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1 ANSWER KEY Packet # AP Calculus AB Eam Multiple Choice Questions Answers are on the last page. NO CALCULATOR MAY BE USED IN THIS PART OF THE EXAMINATION. On the AP Eam, you will have minutes to answer these 8 questions.. What is the -coordinate of the point of inflection on the graph (B) (C) y 0 0 = +? (E) 0 0. The graph of a piecewise-linear function f, for, is shown. What is the value of f ( ) d? 6 (B) (C) (E). d = (B) 7 (C) 6 7 (E) 7 ln 8. f is a continuous function for a b but not differentiable for some c such that a < c < b. Which of the following could be true? = cis a vertical asymptote of the graph of f. NOT CONTINUOUS f f c NOT CONTINUOUS (B) lim ( ) ( ) c (C) The graph of f has a cusp at = c. f c is undefined. NOT CONTINUOUS ( ) (E) None of the above.. cos t dt = π sin (B) sin (C) sin + sin (E) sin
2 6. If + = 7, then when =, dy d = y y 9 (B) 0 (C) 8 (E) 7 7. e + d = 8 6 e + (B) 6 e + (C) e e e e + 6 (E) 6 7 e + 8. Let f and g be differentiable functions with the following properties: I. f ( ) < 0 for all II. g ( ) = f ( ) f '( ) If h( ) = and h' ( ) =, then g( ) = g( ) g( ) (B) f ( ) (C) f ( ) f '( ) 0 (E) 9. The production rate of cola, in thousands of gallons per hour, at the production plant on July is shown on the graph. Of the following, which best approimates the total number of thousands of gallons of cola that were produced that day? 800 (B) 00 (C) (E) What is the instantaneous rate of change at = of the function f given by f ( ) = +? 7 (B) 6 (C) (E) 7 6
3 . If f is a linear function and 0 a b b < <, then '' ( ) a f d = 0 (B) (C) ab m( a b) (E) a b. If f ( ) = ln, 0< ln, <, then lim f ( ) is ln 9 (B) ln 7 (C) ln + ln (E) noneistent. The graph of the function f shown in the figure has a horizontal tangent at the point (, ) and a cusp at (,0 ). For what values of, < <, is f not differentiable? 0 only (B) 0 and only (C) - and 0 only -, 0, and (E) - and only. A particle moves along the -ais so that its position at time t is given by ( ) velocity of the particle zero?. (B) (C). (E). t = t 7t +. For what value of t is the. If ( ), then ' ( ) F = t + dt F = 9 (B) 9 (C) 7 7 (E) 7 6. If f ( ) = cos e, then '( ) f = sin e (B) sin e (C) sin e sin e (E) e sin e
4 7. The graph of a twice differentiable function f is shown. Which of the following is true? f ( ) < f ' ( ) < f '' ( ) (B) f ( ) < f '' ( ) < f ' ( ) (C) f ' ( ) < f ( ) < f '' ( ) f '' ( ) < f ( ) < f ' ( ) (E) f '' ( ) < f ' ( ) < f ( ) 8. An equation of the line tangent to the graph of y = cos at = 0 is y = (B) y = (C) y = + y = (E) y = 9. If f ''( ) ( )( ) ( ) = +, then the graph of f has inflection points when = - only (B) only (C) and only - and only (E) -, and 0. What are the values of k for which k d = 0? - (B) 0 (C) - and (E) -, 0 and. If dy = my and m is a nonzero constant, then y could be dt e mty (B) e mt mt (C) e + mty + (E) m y +. The function f is given by ( ) 6 f = +. On which of the following intervals is f decreasing? (B), 0, (E), (,0) ( ) (C) 0,
5 . The graph of f is shown in the figure at the right. Which of the following could be the graph of the derivative of f?. The minimum acceleration attained on the interval 0 t by the particle whose velocity is given by v t = t t t + is ( ) 6 (B) 0 (C) 8 (E). What is the area of the region between the graphs of y y = from = 0 to =? = and 0 (B) (C) 8 (E) 0 6. The function f is continuous on the closed interval [, ] and has the values given in the table. The equation ( ) must have at least two solutions in the interval [, ] if k = f() k f = (B) (C) 9 (E)
6 7. What is the average value of y 0,? = + 9 on the interval [ ] 98 (B) 9 (C) 7 8 (E) If f ( ) tan π =, then f ' = 9 (B) (C) 6 (E) 6
7 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS IN THIS PART OF THE EXAM. On the AP Eam, you will have 0minutes to answer these 7 questions. 9. The graph of a function f is shown. Which of the following statements about f is false? lim f ( ) a eists (B) lim f ( ) lim f ( ) + a a (C) f ( a ) eists = eists f is continuous at = a (E) f has a relative minimum at = a g 0. Let f ( ) = e and ( ) =. At what value of do the graphs of f and g have parallel tangents? 0. (B) 0.66 (C) (E) 0.6. The side of a cube is increasing at a constant rate of 0. cm per second. In terms of the surface area S, what is the rate of change of the volume of the cube, in cubic centimeters per second? 0.S (B) 0.S (C) 0.6S 0.S (E) 0.008S. The graphs of the derivatives of the functions f, g and h are shown. Which of the functions has a relative minimum on the open interval a < < b? y = f '( ) y = g' ( ) y = h' ( ) f only (B) g only (C) h only f and h only (E) f, g and h. The first derivative of the function f is given by f '( ) interval ( 0,0 )? sin =. How many critical values does f have on the open 9 One (B) Two (C) Three Four (E) Si
8 . Let f be the function given by ( ) I. f is continuous at = 0 II. f is differentiable at = 0 III. f has an absolute minimum at = 0 f =. Which of the following statements about f are true? I only (B) II only (C) III only I and II only (E) I and III only. If f is a continuous function and if F' ( ) = f ( ) for all real numbers, then f ( ) F 6 F F( ) F( ) (B) F( ) F( ) (C) F( 6) F( ) F( 6) F( ) (E) ( ) ( ) d = 6. If a 0, then a lim a a 6 6 is noneistent (B) 0 (C) a (E) a a 7. Population P grows according to the equation dp kp dt =, where k is a constant and t is measured in years. If the population doubles every years, the value of k is approimately.8 (B).79 (C) (E) The function f is continuous on the closed interval [, 9] and has the values given in the table. Using the subintervals 9 [, ], [, 6] and [ 6,9 ], what is the value of the trapezoid approimation of ( ) 6 9 f() 0 0? f d 0 (B) 0 (C) 7. (E) 6. **9. The base of a solid region in the first quadrant bounded by the -ais, the y-ais, and the line + y = 9, is shown in the figure. If cross sections of the solid perpendicular to the y-ais are isosceles right triangles with the hypotenuses in the y-plane, what is the volume of the solid? 6.7 (B). (C) (E) 0.
9 6 0. Which of the following is an equation of the line tangent to the graph of f ( ) = at the point where f '( ) =? y =.0 (B) y = 0.86 (C) y = y = (E) y = +.0 ( ) ln. Let F() be an antiderivative of. If F ( ) = 0, then F ( 8) =.6 (B) (C) (E) If g is a differentiable function such that ( ) 0 of the following is true? f has a relative maimum at = - and a relative minimum at =. (B) f has a relative minimum at = - and a relative maimum at =. (C) f has a relative maimum at = and a relative minimum at = -. f has a relative minimum at = and a relative minimum at = -. (E) It cannot be determined if f has any relative etrema. g < for all real numbers and if f '( ) ( ) g( ) =, which NEGATIVE. If the length l of a rectangle is decreasing at a rate of inches per minute while its width w is increasing at a rate of inches per minute, which of the following must be true about the area A of the rectangle? A is always increasing. (B) A is always decreasing. (C) A is increasing only when l A is increasing only when l (E) A remains constant. > w. < w.. Let f be a function that is differentiable on the open interval (, 7). If f ( ) =, f ( ) = and ( ) which of the following must be true? I. f has at least two zeros. IVT II. f has a relative minimum at =. < c< 6, f c =. IVT III. For some c, ( ) f 6 = 8, I only (B) II only (C) I and II only I and III only (E) I, II and III π π. If 0 k and the area under the curve y = sin from = k to = is 0.7, then k =.8 (B) 0.88 (C) (E) 0.
10 AP Calculus AB Eam Multiple Choice Questions Packet # Answers. B 9. D. E 0. B. D. A. C. D. D. B 6. A. E 7. B. E 8. E 6. E 9. C 7. E 0. E 8. D. A 9. D. E 0. A. B. A. C. B. A. C 6. E. D 7. C. C 8. D 9. D 0. D. B. E. A. D. D 6. A 7. E 8. D
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