### Transcription

AP Calculus AB Practice ExamBy: Patrick CoxOriginal non-secure materials written based on previous secure multiple choice and FRQquestions from the past three years. I wrote this as a way for my students to have access tomultiple choice and FRQ since secure materials can’t be used outside of class.Feel free to use in your class, post to the internet/classroom, you will find the answer keyto the multiple choice and FRQ at the end. Below, you can find which problems can beanswered after each unit in the CED (although questions definitely can span acrossmultiple units in the CED). I do not work for Collegeboard so these categorizations are tothe best of my knowledge using the public CED.Pages 2-15 Non-Calculator MCPages 16-23 Calculator MCPages 24-27 Calculator FRQPages 28-34 Non-Calculator FRQPages 36-42 SolutionsQuestions By Unit in CED:Unit 1: 21, 24, 76, 90, FRQ 1(a), FRQ 5 (d)Unit 2: 6, 8, 25, 28, 80, FRQ 5 (c), FRQ 6 (a)Unit 3: 14, 16, 81, FRQ 4 (c), FRQ 6 (b)Unit 4: 5, 10, 12, 18, 23, 87, 88, FRQ 1(d), FRQ 2 (b) (c), FRQ 5 (b) (c), FRQ 6 (d)Unit 5: 9, 13, 15, 82, 83, 84, 85, FRQ 3 (b) (c)Unit 6: 3, 4, 11, 19, 20, 22, 29, 78, 79, 86, FRQ 3 (a) (d), FRQ 5 (a) (b), FRQ 6 (c)Unit 7: 2, 7, 27, FRQ 4 (a) (b)Unit: 8: 1, 17, 26, 30, 77, 89, FRQ 1 (b) (c), FRQ 2 (a) (d)

Non-Calculator Multiple Choice1) A particle moves along a straight line so that at time t 0 its acceleration isgiven by the function a(t) 4t . At time t 0, the velocity of the particle is 4and the position of the particle is 1. Which of the following is an expression forthe position of the particle at time t 0?(a) 23 t3 4t 1(b) 2t3 4t 1(c) 13 t3 4t 1(d) 23 t3 42)Shown above is a slope field for which of the following differential equations?(a)dydx xy(b)dydx xy(c)dydx x y(d)dydx x y

3)The graph of a piecewise linear function f(x) is above. Evaluate(a) 2(b) 2(c) 5(d) 0(b) 4 ln 5(c) 2 ln 5(d) 1 ln 54)(a) 5 ln 5

5)is(a) 2e(b) 1(c) 0(d) nonexistent6) Let f be the function defined above. Which of the following statementsabout f is true?(a) f is continuous and differentiable at x -2.(b) f is continuous but not differentiable at x -2.(c) f is differentiable but not continuous at x -2.(d) f is defined but is neither continuous nor differentiable at x -2.7) The equation y e2x is a particular solution to which of the followingdifferential equations?(a)dydx 1(b)dydx y(c)dydx y 1(d)dydx y 1

8) For any real number x,(a) cos(x2 )limh 0(b) 2xcos(x2 )cos((x h)2 ) cos(x2 )h (c) sin(x2 )(d) 2xsin(x2 )9) What is the value of x at which the maximum value ofoccurs on the closed interval [0, 4]?(a) 0(b)32(c)52(d) 4

10) At time t 0, a reservoir begins filling with water. For t 0 hours, thedepth of the water in the reservoir is increasing at a rate of R(t) inches perhour. Which of the following is the best interpretation of R′(2) 4 ?(a) The depth of the water is 4 inches, at t 2 hours.(b) The depth of the water is increasing at a rate of 4 inches per hour, at t 2hours.(c) The rate of change of the depth of the water is increasing at a rate of 4inches per hour per hour at t 2 hours(d) The depth of the water increased by 4 inches from t 0 to t 2 hours.11) If(a) 7and f (2) 3 , then f (1) (b) -4(c) 7(d) 10

12) A particle moves along the x-axis so that at time t 0 its velocity is givenby v (t) et 1 3sin(t 1) . Which of the following statements describes themotion of the particle at time t 1 ?(a) The particle is speeding up at t 1 .(b) The particle is slowing down at t 1 .(c) The particle is neither speeding up nor slowing down at t 1 .(d) The particle is at rest at t 1 .13)x02468f (x)1-1-575The table above gives selected values for the twice-differentiable function f.In which of the following intervals must there be a number c such that(a) (0, 2)(b) (2, 4)(c) (4, 6)(d) (6, 8)

14)(a)ddx (tan(ln(x)))tan(ln(x))x (b) sec2 (ln(x))(c)sec2 (ln(x))x( d) tan( 1x )15) The function f is given by f (x) x3 2x2 . On what interval(s) is f(x)concave down?(a) (0, 43 )16) If(a) 116(b) ( , 0) and ( 43 , )what is(b)116dydx(c) ( ,23) (d) ( 23 , )at the point (4,0)?(c) 14(d)14

17) Let R be the region bounded by the graphs of y 2x and y x2 . What isthe area of R?(a) 0(b) 4(c)23(d)4318) A block of ice in the shape of a cube melts uniformly maintaining its shape.The volume of a cube given a side length is given by the formula V S 3 . Atthe moment S 2 inches, the volume of the cube is decreasing at a rate of 5cubic inches per minute. What is the rate of change of the side length of thecube with respect to time, in inches per minute, at the moment when S 2inches?(a) 512(b)512(c) 125(d)125

x19) Let g be the function given by f (x) (3t 6t2 )dt .What is the x-coordinate1of the point of inflection of the graph of f?(a) 14(b)14(c) 0(d)1220) sin(3x)dx (a) 3cos(3x) C(b) 13 cos(3x) C(c) 3cos(3x) C(d) 13 cos(3x) C21) How many removable discontinuities does the graph of(a) one(b) two(c) threehave?(d) four

622) If f (x)dx 5 and4(a) 124 f (x)dx 810(b) 1210then what is the value of (4f (x) 10)dx6(c) 52(d) 6223) What is the equation of the line tangent to the graph y e2x at x 1 ?(a) y 2e2 e2 (x 1)(b) y e2 2e2 (x 1)(c) y 2e2 e2 (x 1)(d) y e2 2e2 (x 1)

24) (a) 2(b) 2(c) 0(d) nonexistent25) The graph of a function, f is shown above. Let h(x) be defined ash(x) (x 1) · f (x) . Find h′(4).(a) 6(b) 2(c) 4(d) 14

26) A region R is the base of a solid where f (x) g (x) for all x a x b . Forthis solid, each cross section perpendicular to the x-axis are rectangles withheight 5 times the base. Which of the following integrals gives the volume ofthis solid?b(a) 25 (g(x) f (x))2 dxab(b) 5 (g(x) f (x))dxab(c) 5 (f (x) g (x))dxab(d) 5 (f (x) g (x))2 dxa27) If(a) and if y 4 when x 2, then y 1 22x 14(b) 2x2 8(c) x2 6(d) x2 12

28) Ifthen f ′(x) (a)(b)(c)(d)29) (2x 3)(x2 3x)4 dx (a) 15 (x2 3x)5 C1(b) 10(x2 3x)5 C(c) (x2 3x)5 C(d) 5(x2 3x)5 C

x1467f(x)3528g(x)210530) Two differentiable functions, f and g have the property that f (x) g (x) forall real numbers and form a closed region R that is bounded from x 1 tox 7. Selected values of f and g are in the table above. Estimate the areabetween the curves f and g between x 1 and x 7 using a Right Riemannsum with the three sub-intervals given in the table.(a) 13(b) 19(c) 21(d) 27

Calculator Active Multiple Choice76) Let f be the function defined above, where k is a constant. For what valueof k, is f(x) continuous at x 1?(a) 92(b) -4(c) 4(d)9277) At time t, 0 t 2, the velocity of a particle moving along the x-axis is2given by v (t) et 2 . What is the total distance traveled by the particleduring the time interval 0 t 2?a) 12.453(b) 13.368(c) 51.598(d) 53.598

552878) Let f be a continuous function such that f (x) dx 4 and f (x) dx 32then f (x) dx 8(a) 7(b) -1(c) 1(d) 779) Let f be a twice-differentiable function defined by the differentiablexfunction g such that f (x) g (x) dx . It is also known that g(x) is always 2concave up, decreasing, and positive for all real numbers. Which of thefollowing could be false about f(x)?(a) f(x) is concave down for all x(b) f(x) is increasing for all x(c) f(x) is negative for all x(d) f(x) 0 for some x in the real numbers

80) Let f be the function defined by f (x) ecos(x) sin(x) . For what value of x,on the interval (0,4), is the average rate of change of f(x) equal to theinstantaneous rate of change of f(x) on [0,4]?(a) 0.723(b) 1.901(c) 1.966(d) 2.11081) Let f and g be differentiable functions such that f (g(x)) x for all x.If f (1) 3 and f ′(1) - 4, what is the value of g’(3)?(a)13(b) 13(c)14(d) 14

82) The graph of y f (x) is shown above. Which of the following could be thegraph of y f ′(x) ?(a)(b)(c)(d)

x-503f(x)64-283) The table above gives values of a differentiable function f(x) at selected xvalues. Based on the table, which of the following statements about f(x) couldbe false?(A) There exists a value c, where -5 c 3 such that f(c) 1(B) There exists a value c, where -5 c 3 such that f’(c) 1(C) There exists a value c, where -5 c 3 such that f(c) -1(D) There exists a value c, where -5 c 3 such that f’(c) -184) The function f is the antiderivative of the function g defined byg (x) ex ln(x) 2x2 . Which of the following is the x-coordinate of locationof a relative maximum for the graph of y f(x).(a) 1.312(b) 2.242(c) 2.851(d) 2.970

85) The function f is continuous on the closed interval [-2,2]. The graph of f’,the derivative of f, is shown above. On which interval(s) is f(x) increasing?(a) [-1,1](b) [-2, -1] and [1, 2](c) [0, 2](d) [-2, 0]86) Let f be the function with the first derivative f ′(x) If g(3) 4, what is the value of g(6)?(a) 2.328(b) 3.918(c) 6.328 sin(x) cos(x) 2(d) 7.918.

87) The velocity of a particle for t 0 is given by v (t) ln(t3 1) . What is theacceleration of the particle at t 4 ?(a) 0.738(b) 3.436(c) 4.174(d) 8.23288) The function f is differentiable and f(4) 3 and f’(4) 2. What is theapproximation of f(4.1) using the tangent line to the graph of f at x 4 ?(a) 2.6(b) 2.8(c) 3.2(d) 3.489) Patrick is climbing stairs and the rate of stair climbing is given by thedifferentiable function s, where s(t) is measured in stairs per second and t ismeasured in seconds. Which of the following expressions gives Patrick'saverage rate of stairs climbed from t 0 to t 20 seconds?20(a) s(t)dt0(b)12020 s(t)dt020(c) s′(t)dt0(d)12020 s′(t)dt0

90) The graph of f is shown above. Which of the following statements is false?(a) f (1) lim f (x)x 1(b) f (3) lim f (x)x 3(c) f(x) has a jump discontinuity at x 2(d) f(x) has a removable discontinuity at x 4

Free Response Question with Calculator1) A water bottle has a height of 18 centimeters and has circular crosssections. The radius, in centimeters, of a circular cross section of the bottle atheight h centimeters is given by the piecewise function:R(h) (a) Is R(h) continuous at h 12? Justify your response.(b) Find the average value of the radius from h 12 to h 18.

(c) Find the volume of the water bottle. Include units.(d) The water bottle is being filled up at a hydration station. At the instantwhen the height of the liquid is h 14 centimeters, the height is increasing ata rate of 34 centimeters per second. At this instant, what is the rate of changeof the radius of the cross section of the liquid with respect to time?

2) Coal is burning in a furnace, thus exhausting the resource. The rate at whichcoal is burning, measured in pounds per hour, is given by B (t) 4sin( 2t ). Att 2 hours, a worker starts supplying additional coal into the furnace. Therate at which coal is being added, measured in pounds per hour, is given byS (t) 12 10sin( 4πt) . The worker stops adding coal at t 6 hours. At t 025there are 500 pounds of coal in the furnace.(a) Find the total amount of coal added by the worker. Include units.(b) Is the amount of coal in the furnace increasing or decreasing at t 5hours? Explain.

(c) Find S’(4). Explain, with units, the meaning of this in the context of theproblem.(d) Find the amount of coal, in pounds, in the furnace at t 6 hours.

Free Response Question without Calculator3) Graph of f ’The figure above represents the function f ’ the derivative of f over the interval[-4, 6] and satisfies f(4) 2. The graph of f ’ consists of three line segmentsand a semi-circle.(a) Find the value of f(-4).(b) On what interval(s) is f decreasing and concave up? Justify your answer.

(c) State all x-values where f(x) has a horizontal tangent on the open interval(-4, 6). Explain whether f has a relative minimum, relative maximum, orneither at each of those x-values.3(d) Evaluate f ′′(2x)dx2

4) Consider the differential equation, the derivative of f(x),andwhere f(2) 1.(a) On the axes below, sketch a slope field for the given differential equation atthe six points indicated.(b) Find an expression for f(x) given that f(2) 1.(c) Find

x1469f(x)103525) Let g(x) be a twice-differentiable function defined by a differentiablex2function f, such that g (x) 2x f (t)dt . Selected values of f(x) are given in1the table above.(a) Use a Left Riemann sum using the subintervals indicated by the table toapproximate g(3).(b) Find g’(x) and evaluate g’(3).

(c) Using the data in the table, estimate f’(3).(d) Explain why there must be a value of f, on 1 x 9 such that f (c) 4

6)xg(x)g’(x)02-411-25/24-35-13Graph of f(x)Let f be a continuous function defined on the interval [-1, 3], and whose graphis given above. Let g be a differentiable function with derivative g’. The tableabove gives the value of g(x) and g’(x) at selected x-values.(a) Let h be the function defined by h(x) f (x) · g (x) . Find h′(0)(b) Let k be the function defined by k (x) g (f (x)) . Find the slope of the linetangent to k(x) at x 52 .

x(c) Let m(x) g ′(t)dt . Find m(5) and m’(5)1(d) Evaluate limx 0g(x) 2f (2x)

Multiple Choice Answers1) A24) C2) C25) A3) B26) D4) B27) D5) A28) B6) B29) A7) C30) B8) D9) D76) C10) C77) B11) B78) D12) B79) C13) B80) B14) C81) D15) C82) D16) B83) B17) D84) A18) A85) C19) B86) D20) D87) A21) A88) C22) A89) B23) D90) A

Free Response Questions Solutions1) A water bottle has a height of 18 centimeters and has circular cross sections. The radius,in centimeters, of a circular cross section of the bottle at height h centimeters is given bythe piecewise function:R(h) (a) Is R(h) continuous at h 12? Justify your response.lim R(h) 3lim R(h) R(12) 3h 12h 12Since lim R(h) lim R(h) R(12) 3, R(h) is continuous at h 12.h 12h 12(b) Find the average value of the ra