1998 AP Calculus AB: Section I, Part A

Transcription

1 55 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number.. What is the -coordinate of the point of inflection on the graph of y 5 4? 5 (B) 0 (C) 0 (D) 5 (E) 0. The graph of a piecewise-linear function f, for 4, is shown above. What is the value of 4 f ( ) d? (B).5 (C) 4 (D) 5.5 (E) 8. d (B) 7 4 (C) (D) (E) ln AP Calculus Multiple-Choice Question Collection 5 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

2 4. If f is continuous for a b and differentiable for a b, which of the following could be false? f ( b) f( a) f() c for some c such that ac b. b a (B) f( c) 0 for some c such that ac b. (C) f has a minimum value on a b. (D) f has a maimum value on a b. b (E) f ( d ) eists. a 5. sin tdt 0 sin (B) cos (C) cos (D) cos (E) cos 6. If y 0, then when, dy d 7 (B) (C) 7 (D) (E) 7 7. e d e (B) e e e (C) e e (D) e (E) e 8. Let f and g be differentiable functions with the following properties: (i) g ( ) 0 for all (ii) f (0) If h ( ) f( g ) ( ) and h( ) f( ) g( ), then f ( ) f ( ) (B) g ( ) (C) e (D) 0 (E) AP Calculus Multiple-Choice Question Collection 6 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

3 9. The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph shown above. Of the following, which best approimates the total number of barrels of oil that passed through the pipeline that day? 500 (B) 600 (C),400 (D),000 (E) 4, What is the instantaneous rate of change at of the function f given by f( )? (B) 6 (C) (D) (E) 6 b f a. If f is a linear function and 0 a b, then ( d ) 0 (B) (C) ab (D) b a (E) b a. If ln for 0 f( ) ln for 4, then lim f ( ) is ln (B) ln 8 (C) ln6 (D) 4 (E) noneistent AP Calculus Multiple-Choice Question Collection 7 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

4 . The graph of the function f shown in the figure above has a vertical tangent at the point,0 and horizontal tangents at the points, and,. For what values of, 4, is f not differentiable? 0 only (B) 0 and only (C) and only (D) 0,, and only (E) 0,,, and 4. A particle moves along the -ais so that its position at time t is given by what value of t is the velocity of the particle zero? () t t 6t 5. For (B) (C) (D) 4 (E) 5 5. If then F() 0 F( ) t dt, (B) (C) (D) (E) 8 ( ) sin, then f ( ) = 6. If f e cos( e ) (B) cos( e ) e (C) cos( e ) e (D) e cos( e ) (E) e cos( e ) AP Calculus Multiple-Choice Question Collection 8 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

5 7. The graph of a twice-differentiable function f is shown in the figure above. Which of the following is true? f f f (B) f f f (C) f f f (D) f f f (E) f f f 8. An equation of the line tangent to the graph of y cos at the point 0, is y (B) y (C) y (D) y (E) y 0, then the graph of f has inflection points when 9. If f ( ) only (B) only (C) and 0 only (D) and only (E), 0, and only 0. What are all values of k for which k d 0? (B) 0 (C) (D) and (E), 0, and. If dy ky and k is a nonzero constant, then y could be dt e kty (B) e kt kt (C) e (D) kty 5 (E) ky AP Calculus Multiple-Choice Question Collection 9 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

6 . The function f is given by increasing? 4 f( ). On which of the following intervals is f, (B) (C) 0, (D),0, (E), AP Calculus Multiple-Choice Question Collection 0 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

7 . The graph of f is shown in the figure above. Which of the following could be the graph of the derivative of f? AP Calculus Multiple-Choice Question Collection Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

8 4. The maimum acceleration attained on the interval 0t by the particle whose velocity is given by vt () t t t 4is 9 (B) (C) 4 (D) (E) What is the area of the region between the graphs of y and y from = 0 to =? (B) 8 (C) 4 (D) 4 (E) 6 0 f ( ) k 6. The function f is continuous on the closed interval 0, and has values that are given in the table above. The equation f( ) must have at least two solutions in the interval 0, if k 0 (B) (C) (D) (E) 7. What is the average value of on the interval y 0,? 6 9 (B) 5 9 (C) 6 (D) 5 (E) 4 8. If f ( ) tan( ), then f 6 (B) (C) 4 (D) 4 (E) 8 AP Calculus Multiple-Choice Question Collection Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

9 Section I, Part B 50 Minutes Graphing Calculator Required Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approimates the eact numerical value. () Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number. 76. The graph of a function f is shown above. Which of the following statements about f is false? f is continuous at a. (B) f has a relative maimum at a. (C) a is in the domain of f. (D) lim f ( ) a is equal to lim f ( ). a (E) lim f ( ) eists. a 77. Let f be the function given by f ( ) e and let g be the function given by value of do the graphs of f and g have parallel tangent lines? g ( ) 6. At what 0.70 (B) (C) 0.9 (D) 0.0 (E) 0.58 AP Calculus Multiple-Choice Question Collection Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

10 Section I, Part B 78. The radius of a circle is decreasing at a constant rate of 0. centimeter per second. In terms of the circumference C, what is the rate of change of the area of the circle, in square centimeters per second? 0. C (B) 0.C 0.C (C) (D) 0. C (E) 0. C 79. The graphs of the derivatives of the functions f, g, and h are shown above. Which of the functions f, g, or h have a relative maimum on the open interval a b? f only (B) g only (C) h only (D) f and g only (E) f, g, and h 80. The first derivative of the function f is given by does f have on the open interval 0,0? cos f( ). How many critical values 5 One (B) Three (C) Four (D) Five (E) Seven AP Calculus Multiple-Choice Question Collection 4 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

11 8. Let f be the function given by f ( ) Section I, Part B. Which of the following statements about f are true? I. f is continuous at 0. II. f is differentiable at 0. III. f has an absolute minimum at 0. I only (B) II only (C) III only (D) I and III only (E) II and III only 8. If f is a continuous function and if F ( ) f( ) F() F() (B) F() F() (C) F(6) F() (D) F(6) F() for all real numbers, then f d (E) F(6) F() 8. If a 0, then lim a a a 4 4 is a (B) a (C) 6a (D) 0 (E) noneistent 84. Population y grows according to the equation dy ky dt, where k is a constant and t is measured in years. If the population doubles every 0 years, then the value of k is (B) 0.00 (C) 0.0 (D). (E) AP Calculus Multiple-Choice Question Collection 5 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

12 Section I, Part B f ( ) The function f is continuous on the closed interval,8 and has values that are given in the table above. Using the subintervals,5, 5,7, and 7,8, what is the trapezoidal approimation of 8 f ( d )? 0 (B) 0 (C) 60 (D) 90 (E) The base of a solid is a region in the first quadrant bounded by the -ais, the y-ais, and the line y 8, as shown in the figure above. If cross sections of the solid perpendicular to the -ais are semicircles, what is the volume of the solid?.566 (B) 4.66 (C) (D) 67.0 (E) Which of the following is an equation of the line tangent to the graph of point where f( )? 4 f ( ) at the y 8 5 (B) y 7 (C) y 0.76 (D) y 0. (E) y Let F( ) be an antiderivative of ln. If F() 0, then F(9) (B) 0.44 (C) 5.87 (D).08 (E), AP Calculus Multiple-Choice Question Collection 6 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

13 89. If g is a differentiable function such that g ( ) 0 for all real numbers and if f ( ) 4 g( ), which 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 relative minima at and at. (D) f has relative maima at and at. (E) It cannot be determined if f has any relative etrema. Section I, Part B 90. If the base b of a triangle is increasing at a rate of inches per minute while its height h is decreasing at a rate of inches per minute, which of the following must be true about the area A of the triangle? A is always increasing. (B) A is always decreasing. (C) A is decreasing only when b (D) A is decreasing only when b (E) A remains constant. h. h. 9. Let f be a function that is differentiable on the open interval f (9) 5, which of the following must be true?,0. If f() 5, f(5) 5, and I. f has at least zeros. II. The graph of f has at least one horizontal tangent. III. For some c, c5, f( c). None (B) I only (C) I and II only (D) I and III only (E) I, II, and III 9. If 0 k and the area under the curve y cos from k to is 0., then k.47 (B).44 (C).77 (D).0 (E) 0.46 AP Calculus Multiple-Choice Question Collection 7 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.