2008 AP Calculus AB Multiple Choice Exam

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1 008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active

2 AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active

3 AP Calculus 008 Multiple Choice. ( )( ) ( )( + ) lim is (A) (B) (C) (D) (E) noneistent. d = (A) ln + C (B) ln + C (C) + C (D) + C (E) + C

4 AP Calculus 008 Multiple Choice. If ( ) ( )( ) f, = + then '( ) (A) 6( + ) (B) 6( )( + ) (C) ( + ) ( + ) (D) ( + ) ( ) (E) ( )( + ) f = ( ) 4. sin ( ) cos( ) + d = + + C (A) cos( ) sin ( ) + + C (B) cos( ) sin ( ) (C) cos( ) sin ( ) + + C (D) cos( ) sin ( ) + C (E) cos( ) sin ( ) + + C

5 AP Calculus 008 Multiple Choice lim is (A) (B) 0 (C) (D) 5 + (E) noneistent 4 if f ( ) = if = 6. Let f be the function defined above. Which of the following statements about f are true? I. f has a limit at =. II. f is continuous at =. III. f is differentiable at =. (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III

6 AP Calculus 008 Multiple Choice 7. A particle moves along the -ais with velocity given by v( t) = t + 6t for time t 0. If the particle is at position = at time t = 0, what is the position of the particle at t =? (A) 4 (B) 6 (C) 9 (D) (E) 8. If f ( ) ( ) π = cos, then f ' = 9 (A) (B) (C) (D) (E)

7 AP Calculus 008 Multiple Choice 0 9. The graph of the piecewise linear function f is shown in the figure above. If g = f t dt, which of the following values is greatest? ( ) ( ) Graph of f (A) g ( ) (B) g ( ) (C) g ( 0) (D) g ( ) (E) g ( )

8 AP Calculus 008 Multiple Choice y Graph of f 0. The graph of function f is shown above for 0. f the following, which has the least value? length (A) f ( ) d (B) Left Riemann sum approimation of f ( ) d with 4 subintervals of equal length (C) Right Riemann sum approimation of f ( ) d with 4 subintervals of equal length (D) Midpoint Riemann sum approimation of ( ) f d with 4 subintervals of equal (E) Trapezoidal sum approimation of f ( ) d with 4 subintervals of equal length

9 AP Calculus 008 Multiple Choice y Graph of f. The graph of a function f is shown above. Which of the following could be the graph of f ', the derivative of f? (A) y (B) y (C) y (D) y (E) y

10 AP Calculus 008 Multiple Choice ( / ). If f ( ) = e, then '( ) f = (A) ( / ) e ln (B) ( / ) e (C) ( / ) e (D) ( / ) e (E) e ( / ) d d. If f ( ) = +, then ( f ( ln ) ) = (A) ln + (B) ln + (C) ln + (D) ln + (E) +

11 AP Calculus 008 Multiple Choice 0 f "( ) The polynomial function f has selected values of its second derivative f " given in the table above. Which of the following statements must be true? (A) f is increasing on the interval ( 0, ). (B) f is decreasing on the interval ( 0, ). (C) f has a local maimum at =. (D) The graph of f has a point of inflection at =. (E) The graph of f changes concavity in the interval ( 0, ). 5. d = 4 (A) ( ) C (B) ( ) 4 + C (C) (D) ln 4 ln 4 + C + C (E) arctan + C

12 AP Calculus 008 Multiple Choice 6. If ( y) sin =, then dy d = (A) cos( y) (B) (C) (D) (E) cos cos cos ( y) ( y) ( y) y cos y cos ( y) ( y) ( cos( y) ) y Graph of f 7. The graph of the function f shown above has horizontal tangents at = and = 5. Let g be the function defined by g ( ) ( ) have a point of inflection? = f t dt. For what values of does the graph of g 0 (A) only (B) 4 only (C) and 5 only (D), 4, and 5 (E) 0, 4, and 6

13 AP Calculus 008 Multiple Choice 8. In the y -plane, the line + y = k, where k is a constant, is tangent to the graph of y = + +. What is the value of k? (A) (B) (C) (D) 0 (E) 9. What are all horizontal asymptotes of the graph of 5 + y = in the y -plane? (A) y = only (B) y = 0 only (C) y = 5 only (D) y = and y = 0 (E) y = and y = 5

14 AP Calculus 008 Multiple Choice 0. Let f be a function with a second derivative given by f ( ) ( )( ) the -coordinates of the points of inflection of the graph of f? (A) 0 only (B) only (C) 0 and 6 only (D) and 6 only (E) 0,, and 6 " = 6. What are ( t ) t. A particle moves along a straight line. The graph of the particle s position ( t ) at time t is shown above for 0 < t < 6. The graph has horizontal tangents at t = and t = 5 and a point of inflection at t =. For what values of t is the velocity of the particle increasing? (A) 0 < t < (B) < t < 5 (C) < t < 6 (D) < t < 5 only (E) < t < and 5 < t < 6

15 AP Calculus 008 Multiple Choice. A rumor spreads among a population of N people at a rate proportional to the product of the number of people who have heard the rumor and the number of people who have not heard the rumor. If p denotes the number of people who have heard the rumor, which of the following differential equations could be used to model this situation with respect to time t, where k is a positive constant? (A) dp kp dt = dp dt = (B) kp ( N p) dp dt = (C) kp ( p N ) dp dt = (D) kt ( N t) dp dt = (E) kt ( t N )

16 AP Calculus 008 Multiple Choice. Which of the following is the solution to the differential equation condition y ( ) =? dy = with the initial d y (A) (B) 9 / y = e + 9 / y = e + (C) y = (D) (E) y = 4 y = 4 4. The function f is twice differentiable with f ( ) =, f '( ) = 4, and "( ). value of the approimation of f (.9) using the line tangent to the graph of f at =? (A) 0.4 (B) 0.6 (C) 0.7 (D). (E).4 f = What is the

17 AP Calculus 008 Multiple Choice f ( ) = c + d for c for > 5. Let f be the function defined above, where c and d are constants. If f is differentiable at =, what is the value of c + d? (A) 4 (B) (C) 0 (D) (E) 4 6. What is the slope of the line tangent to the curve y arctan ( 4) =? 4 = at the point at which (A) (B) (C) 0 (D) (E)

18 AP Calculus 008 Multiple Choice 5 y Shown above is a slope field for which of the following differential equations? (A) dy y d = (B) dy y y d = (C) dy y y d = + (D) dy y d = + dy d = + (E) ( )

19 AP Calculus 008 Multiple Choice 8. Let f be a differentiable function such that f ( ) = 5, f ( 6) =, ( ) f '( 6) =. The function g is differentiable and g ( ) f ( ) g '( )? (A) (B) 8 f ' = 8, and = for all. What is the value of (C) 6 (D) (E) The value of g '( ) cannot be determined from the information given.

20 AP Calculus 008 Multiple Choice y 4 5 Graph of f ' 76. The graph of f ', the derivative f, is shown above for 5. n what intervals is f increasing? (A) [, ] only (B) [, ] (C) [, 5] only (D) [0,.5] and [, 5] (E) [, ], [, ], and [4, 5]

21 AP Calculus 008 Multiple Choice y 4 Graph of f 77. The figure above shows the graph of a function f with domain 0 4. Which of the following statements are true? I. lim f ( ) II. lim f ( ) + eists. eists. III. lim f ( ) eists. (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III 78. The first derivative of the function f is defined by f '( ) sin ( ) what interval(s) is f increasing? (A).445 (B).69 (C) (D) and.875 (E) 0 and.69 = for 0. n

22 AP Calculus 008 Multiple Choice If f ( ) d = 7 and f ( ) d = 4, 5 what is the value of ( ) 5 5 (A) (B) (C) 0 (D) (E) f d? 80. The derivative of the function f is given by f ( ) ( ) inflection does the graph of f have on the open interval (, )? ' = cos. How many points of (A) ne (B) Two (C) Three (D) Four (E) Five

23 AP Calculus 008 Multiple Choice 8. If G ( ) is an antiderivative for f ( ) and G ( ) = 7, then ( 4) (A) f '( 4) (B) 7 + f '( 4) 4 (C) f ( t ) dt ( ) 4 (D) + ( ) 7 f t dt 4 (E) ( ) + 7 f t dt G = 8. A particle moves along a straight line with velocity given by v( t) 7 (.0) t t 0. What is the acceleration of the particle at time t =? (A) 0.94 (B) (C) (D) (E) = at time

24 AP Calculus 008 Multiple Choice 8. What is the area enclosed by the curves = and y = + 5? y (A) (B).8 (C) 4.58 (D). (E) y 4 5 Graph of f ' 84. The graph of the derivative of a function f is shown in the figure above. The graph has horizontal tangent lines at =, =, and =. At which of the following values of does f have a relative maimum? (A) only (B) only (C) 4 only (D) and only (E),, and 4

25 AP Calculus 008 Multiple Choice 4 f ( ) f '( ) The table above gives values of a function f and its derivative at selected values of. If is continuous on the interval [ 4, ], what is the value of f ( ) 4 ' d? (A) 4.5 (B).5 (C) 0 (D).5 (E) 4.5 f '

26 AP Calculus 008 Multiple Choice t 0 4 v( t ) The table gives selected values of the velocity, v( t ), of a particle moving along the -ais. At time t = 0, the particle is at the origin. Which of the following could be the graph of the position, ( t ), of the particle for 0 t 4? (A) ( t ) (B) ( t ) t t (C) ( t ) (D) ( t ) t t (E) ( t ) t

27 AP Calculus 008 Multiple Choice 87. An object traveling in a straight line has position ( t ) at time t. If the initial position is ( 0) = and the velocity of the object is v( t) t time t =? = +, what is the position of the object at (A) 0.4 (B).54 (C) 4.5 (D) 6.5 (E) The radius of a sphere is decreasing at a rate of centimeters per second. At the instant when the radius of the sphere is centimeters, what is the rate of change, in square centimeters per second, of the surface area of the sphere? (The surface area S of a sphere with radius r is S = 4π r ) (A) 08π (B) 7π (C) 48π (D) 4π (E) 6π

28 AP Calculus 008 Multiple Choice 89. The function f is continuous for and f ( ) f ( ) = = 0. If there is no c, where < c <, for which f '( c ) = 0, which of the following statements must be true? (A) For k, < < ( ) (B) For k, f ' k > 0. < < ( ) (C) For k, f ' k < 0. < < f '( ) (D) For k, < < '( ) k eists. (E) For some k, where k, f k eists, but f ' is not continuous. < < f '( ) k does not eist. 90. The function f is continuous on the closed interval [, 4] and twice differentiable on the open interval (, 4). If f '( ) = and ( ) following could be a table of values for f? (A) (B) (C) f " < 0 on the open interval (, 4), which of the f ( ) f ( ) f ( ) (D) (E) f ( ) f ( )

29 AP Calculus 008 Multiple Choice cos 9. What is the average value of y = + + on the closed interval [, ]? (A) (B) (C) 0.8 (D) 0.44 (E) miles River City 7 miles 9. A city located beside a river has a rectangular boundary as shown in the figure above. The population density of the city at any point along a strip miles from the river s edge is f persons per square mile. Which of the following epressions gives the population of ( ) the city? 4 (A) f ( ) d 0 4 (B) ( ) 7 f d 0 4 (C) ( ) 8 f d 7 (D) f ( ) d (E) ( ) 4 f d 0

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