1993 AP Calculus AB: Section I

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1 99 AP Calculus AB: Section I 90 Minutes Scientific Calculator 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.. If f( ) =, then f () = Which of the following represents the area of the shaded region in the figure above? d ( ( )) f ( ydy ) c ( b a) [ f( b) f( a) ] ( d c) [ f( b) f( a) ] b d f d f ( b) f ( a) a. n 5n lim n n n + is 5 noneistent AP Calculus Multiple-Choice Question Collection 78 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

2 99 AP Calculus AB: Section I. If + y+ y = 7, then in terms of and y, dy d = + y + y + y + y + y + y + y y + y 5. If the function f is continuous for all real numbers and if then f ( ) = f( ) = + when, 0 6. The area of the region enclosed by the curve y =, the -ais, and the lines = and = is 5 6 ln ln ln ln 6 7. An equation of the line tangent to the graph of + y = at the point (, 5 ) is y = 8 + y = 8 y = 6 + y = 66 + y = AP Calculus Multiple-Choice Question Collection 79 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

3 99 AP Calculus AB: Section I 8. If y = tan cot, then dy d = sec csc sec csc sec + csc sec csc sec + csc 9. If h is the function given by h ( ) = f( g ( )), where f( ) = and g ( ) =, then h ( ) = 0. If f ( ) = ( ) sin, then f (0) = 0. The acceleration of a particle moving along the -ais at time t is given by at () = 6t. If the velocity is 5 when t = and the position is 0 when t =, then the position () t = 9t + t t+ t t + t+ 6 t t + 9t 0 6t t 77t+ 55. If f and g are continuous functions, and if f( ) 0 for all real numbers, which of the following must be true? b b b I. f ( gd ) ( ) f( d ) gd ( ) = a a a b b b a a a II. ( ) f ( ) + g ( ) d= f( d ) + gd ( ) b III. f ( ) d= f( ) d a b a I only II only III only II and III only I, II, and III AP Calculus Multiple-Choice Question Collection 80 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

4 99 AP Calculus AB: Section I. The fundamental period of cos( ) is π π 6π. d = C + + C + + C ln + + C ln( + ) + C 5. For what value of does the function f( ) = ( )( ) have a relative maimum? The slope of the line normal to the graph of y = ln(sec ) at π = is noneistent AP Calculus Multiple-Choice Question Collection 8 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

5 99 AP Calculus AB: Section I 7. ( + ) d = ( + ) ( + ) 6 + C + C + + C ( + ) 5 + C C 5 8. If f( ) = sin, then there eists a number c in the interval π π < < that satisfies the conclusion of the Mean Value Theorem. Which of the following could be c? π π 5π 6 π π 9. Let f be the function defined by about f is true? f( ) = for 0, for > 0. Which of the following statements f is an odd function. f is discontinuous at = 0. f has a relative maimum. f (0) = 0 f ( ) > 0 for 0 AP Calculus Multiple-Choice Question Collection 8 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

6 99 AP Calculus AB: Section I 0. Let R be the region in the first quadrant enclosed by the graph of y = ( + ), the line = 7, the -ais, and the y-ais. The volume of the solid generated when R is revolved about the y -ais is given by 7 π ( + 0 ) d 7 0 π ( + ) d π ( + 0 ) d 0 π ( + ) d 7 π ( y ) dy 0. At what value of does the graph of y = have a point of inflection? 0 At no value of. An antiderivative for + is ( + ) ln( + ) ln + arcsec( ) arctan( ). How many critical points does the function 5 f( ) = ( + ) ( ) have? One Two Three Five Nine. If f( ) = ( ), then f (0) is 0 AP Calculus Multiple-Choice Question Collection 8 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

7 99 AP Calculus AB: Section I d 5. ( ) d = ( ) ( )ln ( )ln ln 6. A particle moves along a line so that at time t, where 0 t π, its position is given by t st ( ) = cost + 0. What is the velocity of the particle when its acceleration is zero? The function f given by f( ) = + is increasing for <, decreasing for < <, increasing for > decreasing for < 0, increasing for > 0 increasing for all decreasing for all decreasing for <, increasing for < <, decreasing for > ( ) 500 d ( ) + d = cos lim θ 0 sin θ θ is 0 8 noneistent 0. The region enclosed by the -ais, the line =, and the curve y = is rotated about the -ais. What is the volume of the solid generated? π π 9 6 π 9 π 5 π AP Calculus Multiple-Choice Question Collection 8 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

8 99 AP Calculus AB: Section I. If ln( ) f( ) = e, then f ( ) = ln( ) e ln( ) e ln( ) 6(ln e ) d 0 = π π π 6 ln ln. If dy = y and if y = when =, then when =, y = d 0. The top of a 5-foot ladder is sliding down a vertical wall at a constant rate of feet per minute. When the top of the ladder is 7 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall? 7 feet per minute 8 7 feet per minute feet per minute feet per minute feet per minute 5. If the graph of then a+ c= a + b y = + c has a horizontal asymptote y = and a vertical asymptote =, AP Calculus Multiple-Choice Question Collection 85 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

9 6. If the definite integral 0 99 AP Calculus AB: Section I e d is first approimated by using two inscribed rectangles of equal width and then approimated by using the trapezoidal rule with n =, the difference between the two approimations is If f is a differentiable function, then f ( a) is given by which of the following? I. II. III. lim h 0 lim a lim a f ( a+ h) f( a) h f ( ) f( a) a f ( + h) f( ) h I only II only I and II only I and III only I, II, and III 8. If the second derivative of f is given by f ( ) = cos, which of the following could be f ( )? + cos + cos + + cos + sin + + sin + 9. The radius of a circle is increasing at a nonzero rate, and at a certain instant, the rate of increase in the area of the circle is numerically equal to the rate of increase in its circumference. At this instant, the radius of the circle is π π AP Calculus Multiple-Choice Question Collection 86 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

10 99 AP Calculus AB: Section I 0. The graph of y = f( ) is shown in the figure above. Which of the following could be the graph of y = f ( )?. d cos( udu d π ) is 0 0 sin π cos( π ) π cos( π ) πcos( π ). A puppy weighs.0 pounds at birth and.5 pounds two months later. If the weight of the puppy during its first 6 months is increasing at a rate proportional to its weight, then how much will the puppy weigh when it is months old?. pounds.6 pounds.8 pounds 5.6 pounds 6.5 pounds AP Calculus Multiple-Choice Question Collection 87 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

11 99 AP Calculus AB: Section I. f( ) d= f( ) f ( ) d f ( ) ( ) f d f( ) f( ) + C f( ) f ( ) d f ( d ). What is the minimum value of f ( ) = ln? e 0 f ( ) has no minimum value. e 5. If Newton s method is used to approimate the real root of + = 0, then a first approimation = would lead to a third approimation of = AP Calculus Multiple-Choice Question Collection 88 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

12 99 Answer Key 99 AB 99 BC. C. B. D. A 5. A 6. D 7. B 8. E 9. E 0. D. C. B. A. A 5. D 6. B 7. E 8. D 9. E 0. B. C. E. C. A 5. C 6. D 7. C 8. B 9. C 0. C. E. A. B. D 5. E 6. D 7. C 8. A 9. D 0. C. D. B. B. C 5. B. A. C. E. B 5. D 6. A 7. A 8. B 9. D 0. E. E. E. C. B 5. D 6. A 7. A 8. B 9. B 0. E. A. B. D. C 5. D 6. B 7. C 8. A 9. E 0. C. A. B. A. E 5. A 6. E 7. B 8. C 9. C 0. C. C. E. A. E 5. D AP Calculus Multiple-Choice Question Collection 57 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

13 99 Calculus AB Solutions. C f ( ) = ; f () = = =. B Summing pieces of the form: (vertical) (small width) b Area = ( ( )) a d f d. D Divide each term by. A ( ) n., vertical ( d f( ) ) 5 n 5n lim = lim n = n n n + n + n n + y+ y + 6y y = 0; y (+ 6 y ) = ( + y) + y + y y = = + 6y + y =, width = 5. A ( + )( ) lim = lim = lim ( ) =. + + For continuity f ( ) must be. Area = = ln = ln ln = ln 6. D d ( ) 7. B 8. E ( ) (+ ) y = ; y () =. Tangent line: y 5 = ( ) + y = 8 ( ) y = sec + csc 9. E ( ) h ( ) = f = = 0. D. C f ( ) = ( ) sin + ( ) cos ; f (0) = ( ) 0 + = at ( ) = 6t ; vt ( ) = t t+ Cand v() = 5 5 = C; vt ( ) = t t+ () t = t t + t+ K;Since () = 0, K = 6; () t = t t + t+ 6. AP Calculus Multiple-Choice Question Collection 06 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

14 99 Calculus AB Solutions. B The only one that is true is II. The others can easily been seen as false by eamples. For eample, let f( ) = and g= ( ) with a = 0 and b =. Then I gives = and III gives =, both false statements.. A period = π π = B. A Let u = +. Then / / d = u du = u + C = + + C + 5. D 6. B f ( ) = ( ) + ( )( ) = ( )( 7); f ( ) changes from positive to negative at 7 =. sec tan y = = tan ; y ( π ) = tan( π ) =. The slope of the normal line sec = y ( π ) 7. E Epand the integrand. 5 ( + ) d = ( + + ) d = C 5 π π π π f f sin sin 0 8. D Want c so that f () c = = =. π π π π c f () c = cos = 0 c=π 9. E The only one that is true is E. A consideration of the graph of y = f( ), which is a standard cubic to the left of 0 and a line with slope to the right of 0, shows the other options to be false. AP Calculus Multiple-Choice Question Collection 07 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

15 99 Calculus AB Solutions 0. B Use Cylindrical Shells which is no part of the AP Course Description. The volume of each 7 shell is of the form ( πrh) with r = and h= y. Volume = π ( + ) d. 0. C. E 5 5 y = ; y = + ; y = 6 = 6 ( ). The only domain value at which there is a sign change in y is =. Inflection point at =. d = d = d = tan ( ) + C + ( + ) + + ( ). C A quick way to do this problem is to use the effect of the multiplicity of the zeros of f on the graph of y = f( ). There is point of inflection and a horizontal tangent at =. There is a horizontal tangent and turning point at =. There is a horizontal tangent on the interval (,). Thus, there must be critical points. Also, f ( ) = ( ) ( + ) (9 7). f ( ) =, f (0) = ( ) ( ) =. A ( ) ( ) d 5. C ( ) = ln d 6. D 7. C 8. B vt ( ) = sin t t; at ( ) = cost = 0 at t= cos ( ) =.8; v(.8) =.5587 f ( ) = + > 0. Thus f is increasing for all = ( ) d ( ) d ( ) d ( ) d d = ( ) = = =.96 ln ln ln ln AP Calculus Multiple-Choice Question Collection 08 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

16 99 Calculus AB Solutions 9. C Use L Hôpital s Rule (which is no longer part of the AB Course Description). cosθ sinθ lim = lim = lim = sin θ sinθcosθ cosθ θ 0 θ 0 θ 0 A way to do this without L Hôpital s rule is the following cosθ cosθ cosθ lim = lim = lim = lim = sin θ ( cos θ) ( cos θ )( + cos θ ) ( + cos θ) θ 0 θ 0 θ 0 θ 0 0. C Each slice is a disk whose volume is given by. E π d d Volume =π ( ) =π = = π. 6 ln( ) ln( ) 6 5 f ( ) = e = e = ; f ( ) = 6 π r, where r =.. A a du u u = sin + C, a> 0 a π = sin = sin sin (0) = d 0 0. B Separate the variables. y dy = d; = + C; y =. Substitute the point (, ) y + C to find the value of C. Then = C =, so y =. When =, y =. + C. D Let and y represent the horizontal and vertical sides of the triangle formed by the ladder, the wall, and the ground. d dy d d 7 + y = 5; + y = 0; () + (7)( ) = 0; =. dt dt dt dt 8 5. E For there to be a vertical asymptote at =, the value of c must be. For y = to be a horizontal asymptote, the value of a must be. Thus a+ c= D Rectangle approimation = 0 e + e = + e + e+ e /. Trapezoid approimation. = ( ) Difference = ( e ) / = AP Calculus Multiple-Choice Question Collection 09 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

17 99 Calculus AB Solutions 7. C I and II both give the derivative at a. In III the denominator is fied. This is not the derivative of f at a, ( ) a+ h, f( a+ h). =. This gives the slope of the secant line from ( a f a ) to ( ) 8. A f ( ) = sin + C, f( ) = + cos+ C+ K. Option A is the only one with this form. 9. D da dr dc dr A=πr and C= π r; = π r and = π. For dt dt dt dt da dc =, r =. dt dt 0. C The graph of y = f ( ) is symmetric to the y-ais. This leaves only options C and E. For > 0, and are the same, so the graphs of f ( ) and ( ) option C.. D Answer follows from the Fundamental Theorem of Calculus. f must be the same. This is.5. B This is an eample of eponential growth. We know from pre-calculus that w = is an eponential function that meets the two given conditions. When t =, w =.60. Using calculus the student may translate the statement increasing at a rate proportional to its weight to mean eponential growth and write the equation w= e kt. Using the given conditions,.5 = e k ; ln(.75) t ln(.75) ln(.75) = k; k = ; w= e. When t =, w =.60.. B Use the technique of antiderivative by parts, which is no longer in the AB Course Description. The formula is udv= uv vdu u = f and dv = d. This leads to f ( ) d = f ( ) f ( ) d.. Let ( ) t. C f ( ) = ln + ; f ( ) changes sign from negative to positive only at f( e ) = e =. e = e. AP Calculus Multiple-Choice Question Collection 0 Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

18 99 Calculus AB Solutions 5. B Let f ( ) = +. Then Newton s method (which is no longer part of the AP Course Description) gives f n n n n+ = n = n f n n + ( ) + ( ) + = = = = = AP Calculus Multiple-Choice Question Collection Copyright 005 by College Board. All rights reserved. Available at apcentral.collegeboard.com.