2008 CALCULUS AB SECTION I, Part A Time 55 minutes Number of Questions 28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

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1 8 CALCULUS AB 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. After eamining the form of the choices, decide which is best of the choices given and fill in the corresponding oval on the answer sheet. In this test: () Unless otherwise specified, the domain of a function f is assumed to be the set of all real by numbers for which f() is a real number. () The inverse of a trigonometric function f may be indicated using the inverse function notation f or with the prefi arc (e.g. sin = arcsin ).. lim ( )( ) ( )( + ) is (A) - (B) - (C) (E) noneistent. d = (A) ln (B) ln (C) (E) + C. If f( ) = ( )( + ), then f ()= (A) 6 ( + ) (B) 6 ( )( + ) (C) ( + ) (7 6+ ) (E) ( )( + ) ( + ) ( + ). (sin( ) + cos( )) d= (A) cos( ) + sin( ) (B) cos( ) + sin( ) (C) cos( ) + sin( ) cos( ) sin( ) (E) cos( ) + sin( ) lim 6 (A) is (B) (C) 5 (E) noneistent 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 I and II only (E) I, II, and III

2 7. A particle moves along the ais with velocity given by vt () = t + 6t for time t. If the is at position = at time t =, what is the position of the particle at time t =? (A) (B) 6 (C) 9 (E) π 8. If f ( ) = cos( ), then f '( 9 ) = (A) (B) (C) (E) 9. The graph of the piecewise linear function f is shown in the figure above. If g ( ) = ft ( ), which of the following values is greatest? (A) g( ) (B) g( ) (C) g() g() (E) g(). The graph of the function f is shown above for. Of the following, which has the least value? (A) f ( d ) (B) Left Riemann sum approimation of f ( d ) with subintervals of equal length (C) Right Riemann sum approimation of f ( d ) with subintervals of equal length Midpoint Riemann sum approimation of f ( d ) with subintervals of equal length (E) Trapezoidal sum approimation of f ( d ) with subintervals of equal length

3 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) (B) (C) (E). If ( / ) f ( ) = e, then f () = (A) (/ ) ln e (B) (/ ) e (C) ( / ) e (/ ) e (E) e (/ ) d. If f ( ) = +, then d ( f(ln )) = (A) ln + (B) ln+ (C) ln + ln + (E) + f () 5 7. 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 (, ). The graph of f has a point of inflection at =. (B) f is decreasing on the interval (,). (E) The graph of f changes concavity in the interval (,). (C) f has a local maimum at =. 5. d = (A) (B) ( ) ( ) (C) ln ln (E) arctan( ) 6. If sin( y) =, then dy = d (A) (B) cos( y ) cos( y ) cos( y) (C) cos( y) ycos( y) cos( y) (E) y( cos( y)) 7. The graph of the function f shown to the right has horizontal tangents at = and = 5. Let g be the function defined by g ( ) = ft ( ). For what values of does the graph of g have a point of inflection? (A) only (B) only (C) and 5 only,, and 5 (E),, and 6 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) (E)

4 5+ 9. What are all horizontal asymptotes of the graph of y = in the y plane? (A) y = only (B) y = only (C) y = 5 only y = & y= (E) y= & y = 5. Let f be a function with a second derivative given by f ''( ) = ( )( 6). What are the coordinates of the points of inflection of the graph of f? (A) only (B) only (C) and 6 only and 6 only (E),, and 6. A particle moves along a straight line. The graph of the particle s position (t) at time t is shown to the right for < 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) < t < (B) < t < 5 (C) < t < 6 < t < 5 only (E) < t < and 5 < t < 6. 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 dp dp dp dp = kp (B) = kp( N p) (C) = kp( p N) = kt( N t) (E) = kt( t N) dy. Which of the following is the solution to the differential equation d = y with the initial condition y() =? 9 / (A) y = e + 9 / (B) y = e + (C) y = y = (E) y =. The function f is twice differentiable with f() =, f () =, and f () =. What is the value of the approimation of f(.9) using the line tangent to the graph of f at =? (A). (B).6 (C).7. (E). c + d, for f( ) = 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) (B) (C) (E) 6. What is the slope of the line tangent to the curve y = arctan( ) at the point at which = ¼? (A) (B) ½ (C) ½ (E) 7. Shown to the right is a slope field for which of the following differential equations? (A) dy d = y (B) dy d = y y (C) dy d = y+ y dy dy d = y+ (E) d = ( + )

5 8. Let f be a differentiable function such that f() = 5, f(6) =, f () = 8, and f (6) =. The function g is differentiable and g() = f () for all. What is the value of g ()? (A) ½ (B) 8 (C) 6 (E) The value of g () cannot be determined from the information given. END OF PART A OF SECTION I

6 8 CALCULUS AB SECTION I, Part B Time 55 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 scratch work. After eamining the form of the choices, decide which is best of the choices given and fill in the corresponding oval on the answer sheet. In this eam: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from the choices the number that best approimates the eact numerical value. () Unless otherwise specified, the domain of 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 or with the prefi arc (e.g. sin = arcsin ). 76. The graph of f, the derivative of f, is shown to the right for 5. On what intervals is f increasing? (A) [,] only (B) [, ] (C) [,5] [,.5] and [,5] (E) [, ], [,], and [,5] 77. The figure to the right shows the graph of a function f with domain. Which of the following statements are true? I. lim f ( ) eists II. lim f ( ) eists III. lim f ( ) eists + (A) I only (B) II only (C) I and II only I and III only (E) I, II, and III 78. The first derivative of the function f is defined by f '( ) = sin( ) for. On what intervals is f increasing? (A).5 only (B).69 (C) and.875 (E) and If f( ) d= 7 and 5 f( ) d=, what is the value of 5 f ( d )? 5 (A) (B) (C) (E) 8. The derivative of the function f is given by f '( ) = cos( ). How many points of inflection does the graph of f have on the open interval (,)? (A) (B) (C) (E) If G() is an antiderivative for f() and G() = 7, then G() = (A) f () (B) 7 + f () (C) f () t ( 7 + f ( t)) (E) 7 + f () t

7 8. A particle is moving along a straight line with velocity given by vt ( ) = 7 (.) t at time t. What is the acceleration of the particle at time t =? (A).9 (B).55 (C) (E) What is the area enclosed by the curves y = and y = + 5? (A).667 (B).8 (C).58. (E) 8. The graph of the derivative of a function f is shown in the figure to the right. 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) only and only (E),, and f() f () The table above gives values of a function f and its derivative at selected values of. If f is continuous on the interval [, ], what is the value of f '( d )? (A).5 (B).5 (C).5 (E).5 t v(t) 86. The table above gives selected values of the velocity, v(t), of a particle moving along the -ais. At time t =, the particle is at the origin. Which of the following could be the graph of the position, (t), of the particle for t? (A) (B) (C) (E) 87. An object traveling in a straight line has position (t) at time t. If the initial position is () = and the velocity of the object is vt () = + t, what is the position of the object at time t =? (A). (B).5 (C) (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 = π r.) (A) 8π (B) 7π (C) 8π π (E) 6π

8 89. The function f is continuous for and f( ) = f() =. If there is no c, where < c <, for which f (c) =, which of the following statements must be true? (A) For < k <, f (k) >. For < k <, f (k) eists, but f is not continuous (B) For < k <, f (k) <. (E) For some k, where < k <, f (k) does not eist (C) For < k <, f (k) eists 9. The function f is continuous on the closed interval [,] and twice differentiable on the open interval (,). If f () = and f () < on the open interval (,), which of the following could be a table of values for f? (A) (B) (C) (E) cos 9. What is the average value of y = on the closed interval [,]? + + (A).85 (B).9 (C).8. (E).7 9. A city located beside a river has a rectangular boundary as shown in the figure to the right. 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? (A) f ( d ) (B) 7 f ( d ) (C) 8 f ( d ) 7 f ( d ) (E) f() f() f ( d ) f() f() 5 7 f() END OF SECTION I