Banco de preguntas para el ap
- 1. CALCULUS AB SECTION I, Part A Time β 60 minutes Number of questions β 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using the available space for scratchwork. After examiningthe formof the choices,decidewhichisthe bestof the choicesgivenandfill inthe correspondingoval onthe answersheet. No creditwill be givenfor anythingwritteninthe test book. Do not spend too much time on any one problem. In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x ) is a real number. 1. If f ( x ) = 5π₯ 4 3 , then fΒ΄( 8 ) = ( A ) 10 ( B ) 40 3 ( C ) 80 ( D ) 160 3 2. limβ‘β‘ π₯β‘βββ‘ 5π₯2β3π₯+1 4π₯2+2π₯+5 is ( A ) 0 ( B ) 4 5 ( C ) 5 4 ( D ) β 3. If f ( x ) = 3π₯2+π₯ 3π₯2βπ₯ , then fΒ΄( x ) is ( A ) 1 ( B ) 6π₯2+1 6π₯2β1 ( C ) ββ‘6 (β‘3π₯β1β‘)2 ( D ) ββ‘2π₯2 (β‘π₯2βπ₯β‘)2
- 2. 4. lim π₯β‘β0 πππβ‘π₯2 π₯ = ( A ) 1 ( B ) 0 ( C ) π 2 ( D ) The limit does not exist 5. If y = sin ( xy ) , find ππ¦ ππ₯ ( A ) Cos ( xy ) ( B ) π¦β‘πππ β‘(β‘π₯π¦β‘) 1βπ₯β‘πππ β‘(β‘π₯π¦β‘) ( C ) x cos ( xy ) ( D ) π¦β‘πππ β‘(β‘π₯π¦β‘) 1βπ₯β‘ 6. Whichof the followingintegralscorrectlycorrespondstothe areaof the shadedregioninthe figure above? f ( x ) = 5 ; g ( x ) = 1 + x2 ( A ) β« (β‘π₯2 β 4β‘)β‘ππ₯ 2 1 ( B ) β« (β‘4 ββ‘ π₯2β‘)β‘ππ₯ 2 1 ( C ) β« (β‘π₯2 β 4β‘)β‘ππ₯ 5 1 ( D ) β« (β‘4 ββ‘ π₯2β‘)β‘ππ₯ 5 1 7. If h ( x ) = f ( x ) g ( 4x ) , then hΒ΄( 2 ) = ( A ) fΒ΄( 2 ) gΒ΄( 8 ) ( B ) fΒ΄( 2 ) g ( 8 ) + 4 f ( 2 ) gΒ΄( 8 )
- 3. ( C ) fΒ΄( 2 ) g ( 8 ) + f ( 2 ) gΒ΄( 8 ) ( D ) 4 fΒ΄( 2 ) gΒ΄( 8 ) 8. An equation of the line normal to the graph of y = ββ‘3π₯2 + 2π₯β‘ at ( 2 , 4 ) is ( A ) 4x + 7y = 20 ( B ) - 7x + 4y = 2 ( C ) 7x + 4y = 30 ( D ) 4x + 7y = 36 9. β« (β‘3π₯ + 1β‘)2 = 2 0 ( A ) 38 ( B ) 49 ( C ) 343 3 ( D ) 343 10. If f ( x ) = πππ 2 π₯β‘,β‘then f ´´( Ο ) = ( A ) - 2 ( B ) 0 ( C ) 1 ( D ) 2 11. If f ( x ) = 5 π₯2+1 and g ( x ) = 3x , then g ( f ( 2 ) ) = ( A ) 5 37 ( B ) 3 ( C ) 5 ( D ) 37 5 12. β« π₯β‘ββ‘5π₯2 β 4β‘β‘dx =
- 4. ( A ) 1 10 β‘(β‘5π₯2 β 4β‘) 3 2 + C ( B ) 1 15 β‘(β‘5π₯2 β 4β‘) 3 2 + C ( C ) 20 3 β‘(β‘5π₯2 β 4β‘) 3 2 + C ( D ) 3 20 β‘(β‘5π₯2 β 4β‘) 3 2 + C 13. The slope of the line tangent to the graph of 3π₯2 + 5πππ¦ = 12 at ( 2 , 1 ) is ( A ) - 12 5 ( B ) 12 5 ( C ) 5 12 ( D ) - 7 14. The equation y = 2 β 3 Sin π 4 ( x β 1 ) has a fundamental period of ( A ) 1 8 ( B ) 4 π ( C ) 8 ( D ) 2π 15. If f ( x ) = { β‘π₯2 + 5β‘β‘β‘ππβ‘πβ‘ < π 7π₯ β 5β‘β‘β‘β‘ππβ‘β‘πβ‘ β₯ π β‘β‘,β‘ for all real numbers x, which of the following must be true ? I f ( x ) is continuous everywhere II f ( x ) is differentiable everywhere III f ( x ) has a local minimum at x = 2 ( A ) I only ( B ) I and II only ( C ) II and III only ( D ) I , II and III 16. For what value of x does the function f ( x ) = π₯3 β 9π₯2 β 120π₯ + 6 have a local minimum ?
- 5. ( A ) 10 ( B ) 4 ( C ) - 4 ( D ) - 10 17. The accelerationof a particle movingalongthe x β axisat time t is givenby a ( t ) = 4t β 12 . If the velocity is 10 when t = 0 and the position is 4 when t = 0, then the particle is changing direction at ( A ) t = 1 ( B ) t = 3 ( C ) t = 5 ( D ) t = 1 and t = 5 18. What is the area of the region between y = 2x2 and y = 12 β x2 ? ( A ) 0 ( B ) 16 ( C ) 32 ( D ) 48 19 β«β‘(β‘π3β‘πππ₯ +β‘ π3π₯β‘)β‘ππ₯ = ( A ) 3 + π3π₯ 3 + C ( B ) π π₯4 4 + 3π3π₯β‘+ C ( C ) π π₯4 4 +β‘ π3π₯ 3 + C ( D ) π₯4 4 +β‘ π3π₯ 3 + C 20. If f ( x ) = ( x2 + x + 11 ) ββ‘π₯3 + 5π₯ + 121β‘β‘, then fΒ΄( 0 ) = ( A ) 5 2 ( B ) 27 2 ( C ) 22
- 6. ( D ) 247 2 21. If f ( x ) = 53x , then fΒ΄( x ) = ( A ) 53x ( ln 125 ) ( B ) 53π₯ 3ππ5 ( C ) 3 ( 52x ) ( D ) 3 ( 53x ) 22. A solidisgeneratedwhenthe regionin the first quadrant enclosed by the graph of y = ( x2 + 1 )3 ,the line x = 1 ,the x β axis,andthe y β axis. Itsvolume isfoundbyevaluatingwhichof the following integrals? ( A ) πβ‘ β« (β‘π₯2 + 1β‘)3β‘ππ₯ 8 1 ( B ) πβ‘ β« (β‘π₯2 + 1β‘)6β‘ππ₯ 8 1 ( C ) πβ‘ β« (β‘π₯2 + 1β‘)3β‘ππ₯ 1 0 ( D ) πβ‘ β« (β‘π₯2 + 1β‘)6β‘ππ₯ 1 0 23. lim π₯β‘β0 4β‘ sinπ₯cosπ₯βsinπ₯ π₯2 = ( A ) 2 ( B ) 40 3 ( C ) 0 ( D ) undefined 24. If ππ¦ ππ₯ =β‘ 3π₯2+2 π¦ and y = 4 when x = 2, then when x = 3 ; y = ? ( A ) 18 ( B ) 58 ( C ) Β±ββ‘74 ( D ) Β±ββ‘58
- 7. 25. β« ππ₯ 9+β‘π₯2 = ( A ) 3 tan- 1 (β‘ π₯ 3 β‘) + πΆ ( B ) 1 3 tan- 1 (β‘ π₯ 3 β‘) + πΆ ( C ) 1 3 tan- 1 (β‘π₯β‘) + πΆ ( D ) 1 9 tan- 1 (β‘π₯β‘) + πΆ 26. If f ( x ) = cos3 ( x + 1 ) , then fΒ΄( πβ‘) = ( A ) - 3 cos2 ( π + 1β‘) sin β‘(β‘π + 1β‘)β‘ ( B ) 3 cos2 ( π + 1β‘) ( C ) 3 cos2 ( π + 1β‘) sin β‘(β‘π + 1β‘)β‘ ( D ) 0 27. β« π₯β‘ββ‘π₯ + 3 β‘ππ₯ ( A ) 2β‘(β‘π₯+3β‘) 3 2 3 + πΆ ( B ) 2 5 β‘(β‘π₯ + 3β‘) 5 2 β 2β‘(β‘π₯ + 3β‘) 3 2 + C ( C ) 3β‘(β‘π₯+3β‘) 3 2 2 + C ( D ) 4β‘π₯2β‘(β‘π₯β‘+3β‘) 3 2 3 + C 28. If f ( x ) = ln ( ln ( 1 β x ) ) , then fΒ΄( x ) = ( A ) - 1 ln(β‘1βπ₯β‘) ( B ) 1 (β‘1βxβ‘)β‘ln(β‘1βπ₯β‘) ( C ) 1 (β‘1βπ₯β‘)2 ( D ) - 1 (β‘1βxβ‘)β‘ln(β‘1βπ₯β‘) 29. lim π₯β‘β0 π₯π π₯ 1ββ‘π π₯ =
- 8. ( A ) - β ( B ) - 1 ( C ) 1 ( D ) β 30. β« π‘ππ6 π₯β‘π ππ2 π₯β‘ππ₯= ( A ) π‘ππ7 π₯ 7 + C ( B ) π‘ππ7 π₯ 7 + π ππ3 π₯ 3 + C ( C ) π‘ππ7 π₯β‘π ππ3 π₯ 21 + πΆ ( D ) 7 tan7 x + C
- 9. CALCULUS AB SECTION I, Part B Time β 45 Minutes Number of questions β 15 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 scratchwork. After examiningthe formof the choices,decidewhichisthe bestof the choicesgivenandfill inthe correspondingoval onthe answersheet. No creditwill be givenfor anything writteninthe test book. Do not spend too much time on any one problem. In this test: 1. The exactnumerical value of the correct answerdoesnot alwaysappearamongthe choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. 2. Unlessotherwise specified,the domainof the functionf is assumedtobe the set of all real numbers x for which f ( x ) is a real number. 31. β« sin π₯β‘ππ₯ +β‘β« cos π₯β‘ππ₯ 0 ββ‘ π 4 π 4 0 = ( A ) - 1 ( B ) 0 ( C ) 1 ( D ) ββ‘2 32. Boats A and B leave the same place at the same time. Boat A heads due north at 12 km/hr. Boat B headsdue eastat 18 km/hr. After2.5 hours,how fast isthe distance betweenthe boatsincreasing( in km/hr ) ? ( A ) 21.63 ( B ) 31.20 ( C ) 75.00 ( D ) 9.84 33. If β« π(β‘π₯β‘) ππ₯ = π΄β‘β‘β‘πππβ‘β‘β‘ β« π(β‘π₯β‘) ππ₯ 100 50 = π΅β‘, π‘βππ 100 30 β‘β‘β‘β« πβ‘(β‘π₯β‘) ππ₯ 50 30 = ( A ) A + B ( B ) A β B ( C ) B β A
- 10. ( D ) 20 34. If f ( x ) = 3x2 β x and g ( x ) = f β 1 ( x ) , then gΒ΄( 10 ) could be ( A ) 59 ( B ) 1 59 ( C ) 1 10 ( D ) 1 11 35. lim π₯β‘β0 3sin π₯β1 π₯ = ( A ) 0 ( B ) 1 ( C ) ln 3 ( D ) 3 36. The volume generatedbyrevolvingaboutthe y β axis the regionenclosedby the graphs y = 9 β x2 and y = 9 β 3x , for 0 β€ π₯β‘ β€ 2 , is ( A ) - 8π ( B ) 4π ( C ) 8π ( D ) 24π 37. The average value of the function f ( x ) = ln 2 x on the interval [β‘2β‘,4β‘] is ( A ) 1.204 ( B ) 2.159 ( C ) 2.408 ( D ) 8.636 38. π ππ₯ β« cos(β‘π‘β‘)β‘ππ‘ 3π₯ 0 =
- 11. ( A ) sin 3x ( B ) cos 3x ( C ) 3 sin 3x ( D ) 3 cos 3x 39. Find the average value of f ( x ) = 4 cos ( 2x ) on the interval from x = 0 to x = π ( A ) - 2 π ( B ) 0 ( C ) 2 π ( D ) 4 π 40. The radiusof a sphere isincreasingatarate proportional toitself. If the radiusis 4 initially,andthe radius is 10 after two seconds, what will the radius be after three seconds? ( A ) 62.50 ( B ) 15.81 ( C ) 16.00 ( D ) 25.00 41. Suppose f ( x ) = β« (β‘π‘3 + π‘β‘)β‘ππ‘ π₯ 0 . Find fΒ΄( 5 ) ( A ) 130 ( B ) 120 ( C ) 76 ( D ) 74 42. β«β‘ππ2π₯β‘ππ₯= ( A ) ln 2π₯ 2π₯ + πΆ ( B ) xlnx β x + C ( C ) xln2x β x + C ( D ) 2xln2x β 2x + C
- 12. 43. Given f ( x ) = {β‘ππ₯2 + 3ππ₯ + 14β‘β‘β‘;β‘β‘β‘π₯β‘ β€ 12 β‘3ππ₯ + 5πβ‘β‘β‘; β‘β‘β‘π₯β‘ > 2 , findthe valuesof a and b that make f differentiable for all x. ( A ) a = - 6 , b = 2 ( B ) a = 6 ; b = - 2 ( C ) a = 2 ; b = 6 ( D ) a = 2 ; b = - 6 44. Two particles leave the origin at the same time and move along the y β axis with their respective positions determinedby the functions y1 = cos 2t and y2 = 4 sin t for 0 < π‘β‘ < 6. For how many values of t do the particles have the same acceleration? ( A ) 0 ( B ) 1 ( C ) 2 ( D ) 3 45. Find the distance traveled ( to three decimal places ) in the first four seconds, for a particle whose velocity is given by v ( t ) = 7πββ‘π‘2 , where t stands for time. ( A ) 0.976 ( B ) 6.204 ( C ) 6.359 ( D ) 12.720
- 13. SECTION II GENERAL INSTRUCTIONS You may wish to look over the problems before starting to work on them, since it is not expected that everyone will be able to complete all parts of all problems. All problems are given equal weight, but the parts of a particular problem are not necessarily given equal weight. A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THIS SECTION OF THE EXAMINATION ο· You should write all work for each part of each problem in the space provided for that part in the booklet. Be sure to write clearlyandlegibly. If youmake an error, youmay save time by crossingit out rather than trying to erase it. Erased or crossed-out work will not be graded. ο· Show all your work. You will be graded on the correctness and completeness or your methods as well as your answers. Correct answers without supporting work may not receive credit. ο· Justifications require that you give mathematical ( noncalculator ) reasons and that you clearly identify functions, graphs, tables, or other objects you use. ο· You are permittedtouse your calculator to solve an equation,findthe derivative of a functionat a point, or calculate the value of a definite integral. However, you must clearlyindicate the setup of your problem, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results. ο· Your work mustbe expressedinstandardmathematical notationratherthancalculatorsyntax. For example, β« π₯2 ππ₯ 5 1 may not be written as fnInt ( X2 , X , 1 , 5 ). ο· Unlessotherwise specified,answers( numericor algebraic) neednot be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point. ο· Unlessotherwisespecified,the domainof afunction f isassumedtobe the set of all real numbers x for which f ( x ) is a real number. SECTION II, PART A Time β 30 minutes Number of problems β 2 A graphing calculator is required for some problems or parts or problems. During the timed portion for Part A, you may work only on the problems in Part A. On part A, youare permittedtouse your calculatorto solve anequation,findthe derivative of afunctionat a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your problem, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results.
- 14. 1. A particle movesalongthe x-axissothatitsacceleration at any time t > 0 is given by a ( t ) = 12t β 18 . At time t = 1 , the velocity of the particle is v ( 1 ) = 0 and the position is x ( 1 ) = 9 ( a ) Write an expression for the velocity of the particle v ( t ) ( b ) At what values of t does the particle change direction ? ( c ) Write an expression for the position x ( t ) ( d ) Find the total distance traveled by the particle from t = 3 2 to t = 6 2. Let R be the region enclosed by the graphs of y = 2lnx and y = π π , and the lines x = 2 and x = 8 ( a ) Find the area of R ( b ) Set up, but do not integrate, an integral expression, in terms of a single variable,for the volume of the solid generated when R is revolved about the x-axis. ( c ) Set up, but do not integrate, an integral expression, in terms of a single variable,for the volume of the solid generated when R is revolved about the line x = - 1
- 15. SECTION II, PART B Time β 1 hour Number of problems β 4 No calculator is allowed for these problems. Duringthe timed portionforPart B, youmay continue toworkon the problemsinPartA withoutthe use of any calculator. 3. Consider the equation x2 β 2xy + 4y2 = 52 ( a ) Write an expression for the slope of the curve at any point ( x , y ). ( b ) Find the equation of the tangent lines to the curve at the point x = 2 ( c ) Find π2 π¦ ππ₯2 at ( 0 , ββ‘21β‘ ) 4. Waterisdrainingatthe rate of 48π ft3 /secondfromthe vertexatthe bottomof aconical tankwhose diameter at its base is 40 feet and whose height is 60 feet. ( a ) Findan expressionforthe volume of water inthe tank, in termsof its radius,at the surface of the water. ( b ) At what rate is the radius of the water in the tank shrinking when the radius is 16 feet? ( c ) How fast is the height of the water in the tank dropping at the instant that the radius is 16 feet? 5. Let f be the function given by f ( x ) = 2x4 β 4x2 + 1. ( a ) Find an equation of the line tangent to the graph at ( - 2 , 17 ) ( b ) Findthe x-andy-coordinatesofthe relativemaximaandrelative minima. Verifyyouranswer. ( c ) Find the x-and y-coordinates of the points of inflection. Verify your answer. 6. Let F( x ) = β« [πππ ( π‘ 2 ) +β‘( 3 2 )] π₯ 0 dt on the closed interval [β‘0β‘, 2πβ‘] ( a ) Approximate F ( 2π ) using four right hand rectangles. ( b ) Find FΒ΄( 2π ) ( c ) Find the average value of FΒ΄( x ) on the interval [β‘0β‘,4πβ‘].
- 16. CALCULUS AB SECTION I, Part A Time β 60 minutes Number of questions β 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using the available space for scratchwork. After examiningthe formof the choices,decidewhichisthe bestof the choicesgivenandfill inthe correspondingoval onthe answersheet. No creditwill be givenfor anythingwritteninthe test book. Do not spend too much time on any one problem. In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x ) is a real number. 1. If g(x) = 1 32 π₯4 β 5π₯2 , find gΒ΄(4) ( A ) - 72 ( B ) - 32 ( C ) 24 ( D ) 32 2. lim π₯β‘β0β‘ 8π₯2 cosπ₯β1 = ( A ) - 16 ( B ) - 1 ( C ) 8 ( D ) 6 3. lim π₯β‘ββ‘5β‘ π₯2β25 π₯β5 is ( A ) 0 ( B ) 10 ( C ) 5 ( D ) The limit does not exist. 4. If f ( x ) = π₯5βπ₯+2 π₯3+7 , find fΒ΄(x)
- 17. ( A ) 5π₯4β1 3π₯2 ( B ) (β‘π₯3+7β‘)(β‘5π₯4β1β‘)ββ‘(β‘π₯5βπ₯β‘)(β‘3π₯2β‘) π₯3+7 ( C ) (β‘π₯5βπ₯+2β‘)(β‘3π₯2β‘)ββ‘(β‘π₯3+7β‘)(β‘5π₯4β1β‘) (β‘π₯3+7β‘)2 ( D ) ββ‘(β‘π₯5βπ₯+2β‘)(β‘3π₯2β‘)β‘+β‘(β‘π₯3+7β‘)(β‘5π₯4β1β‘) (β‘π₯3+7β‘)2 5. Evaluate lim ββ‘β0 tan( π 4 +ββ‘)β1 β ( A ) 0 ( B ) 1 ( C ) 2 ( D ) This limit does not exist 6. β« π₯β‘ββ‘3π₯β‘β‘ππ₯ =β‘ ( A ) 2β‘ββ‘3β‘ 5 π₯ 5 2 + πΆ ( B ) 5β3 2 π₯ 5 2 + πΆ ( C ) β3 2 π₯ 1 2 + πΆ ( D ) 5β3 2 π₯ 3 2 + πΆ 7. For what value of k is f continuous at x = 1 if f ( x ) = { β‘π₯2 β 3ππ₯ + 2β‘β‘β‘;β‘β‘β‘π₯β‘ β€ 1 5π₯ β ππ₯2β‘β‘;β‘β‘π₯β‘ > β‘1 ( A ) - 1 ( B ) - 1 2 ( C ) 2 ( D ) 8
- 18. 8. Which of the followingintegralscorrectlygivesthe area of the regionconsistingof all points above the x-axis and below the curve y = 8 + 2x β x2 ? ( A ) β« (β‘π₯2 β 2π₯ β 8β‘) ππ₯ 4 ββ‘2 ( B ) β« (β‘ββ‘π₯2 + 2π₯ + 8β‘) ππ₯ 2 ββ‘4 ( C ) β« (β‘ββ‘π₯2 + 2π₯ + 8β‘) ππ₯ 4 ββ‘2 ( D ) β« (β‘π₯2 β 2π₯ β 8β‘) ππ₯ 2 ββ‘4 9. Find π π π π if y = sec ( ππ₯2 ) ( A ) tan ( ππ₯2 ) ( B ) ( 2ππ₯ ) tan ( ππ₯2 ) ( C ) sec ( ππ₯2 ) tan ( ππ₯2 ) ( D ) sec ( ππ₯2 ) tan ( ππ₯2 ) ( 2ππ₯ ) 10. Given the curve y = 5 - (β‘π₯ β 2β‘)β‘ 2 3 , find ππ¦ ππ₯ at x = 2 ( A ) - 2 3 ( B ) - 2 3β‘ ββ‘23 ( C ) 5 ( D ) The limit does not exist. 11. β« β‘ 2 ββ‘1ββ‘π₯2 β‘ππ₯ 1 2 0 = ( A ) π 3 ( B ) - π 3 ( C ) 2π 3 ( D ) - 2π 3 ππ₯ 12 Finda positive value c,forx, the satisfiesthe conclusion of the MeanTheoremfor Derivativesforf ( x ) = 3x2 - 5x + 1 on the interval [β‘2β‘,5β‘] ( A ) 1
- 19. ( B ) 11 6 ( C ) 23 6 ( D ) 7 2 13. Given f ( x ) = 2x2 β 7x β 10 , find the absolute maximum of f ( x ) on [ββ‘1β‘, 3β‘] ( A ) - 1 ( B ) 7 4 ( C ) - 129 8 ( D ) 0 14. Find ππ¦ ππ₯ at ( 1 , 2 ) for y3 = xy β 2x2 + 8 ( A ) - 11 2 ( B ) - 2 11 ( C ) 2 11 ( D ) 11 2 15. lim π₯β‘β0 β‘ π₯β‘.β‘β‘2 π₯ 2 π₯β1 = ( A ) ln 2 ( B ) 1 ( C ) 2 ( D ) 1 ππ2 16. β«β‘π₯π ππ2( 1 + π₯2 ) dx = ( A ) 1 2 tan ( 1 + x2 ) + C ( B ) 2 tan ( 1 + x2 ) + C ( C ) π₯ 2 tan ( 1 + x2 ) + C
- 20. ( D ) 2x tan ( 1 + x2 ) + C 17. Find the equation of the tangent line to 9x2 + 16y2 = 52 through ( 2 , - 1 ) ( A ) - 9x + 8y β 26 = 0 ( B ) 9x β 8y β 26 = 0 ( C ) 9x β 8y β 106 = 0 ( D ) 8x + 9y β 17 = 0 18. A particleΒ΄s position is given by s = t3 β 6t2 + 9t . What is its acceleration at time t = 4 ? ( A ) 0 ( B ) - 9 ( C ) - 12 ( D ) 12 19. If f ( x ) = 3 ππ₯ , then fΒ΄(x) = ( A ) 3 ππ₯ ln3 ( B ) 3 ππ₯ π ( C ) πβ‘(β‘3 ππ₯β‘) ( D ) πβ‘ππ3β‘(β‘3 ππ₯β‘) 20 The average value of f ( x ) = 1 π₯ from x = 1 to x = e is ( A ) 1 π+1 ( B ) 1 1βπ ( C ) e β 1 ( D ) 1 πβ1 21. If f ( x ) = π ππ2 π₯β‘, find f´´´( x )
- 21. ( A ) - sin2 x ( B ) cos 2x ( C ) - 4 sin 2x ( D ) - sin 2x 22. Find the slope of the normal line to y = x + cos xy at ( 0 , 1 ) ( A ) 1 ( B ) - 1 ( C ) 0 ( D ) Undefined 23. β«β‘ ππ π2 β π₯ β π₯ β‘ππ₯ = ( A ) 2cotβ π₯ + C ( B ) - 2cotβ π₯ + C ( C ) ππ π2 β π₯ 3β π₯ + C ( D ) ππ π2 β π₯ 6β π₯ + C 24. lim π₯β‘ββ‘0 π‘ππ3(2π₯) π₯3 = ( A ) - 8 ( B ) 2 ( C ) 8 ( D ) The limit does not exist. 25. A solid is generated when the region in the firstquadrant bounded by the graph of y = 1 + π ππ2 π₯β‘, the line x = π 2 , the x-axis, and the y-axis is revolvedabout the x-axis. Its volume is found by evaluating which of the following integrals? ( A ) πβ‘ β« (β‘1 + π ππ4 π₯β‘) ππ₯ 1 0 ( B ) πβ‘ β« (β‘1 + π ππ2 π₯β‘)2 ππ₯ 1 0
- 22. ( C ) πβ‘ β« (β‘1 + π ππ4 π₯β‘) ππ₯ π 2 0 ( D ) πβ‘ β« (β‘1 + π ππ2 π₯β‘)2 ππ₯ π 2 0 26. If y = (β‘ π₯3ββ‘2 2π₯5β1 ) 4 , find ππ¦ ππ₯ at x = 1 ( A ) - 52 ( B ) - 28 ( C ) 13 ( D ) 52 27. β« π₯ββ‘5 β π₯β‘β‘ππ₯ = ( A ) - 10 3 (β‘5 β π₯β‘β‘) 3 2 ( B ) 10 3 β 5π₯2 2 ββ‘ π₯3 3 + C ( C ) 10 (β‘5 β π₯β‘) 1 2 +β‘ 2 3 β‘(β‘5 β π₯β‘) 3 2 + C ( D ) - 10 3 (β‘5 β π₯β‘) 3 2 +β‘ 2 5 β‘(β‘5 β π₯β‘) 5 2 + C 28. If ππ¦ ππ₯ = π₯3+1 π¦ and y = 2 when x = 2 , y = ( A ) β 27 2 ( B ) β 27 8 ( C ) Β± β 27 8 ( D ) Β± β 27 2 29. The graph of y = 5x4 β x5 has an inflection point ( or points ) at ( A ) x = 3 only ( B ) x = 0 , 3
- 23. ( C ) x = - 3 only ( D ) x = 0 , - 3 30 β« tan π₯β‘ππ₯ 1 0 ( A ) 0 ( B ) ln ( cos ( 1 ) ) ( C ) ln ( sec ( 1 ) ) ( D ) ln ( sec ( 1 ) ) β 1
- 24. CALCULUS AB SECTION I, Part B Time β 45 Minutes Number of questions β 15 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 scratchwork. After examiningthe formof the choices,decidewhich isthe bestof the choicesgivenandfill inthe correspondingoval onthe answersheet. No creditwill be givenfor anythingwritten inthe test book. Do not spend too much time on any one problem. In this test: 1. The exact 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 approximates the exact numerical value. 2. Unlessotherwise specified,the domainof afunction f is assumedto be the set of all real numbersx for which f ( x ) is a real number. 31. The average value of f ( x ) = π4π₯2 on the interval [ββ‘ 1 4 β‘, 1 4 β‘] is ( A ) 0.272 ( B ) 0.545 ( C ) 1.090 ( D ) 2.180 32. π ππ₯ β‘β« π ππ2 π‘β‘ππ‘ π₯2 0 = ( A ) x2 sin2 (x2 ) ( B ) 2xsin2 (x2 ) ( C ) sin2 (x2 ) ( D ) x2 cos2 (x2 ) 33. Find the value (s) of ππ¦ ππ₯ of x2 y + y2 = 5 at y = 1 ( A ) - 2 3 only ( B ) 2 3 only
- 25. ( C ) Β±β‘ 2 3 ( D ) Β± 3 2 34. The graph of y = x3 β 2x2 β 5x + 2 has a local maximum at ( A ) ( 2.120 , 0 ) ( B ) ( 2.120 , - 8.061 ) ( C ) ( - 0.786 , 0 ) ( D ) ( - 0.786 , 4.209 ) 35 Approximate β« π ππ2 π₯β‘ππ₯ 1 0 using the Trapezoid Rule with n = 4 , to three decimal places. ( A ) 0.277 ( B ) 0.555 ( C ) 1.109 ( D ) 2.219 36. The volume generated by revolving about the x-axis the regionabove the curve y = x3 , below the line y = 1 , and between x = 0 and x = 1 is ( A ) π 42 ( B ) 0.143 π ( C ) 0.643 π ( D ) 6π 7 37. A sphere is increasing in volume at the rate of 20 ππ.3 π . At what rate is the radius of the sphere increasing when the radius is 4 in.? ( A ) 0.025 ππ. π ( B ) 0.424 ππ. π ( C ) 0.995 ππ. π ( D ) 0.982 ππ. π
- 26. 38. β« πππ₯ 3π₯ β‘ππ₯ = ( A ) 6ln2 |β‘π₯β‘| + C ( B ) 1 3 β‘ππ2|β‘π₯β‘| + C ( C ) 1 6 β‘ππ2β‘|β‘π₯β‘| + C ( D ) 1 3 ππ|β‘π₯β‘| + C 39. Find two non-negative numbers x and y whose sum is 100 and for which x2 y is a maximum. ( A ) x = 50 and y = 50 ( B ) x = 33.333 and y = 66.667 ( C ) x = 100 and y = 0 ( D ) x = 66.667 and y = 33.333 40. Find the distance traveled ( to three decimal places ) from t = 1 to t = 5 seconds, for a particle whose velocity is given by v ( t ) = t + ln t ( A ) 6.000 ( B ) 1.609 ( C ) 16.047 ( D ) 148.413 41. β«β‘π ππ4(β‘ππ₯β‘)cos(β‘ππ₯β‘)β‘ππ₯ = ( A ) π ππ5(β‘ππ₯β‘) 5π + C ( B ) π ππ5(β‘ππ₯β‘) 2π + C ( C ) ββ‘ πππ 5(β‘ππ₯β‘) 5π + C ( D ) ββ‘ πππ 5(β‘ππ₯β‘) 2π + C 42. The volume of a cube isincreasing ata rate proportional toitsvolume atanytime t. If the volume is 8 ft3 originally, and 12 ft3 after 5 seconds, what is its volume at t = 12 seconds?
- 27. ( A ) 21.169 ( B ) 22.941 ( C ) 28.800 ( D ) 17.600 43. If f ( x ) = (β‘1 +β‘ π₯ 20 β‘) 5 , find f´´ ( 40 ) ( A ) 0.068 ( B ) 1.350 ( C ) 5.400 ( D ) 6.750 44. Find the equation of the line tangent to y = x tan x at x = 1 ( A ) y = 4.983 x + 3.426 ( B ) y = 4.983 x β 3.426 ( C ) y = 4.983 x + 6.540 ( D ) y = 4.983 x β 6.540 45. If f ( x ) is continuous and differentiable and f ( x ) = { β‘ππ₯4 + 5π₯β‘; π₯β‘ β€ 2 ππ₯2 β 3π₯β‘; π₯β‘ > 2 , then b = ( A ) 0 ( B ) 2 ( C ) 6 ( D ) There is no value of b.
- 28. SECTION II GENERAL INSTRUCTIONS You may wish to look over the problems before starting to work on them, since it is not expected that everyone will be able to complete all parts of all problems. All problems are given equal weight, but the parts of a particular problem are not necessarily given equal weight. A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THIS SECTION OF THE EXAMINATION ο· You should write all work for each part of each problem in the space provided for that part in the booklet. Be sure to write clearlyandlegibly. If youmake an error, youmay save time by crossingit out rather than trying to erase it. Erased or crossed-out work will not be graded. ο· Show all your work. You will be graded on the correctness and completeness or your methods as well as your answers. Correct answers without supporting work may not receive credit. ο· Justifications require that you give mathematical ( noncalculator ) reasons and that you clearly identify functions, graphs, tables, or other objects you use. ο· You are permittedtouse your calculator to solve an equation,findthe derivative of a functionat a point, or calculate the value of a definite integral. However, you must clearlyindicate the setup of your problem, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results. ο· Your work mustbe expressedinstandardmathematical notationratherthancalculatorsyntax. For example, β« π₯2 ππ₯ 5 1 may not be written as fnInt ( X2 , X , 1 , 5 ). ο· Unlessotherwise specified,answers( numericor algebraic) neednot be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point. ο· Unlessotherwisespecified,the domainof afunction f isassumedtobe the set of all real numbers x for which f ( x ) is a real number. SECTION II, PART A Time β 30 minutes Number of problems β 2 A graphing calculator is required for some problems or parts or problems. During the timed portion for Part A, you may work only on the problems in Part A. On part A, youare permittedtouse your calculatorto solve anequation,findthe derivative of afunctionat a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your problem, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results. 1. The temperature onNewYearΒ΄s Day in Hinterlandwas givenby T(H) = - A β B cos (β‘ πβ‘π» 12 β‘) , where T esthe temperature indegreesFahrenheitandHisthe numberof hoursfrommidnight ( 0β€ π» < 24) ( a ) The initial temperature atmidnightwas - 15Β°F and at noonof New YearΒ΄sDay was 5Β°F. Find A and B ( b ) Find the average temperature for the first 10 hours ( c ) Use the Trapezoid Rule with 4 equal subdivisions to estimate β« πβ‘(β‘π»β‘)β‘ππ» 8 6 ( d ) Find an expression for the rate that the temperature is changing with respect to H
- 29. 2. Sea grass grows on a lake. The rate of growth of the grass is ππΊ ππ‘ = kG, where k is a constant. ( a ) FindanexpressionforG,the amountof grassinthe lake ( in tons),intermsof t, the number of years, if the amount of grass is 100 tons initially and 120 tons after one year. ( b ) In how many years will the amount of grass available be 300 tons ? ( c ) If fishare now introducedintothe lake andconsume a consistent80 tons/yearof seagrass, how long will it take for the lake to be completely free of sea grass ?
- 30. SECTION II, PART B Time β 1 hour Number of problems β 4 No calculator is allowed for these problems. During the timedportionfor Part B, you may continue to work on the problemsinPart A withoutto use of any calculator. 3. Consider the curve defined by y = x4 + 4x3 . ( a ) Find the equation of the tangent line to the curve at x = - 1 ( b ) Find the coordinates of the absolute minimum. ( c ) Find the coordinates of the point ( s ) of inflection. 4. Water is being poured into a hemispherical bowl of radius 6 inches at the rate of 4 in.3 / sec. ( a ) Giventhatthe volume ofthe waterinthesphericalsegmentshownaboveis V = πβ2 (π β β 3 ), where Risthe radiusof the sphere,findthe rate thatthe water levelisrisingwhen the water is 2 inches deep. ( b ) Finan expressionforr,the radiusof the surface of the spherical segment of water,in terms of h. ( C ) How fast is the circular area of the surface of the spherical segment of water growing ( in in.2 /sec ) when the water is 2 inches deep? 5. Let R be the region in the first quadrant bounded by y2 = x and x2 = y. ( a ) Fin the area of region R ( b ) Find the volume of the solid generated when R is revolved about the x-axis. ( c ) The sectionof a certain solidcutby any plane perpendiculartothe x-axisisa circle withthe endpointsof its diameterlying on the parabolas y2 = x and x2 = y. Findthe volume of the solid
- 31. 6. An object moves with velocity v ( t ) = t2 β 8t + 7 ( a ) Write a polynomial expression for the position of the particle at any time t β₯ 0 ( b ) At what time (s) is the particle changing direction? ( c ) Find the total distance traveled by the particle from time t = 0 to t = 4 CALCULUS AB SECTION I, Part A Time β 60 Minutes Numberof questions β 30 A CALCULATOR MAY NOT USED ON THIS PART OF THE EXAMINATION Directions: Solve each of the following problems, using the available space for scratchwork. After examiningthe formof the choices,decidewhichisthe bestof the choicesgivenandfill inthe correspondingoval onthe answersheet. No creditwill be givenfor anythingwritteninthe test book. Do not spend too much time on any one problem. In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f ( x ) is a real number. 1. β« cos(2π‘) ππ‘ π₯ π 4 = ( A ) cos (2x) ( B ) sin(β‘2π₯β‘)β1 2 ( C ) cos ( 2x ) β 1 ( D ) π ππ2(π₯) 2
- 32. 2. What are the coordinates of the point of inflection on the graph of y = x3 β 15x2 + 33x + 100 ? ( A ) ( 9 , 0 ) ( B ) ( 5 , - 48 ) ( C ) ( 9 , - 89 ) ( D ) ( 5 , 15 ) 3. If 3x2 β 2xy + 3y = 1 , then when x = 2 , ππ¦ ππ₯ = ( A ) - 12 ( B ) - 10 ( C ) - 10 7 ( D ) 12 4. β« 8 π₯3 β‘ππ₯β‘ 3 1 = ( A ) 32 9 ( B ) 40 9 ( C ) 0 ( D ) - 32 9 5. Whichof the followingintegralscorrectlycorrespondstothe areaof the shadedregioninthe figure above? f ( x ) = 5 ; g ( x ) = 1 + x2 ( A ) 1 ( B ) 4 ( C ) 8 ( D ) 10
- 33. 6. lim π₯β‘β0 π₯βπ πππ₯ π₯2 β‘= ( A ) 0 ( B ) 1 ( C ) 2 ( D ) The limit does not exist. 7. If f(x) = x2 ββ‘3π₯ + 1β‘ , then fβ(x) = ( A ) 9π₯2+2π₯ ββ‘3π₯+1β‘ ( B ) ββ‘9π₯2+4π₯ 2ββ‘3π₯+1β‘ ( C ) 15π₯2+4π₯ 2ββ‘3π₯+1β‘ ( D ) ββ‘9π₯2β4π₯ 2ββ‘3π₯+1β‘ 8. What is the instantaneous rate of change at t = - 1 of the function f, if f(t) = π‘3+π‘ 4π‘+1 ? ( A ) 12 9 ( B ) 4 9 ( C ) ββ‘ 4 9 ( D ) ββ‘ 12 9 9. β« (β‘ 4 π₯β1 ) π+1 2 ππ₯= ( A ) 4 ( B ) 4e ( C ) 0 ( D ) - 4
- 34. 10. A carβs velocityisshownonthe graphabove. Which of the followinggivesthe total distancetraveled from t = 0 to t = 16 ( in kilometers ) ? ( A ) 360 ( B ) 390 ( C ) 780 ( D ) 1,000 11. π ππ₯ β‘π‘ππ2 (4x) = ( A ) 8 tan( 4x ) ( B ) 4 sec4 (4x) ( C ) 8 tan (4x)sec2 (4x) ( D ) 4 tan (4x)sec2 (4x) 12. What is the equation of the line tangent to the graph of y = sin2 x at x = π 4 ? ( A ) y - 1 2 =β‘ (β‘π₯ ββ‘ π 4 β‘) ( B ) y - 1 β2 = (β‘π₯ ββ‘ π 4 β‘) ( C ) y - 1 β2 = 1 2 (β‘π₯ ββ‘ π 4 β‘) ( D ) y - 1 2 = 1 2 (β‘π₯ ββ‘ π 4 β‘) 13. If the function f (x) = { β‘3ππ₯2 + 2ππ₯ + 1β‘β‘; β‘β‘π₯β‘ β€ 1 ππ₯4 β 4ππ₯2 β 3π₯β‘β‘;β‘β‘π₯β‘ > 1 β‘ is differentiable for all real values of x, the b = ( A ) - 11 4 ( B ) 11 4 ( C ) 0
- 35. ( D ) - 1 4 14. The graph of y = x4 + 8x3 β 72x2 + 4 is concave down for ( A ) - 6 < π₯β‘ < 2 ( B ) x > 2 ( C ) x <β‘β6 ( D ) - 3 β 3 ββ‘5 < π₯β‘ <β‘β3 + 3β‘ββ‘5 15. lim π₯β‘ββ‘β ln(β‘π₯+1β‘) log2 π₯ = ( A ) 1 ln 2 ( B ) 0 ( C ) 1 ( D ) ln 2 16. The graph of f(x) isshowninthe figure above. Whichof the following could be the graph of fΒ΄(x) ?
- 36. 17. If f(x) = ln ( cos (3x) ) , then fΒ΄(x) = ( A ) 3 sec ( 3x ) ( B ) 3 tan ( 3x ) ( C ) - 3 tan ( 3x ) ( D ) - 3 cot ( 3x ) 18. If f(x) = β« βπ‘2 β 1 3π₯+1 0 dt , then fΒ΄( - 4 ) = ( A ) - 2 ( B ) 2 ( C ) β153 ( D ) 0 19. A particle movesalongthe x-axissothat itspositionat time t, inseconds, is given by x(t) = t2 β 7t + 6. For what value ( s ) of t is the velocity of the particle zero ? ( A ) 1 ( B) 6 ( C ) 1 or 6 ( D ) 3.5 20. β« sin(β‘2π₯β‘)π π ππ2 π₯ π 2 0 ( A ) e β 1 ( B ) 1 β e ( C ) e + 1 ( D ) 1 21.