LSU AP Calculus Practice Test Day

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1 LSU AP Calculus Practice Test Day AP Calculus AB 2018 Practice Exam Section I Part A

3 AP CALCULUS AB: PRACTICE EXAM SECTION I: PART A NO CALCULATORS ALLOWED. YOU HAVE 60 MINUTES. 1. If y = ( 1 + x 5) 3 2, then dy dx = (A) 15 2 x4 1 + x 5 (B) x 5 (C) 10 3 x4 1 + x 5 (D) (1 + x) 3/2 2 x 2. ln 16 ln 4 e x 2 dx = (A) 12 (B) 8 (C) 4 (D) 2 1

4 A A A A A A A A A A A A A A A A A A A 3. The vertical line x = 8 is an asymptote for the graph of the function of f. Which of the following statements must be false? (A) lim x f (x) = 8 (B) lim f (x) = x 8 (C) lim x 8 + f (x) = (D) lim x 8 f (x) = 0 4. If y = 3x2 10 x + 6 dy, then dx = (A) 3x2 + 36x + 10 (x + 6) 2 (B) (C) (D) 6x (x + 6) 6x (x + 6) 2 6x 10 (x 2 + 6x) 2 2

5 A A A A A A A A A A A A A A A A A A A 5. Shown below is a slope field for which of the following differential equations? (A) dy dx = xy (B) dy dx = xy y (C) dy dx = xy + y (D) dy dx = xy + x 6. The second derivative of the function f is given by f (x) = x(x a)(x b) 2. The graph of f is shown below. For what values of x does the graph of f have a point of inflection? (A) 0 and a only (B) 0 and m only (C) b and j only (D) 0, a, and b 3

6 A A A A A A A A A A A A A A A A A A A 7. If f(x) = cos(3x), then f ( π 9 ) = (A) (B) 3 2 (C) 3 2 (D) The graph of f, the derivative of the function f, is shown below. Which of the following statements must be true about f? (A) f is strictly increasing for 0 < x < 3 (B) f (2) = 0 (C) f is strictly increasing for 3 < x < 0 (D) f is strictly increasing for 2 < x < 1 4

7 A A A A A A A A A A A A A A A A A A A 9. The graph of the function f is shown below for 0 x 7. Of the following, which has the greatest value? (A) 7 f(x) dx 0 (B) Left Riemann sum approximation of 7 f(x) dx with 5 subintervals of equal length 0 (C) Right Riemann sum approximation of 7 f(x) dx with 5 subintervals of equal length 0 (D) Midpoint Riemann sum approximation of 7 f(x) dx with 5 subintervals of equal length Which of the following is the solution to the differential equation dy dt condition y(0) = 1? = ty with the initial (A) y = e 1+e t 2 (B) y = e t2 2 (C) y = e 1+e t 2 (D) y = e 1 e t 2 5

8 A A A A A A A A A A A A A A A A A A A 11. The function f has the property that f (x) and f (x) are negative for 10 < x < 10, and the property that f (x) is also negative for all real values x. Which of the following could be the graph of f? (A) (B) (C) (D) 6

9 A A A A A A A A A A A A A A A A A A A 12. lim x 0 5x 4 + 8x 2 3x 4 16x 2 is (A) 1 2 (B) 0 (C) 1 (D) The graph of a function f is shown below. At which value of x is f continuous, but not differentiable? (A) a (B) b (C) c (D) d 7

10 A A A A A A A A A A A A A A A A A A A 14. If y = e 3x sin 3x, then dy dx = (A) 3e 3x (cos 3x + sin 3x) (B) 9e 3x cos 3x (C) e 3x (cos 3x + sin 3x) (D) 9e 3x (cos 3x + sin 3x) 15. Let f be the function with derivative given by f (x) = 3x x On which of the following intervals is f strictly decreasing? (A) ( 8, 4) (B) (, 8) (C) ( 6, ) (D) ( 6, 4) 8

11 A A A A A A A A A A A A A A A A A A A 16. If the line tangent to the graph of the function f at the point (5, 3) passes through the point (2, 6), then f (5) is (A) 3 (B) 1 3 (C) 9 (D) Let f be the function given by f(x) = x 3 9x 2 + x 9. The graph of f is concave up when (A) x > 9 (B) x > 2 3 (C) x < 3 (D) x > 3 9

12 A A A A A A A A A A A A A A A A A A A 18. A curve has slope 6x + 10 at each point (x, y) on the curve. Which of the following is an equation for the curve if it passes through the point ( 2, 7)? (A) y = 3x x + 21 (B) y = 3x 2 10x + 15 (C) y = 10x 2 3x + 2 (D) y = 3x x + 15 { 19. Let f (x) = (A) III only (B) I only 2x if x 1 x if x > 1 I. lim x 1 f (x) does not exist. II. f is continuous at x = 1. III. f is not differentiable at x = 1. (C) I and III only (D) None of the above. Which of the following statements about f is true? 10

13 20. Find lim x 1 (A) 2 (B) -4 (C) -24 (D) -12 A A A A A A A A A A A A A A A A A A A 7x 4 2x 2 5. x x 2 x dx = (A) ln x C (B) x ln x C (C) x 3 ln x C (D) 1 3 ln x C 11

14 A A A A A A A A A A A A A A A A A A A 22. What is the area of the region in the plane between the graphs of y = x 2 and y = 12 x from x = 0 to x = 5? (A) (B) 35 6 (C) 45 3 (D) The derivative g of a function g is continuous and has exactly two zeroes. Selected values of g are given in the table below. If the domain of g is the set of all real numbers, then g is strictly decreasing on which of the following intervals? (A) x > 1 (B) 4 < x < 1 (C) x < 4 (D) x > 4 x g (x)

15 A A A A A A A A A A A A A A A A A A A 24. The function f is twice differentiable with f(2) = 4, f (2) = 5, and f (2) = 4. What is the value of the linear approximation of f(2.1) using the line tangent to the graph of f at x = 2? (A) 4.1 (B) 4.5 (C) 3.5 (D) Which of the following is an equation of the tangent line to the graph of y = 2x 3 + 6x 2 + 7x 10 at the point (0, 10)? (A) y = 7x 1 (B) y = 7x + 10 (C) y = 2x + 1 (D) y = 7x 10 13

16 A A A A A A A A A A A A A A A A A A A 26. A particle moves along the y-axis so that at time t > 0 its position is given by y (t) = 6 ln t 2t At what time t is the particle at rest? (A) 2 (B) ln 3 (C) 1 3 (D) Let f be a function defined by f (x) = 3x 3 10x 2 + 4x + 6. If g (x) = f 1 (x) and f (1) = 3, what is the value of g (3)? (A) 1 7 (B) 1 15 (C) 1 7 (D)

17 A A A A A A A A A A A A A A A A A A A 28. If y = 2x 24, what is the minimum value of the product 2xy? (A) 9 (B) 36 (C) 144 (D) lim x π cos x + sin(2x) + 1 x 2 π 2 = (A) 1 2π (B) 1 π (C) 1 (D) 0 15

18 A A A A A A A A A A A A A A A A A A A 30. The graph of a differentiable function f is shown above for 3 x 3. The graph of f has horizontal tangent lines at x = 1, x = 1, and x = 2. The areas of regions A, B, C, and D are 5, 4, 5, and 3, respectively. Let g be the antiderivative of f such that g(x) + 1 g(3) = 7. Find the value of lim. x 0 2x (A) 1 2 (B) -1 (C) 1 (D) 0 END OF PART A OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. 16

19 A A A A A A A A A A A A A A A A A A A THIS PAGE INTENTIONALLY LEFT BLANK 17

21 LSU AP Calculus Practice Test Day AP Calculus AB 2018 Practice Exam Section I Part B

23 AP CALCULUS AB: PRACTICE EXAM SECTION I: PART B GRAPHING CALCULATORS ARE ALLOWED. YOU HAVE 45 MINUTES. 51. If G(t) is an antiderivative for f(t), f(t) is continuous on [0, 6], and G(0) = 1, then G(6) = (A) f (6) (B) 1 + f (6) (C) f(t) dt 0 (D) 6 f(t) dt Find the equation of the line normal to y = x 2 e 2x + 3x ln(x 2 ) at x = 1. (A) y 1 = 1 4e (x e2 ) (B) y e 2 = (4e 2 + 6)(x 1) (C) y 1 = (4e 2 + 6)(x e 2 ) (D) y e 2 = 1 (x 1) 4e

24 B B B B B B B B B B B B B B B B 53. The function f is continuous for 2 x 1 and differentiable for 2 < x < 1. If f( 2) = 5 and f(1) = 4, which of the following statements could be false? (A) There exists c, where 2 < c < 1, such that f(c) = 0. (B) There exists c, where 2 < c < 1, such that f (c) = 0. (C) There exists c, where 2 < c < 1, such that f(c) = 3. (D) There exists c, where 2 < c < 1, such that f (c) = If 2 5 f(x)dx = 17 and 2 5 f(x)dx = 4, what is the value of 5 5 f(x)dx? (A) 21 (B) 13 (C) 0 (D) 13 2

25 B B B B B B B B B B B B B B B B 55. The radius of a circle is increasing at a constant rate of 1.2 meters per second. What is the rate of change in the circumference of the circle at the instant when the area of the circle is 25π square meters? (A) 2.4π m/s (B) 12π m/s (C) 2.4π m/s (D) 12π m/s 56. Let f be the function with derivative given by f (x) = sin(x 2 1). How many relative extrema does f have on the interval 1 < x < 2? (A) One (B) Two (C) Three (D) Four 3

26 B B B B B B B B B B B B B B B B 57. The rate of change of the altitude of a hot-air balloon is given by r(t) = t 3 + 8t 2 6 for 4 t 10. Which of the following expressions gives the change in altitude of the balloon during the time the altitude is increasing? (A) r (t) dt (B) r(t) dt (C) r (t) dt (D) r(t) dt 58. The velocity, in ft/sec, of a particle moving along the x-axis is given by the function v(t) = 1 5 tet + t 3 2 e 5t. What is the average velocity of the particle from time t = 0 to time t = 4? (A) ft/sec (B) ft/sec (C) ft/sec (D) ft/sec 4

B B B B B B B B B B B B B B B B 59. A pizza, heated to a temperature of 350 degrees Fahrenheit (F ), is taken out of an oven and placed in a 75F room at time t = 0 minutes.

27 B B B B B B B B B B B B B B B B 59. A pizza, heated to a temperature of 350 degrees Fahrenheit (F ), is taken out of an oven and placed in a 75F room at time t = 0 minutes. The temperature of the pizza is changing at a rate of 110e 0.4t degrees Fahrenheit per minute. To the nearest degree, what is the temperature of the pizza at a time t = 5 minutes? (A) 112 F (B) 119 F (C) 147 F (D) 238 F 60. The table below gives values of a function f and its derivative at selected values of x. If f is continuous on the interval [ 4, 1], what is the value of 1 4 f (x)dx? (A) 4.5 (B) 2.25 (C) 0 (D)

28 B B B B B B B B B B B B B B B B 61. If f(x) = e (2/x), then f (x) = (A) 2e (2/x) ln(x) (B) e (2/x) (C) e ( 2/x2 ) (D) 2 e (2/x) x In the xy-plane, the line x + y = k, where k is constant, is tangent to the graph of y = x 2 + 3x + 1. What is the value of k? (A) 1 (B) 0 (C) 1 (D) 3 6

29 B B B B B B B B B B B B B B B B 63. The function f is continuous on the closed interval [2, 4] and twice differentiable on the open interval (2, 4). If f (3) = 2 and f (x) < 0 on the open interval (2, 4), which of the following could be a table of values for f? (A) (C) (B) (D) 64. Let f be a differentiable function with f(9) = 2 and f (9) = 1, and let g be the function defined by g(x) = [f(x)] 3. Which of the following is an equation of the line tangent to the graph of g at the point where x = 9? (A) y + 8 = 12(x 9) (B) y 12 = 12(x 9) (C) y 12 = 12(x 9) (D) y + 8 = 12(x 9) 7

30 B B B B B B B B B B B B B B B B 65. What is the area enclosed by the curves y = x 3 8x x 5 and y = x + 5? (A) (B) (C) (D) END OF PART B OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART B ONLY. 8

31 B B B B B B B B B B B B B B B B THIS PAGE INTENTIONALLY LEFT BLANK 9

33 LSU AP Calculus Practice Test Day AP Calculus AB 2018 Practice Exam Section II Part A Name: School:

35 AP CALCULUS AB PRACTICE EXAM CALCULUS AB SECTION II, PART A TIME - 30 MINUTES NUMBER OF PROBLEMS - 2 GRAPHING CALCULATORS ARE ALLOWED. PLEASE BEGIN YOUR ANSWERS UNDER THE QUESTIONS (IF THERE IS ROOM) AND CONTINUE THEM ON THE BACK OF THE PAPER. 1. Grass clippings are placed in a bin, where they decompose. For 0 t 30, the amount of grass clippings remaining in the bin is modeled by A(t) = 6.687(0.931) t, where A(t) is measured in pounds and t is measured in days. (a) Find the average rate of change of A(t) over the interval 0 t 30. Indicate units of measure. (b) Find the value of A (15). Using correct units, interpret the meaning of the value in the context of the problem. (c) Find the time t for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval 0 t 30. (d) For t > 30, L(t), the linear approximation to A at t = 30, is a better model for the amount of grass clippings remaining in the bin. Use L(t) to predict the time at which there will be 0.5 pounds of grass clippings remaining in the bin. Show the work that leads to your answer. 1

36 1. (cont.) 2

37 2. A particle moves along a straight line. For 0 t 5, the velocity of the particle is given by v(t) = 2 + (t 2 + 3t) 6/5 t 3, and the position of the particle is given by s(t). It is known that s(0) = 10. (a) Find all values of t in the interval 2 t 4 for which the speed of the particle is 2. (b) Write an expression involving an integral that gives the position s(t). Use this expression to find the position of the particle at time t = 5. (c) Find all times t in the interval 0 t 5 at which the particle changes direction. Justify your answer. (d) Is the speed of the particle increasing or decreasing at time t = 4? Give a reason for your answer. 3

38 2. (cont.) END OF PART A OF SECTION II 4

39 LSU AP Calculus Practice Test Day AP Calculus AB 2018 Practice Exam Section II Part B Name: School:

41 AP CALCULUS AB PRACTICE EXAM CALCULUS AB SECTION II, PART B TIME - 60 MINUTES NUMBER OF PROBLEMS - 4 NO CALCULATORS ALLOWED. PLEASE BEGIN YOUR ANSWERS UNDER THE QUESTIONS (IF THERE IS ROOM) AND CONTINUE THEM ON THE BACK OF THE PAPER. 3. There are 700 people in line for a popular amusement-park ride when the ride begins operation in the morning. Once it begins operation, the ride accepts passengers until the park closes 8 hours later. While there is a line, people move onto the ride at a rate of 800 people per hour. The graph above shows the rate, r(t), at which people arrive at the ride throughout the day. Time t is measured in hours from the time the ride begins operation. (a) How many people arrive at the ride between t = 0 and t = 3? Show the computations that lead to your answer. (b) Is the number of people waiting in line to get on the ride increasing or decreasing between t = 2 and t = 3? Justify your answer. (c) At what time t is the line for the ride the longest? How many people are in line at that time? Justify your answers. (d) Write, but do not solve, an equation involving an integral expression of r whose solution gives the earliest time t at which there is no longer a line for the ride. 1

42 3. (cont.) 2

43 4. The function f is defined by f(x) = 25 x 2 for 5 x 5. (a) Find f (x). (b) Write an equation for the line tangent to{ the graph of f at x = 3. f(x) for 5 x 3 (c) Let g be the function defined by g(x) = x + 7 for 3 < x 5. Is g continuous at x = 3? Use the definition of continuity to explain your answer. (d) Find the value of 5 0 x 25 x 2 dx. 3

44 4. (cont.) 4

45 5. At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function W models the total amount of solid waste stored in the landfill. Planners estimate that W will satisfy the differential equation dw = 1 (W 300) for the next 20 dt 25 years. W is measured in tons, and t is measured in years from the start of (a) Use the line tangent to the graph of W at t = 0 to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of 2010 (time t = 1 4 ). (b) Find d2 W in terms of W. Use d2 W to determine whether your answer in part (a) dt 2 dt 2 is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time t = 1. 4 (c) Find the particular solution W = W (t) to the differential equation dw dt with initial condition W (0) = = 1 (W 300) 25 5

46 5. (cont.) 6

47 6. Let f be the function defined by f(x) = k x ln(x) for x > 0, where k is a positive constant. (a) Find f (x) and f (x). (b) For what value of the constant k does f have a critical point at x = 1? For this value of k, determine whether f has a relative minimum, relative maximum, or neither at x = 1. Justify your answer. (c) For a certain value of the constant k, the graph of f has a point of inflection on the x-axis. Find this value of k. 7

48 6. (cont.) STOP END OF EXAM 8

+ 1 for x > 2 (B) (E) (B) 2. (C) 1 (D) 2 (E) Nonexistent

+ 1 for x > 2 (B) (E) (B) 2. (C) 1 (D) 2 (E) Nonexistent dx = (A) 3 sin(3x ) + C 1. cos ( 3x) 1 (B) sin(3x ) + C 3 1 (C) sin(3x ) + C 3 (D) sin( 3x ) + C (E) 3 sin(3x ) + C 6 3 2x + 6x 2. lim 5 3 x 0 4x + 3x (A) 0 1 (B) 2 (C) 1 (D) 2 (E) Nonexistent is 2 x 3x