portion for Part B, you may continue to work on the problems in Part A without the use of any calculator. 4. v(t) Velocity (Kph) Train B Train A t Time (Hours) Three trains, A, B, and C each travel on a straight track for 0 t 6 hours. The graphs above, which consist of line segments, show the velocities, in kilometers per hour, of trains A and B. The velocity of C is given by v(t) = 8t 0.5t. (Indicate units of measure for all answers.) (a) Find the velocities of A and C at time t = 6 hours. (b) Find the accelerations of B and C at time t = 6 hours. (c) Find the positive difference between the total distance that A traveled and the total distance that B traveled in 6 hours. (d) Find the total distance that C traveled in 6 hours. T H E P R I N C E TO N R E V I E W A P C A L C U LU S B C P R A C T I C E T E S T n 8

29 5. Let y be the function satisfying f9(x) = x( f(x)); f(0) = 0. (a) Use Euler s Method, starting at x = 0, with step size of 0.5 to approximate f(x) at x =. (b) Find an exact solution for f(x) when x =. (c) Evaluate x( f ( x)) dx Given f(x) = tan (x) and g(x) =, for x. + x (a) Find the fifth-degree Taylor polynomial and general expression for g(x) about x = 0. d (b) Given that x dx tan = + x, for x, use the result of part (a) to find the fifth-degree Taylor polynomial and general expression for f(x) about x = 0. (c) Use the fifth-degree Taylor polynomial to estimate f 0. END OF EXAMINATION 8 n C R A C K I N G T H E A P C A L C U LU S A B & B C E X A M S

30 Spring Semester - AP Calculus BC Princeton Review - Test D 9 A C 0 D B B 4 B A 5 A B 6 E 4 D 7 B 5 C 8 C 6 D 9 D 7 D 0 D 8 A A 9 B B 40 E D 4 C 4 C 4 B 5 D 4 E 6 B 44 C 7 E 45 E 8 E 9 A 0 B C A D 4 C 5 A 6 D 7 E 8 E

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