AP Calculus AB 2010 Scoring Guidelines Form B

Transcription

1 AP Calculus AB 1 Scoring Guidelines Form B The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 19, he College Board is composed of more han 5,7 schools, colleges, universiies and oher educaional organizaions. Each year, he College Board serves seven million sudens and heir parens,, high schools, and,8 colleges hrough major programs and services in college readiness, college admission, guidance, assessmen, financial aid and enrollmen. Among is widely recognized programs are he SAT, he PSAT/NMSQT, he Advanced Placemen Program (AP ), SpringBoard and ACCUPLACER. The College Board is commied o he principles of excellence and equiy, and ha commimen is embodied in all of is programs, services, aciviies and concerns. 1 The College Board. College Board, ACCUPLACER, Advanced Placemen Program, AP, AP Cenral, SAT, SpringBoard and he acorn logo are regisered rademarks of he College Board. Admied Class Evaluaion Service is a rademark owned by he College Board. PSAT/NMSQT is a regisered rademark of he College Board and Naional Meri Scholarship Corporaion. All oher producs and services may be rademarks of heir respecive owners. Permission o use copyrighed College Board maerials may be requesed online a: AP Cenral is he official online home for he AP Program: apcenral.collegeboard.com.

2 1 SCORING GUIDELINES (Form B) Quesion 1 In he figure above, R is he shaded region in he firs quadran bounded by he graph of y = 4ln( x), he horizonal line y = 6, and he verical line x =. (a) Find he area of R. (b) Find he volume of he solid generaed when R is revolved abou he horizonal line y = 8. (c) The region R is he base of a solid. For his solid, each cross secion perpendicular o he x-axis is a square. Find he volume of he solid. 1 : Correc limis in an inegral in (a), (b), or (c) or : { 1 : inegrand (a) ( 6 4ln ( x )) = ( ) (b) π ( 8 4ln( x) ) ( 8 6) = or : { : inegrand or 6.67 : { : inegrand (c) ( 6 4ln ( x )) = The College Board.

3 The funcion g is defined for x > wih () 1, AP CALCULUS AB 1 SCORING GUIDELINES (Form B) Quesion 1 g = g 1 1 ( x) = sin ( x + ), and g ( x) = ( x + ) x 1 cos. x x (a) Find all values of x in he inerval.1 x 1 a which he graph of g has a horizonal angen line. (b) On wha subinervals of (.1, 1 ), if any, is he graph of g concave down? Jusify your answer. (c) Wrie an equaion for he line angen o he graph of g a x =.. (d) Does he line angen o he graph of g a x =. lie above or below he graph of g for. < x < 1? Why? (a) The graph of g has a horizonal angen line when g ( x) =. This occurs a x =.16 and x = : ses g ( x) = : (b) g ( x) = a x = and x =.74 The graph of g is concave down on (.195,.7 ) because g ( x) < on his inerval. : { 1 : jusificaion (c) g (.) = g(.) = + g ( x) = An equaion for he line angen o he graph of g is y = ( x. ). 1 : g (.) 1 : inegral expression 4 : 1 : g (. ) 1 : equaion (d) g ( x) > for. < x < 1 wih reason Therefore he line angen o he graph of g a x =. lies below he graph of g for. < x < 1. 1 The College Board.

4 1 SCORING GUIDELINES (Form B) Quesion P() The figure above shows an aboveground swimming pool in he shape of a cylinder wih a radius of 1 fee and a heigh of 4 fee. The pool conains 1 cubic fee of waer a ime =. During he ime inerval 1 hours, waer is pumped ino he pool a he rae P () cubic fee per hour. The able above gives values of P () for seleced values of. During he same ime inerval, waer is leaking from he pool a he rae R() cubic fee.5 per hour, where R () = 5 e. (Noe: The volume V of a cylinder wih radius r and heigh h is given by. V = π r h ) (a) Use a midpoin Riemann sum wih hree subinervals of equal lengh o approximae he oal amoun of waer ha was pumped ino he pool during he ime inerval 1 hours. Show he compuaions ha lead o your answer. (b) Calculae he oal amoun of waer ha leaked ou of he pool during he ime inerval 1 hours. (c) Use he resuls from pars (a) and (b) o approximae he volume of waer in he pool a ime = 1 hours. Round your answer o he neares cubic foo. (d) Find he rae a which he volume of waer in he pool is increasing a ime = 8 hours. How fas is he waer level in he pool rising a = 8 hours? Indicae unis of measure in boh answers. 1 : { (a) P () d = 66 f 1 : midpoin sum 1 : { 1 : inegral (b) R () d= f 1 1 (c) 1 + P () d R () d= A ime = 1 hours, he volume of waer in he pool is approximaely 144 f. (d) V () = P() R().4 V ( 8) = P( 8) R( 8) = 6 5e = 4.41 or 4.4 f hr V = π ( 1) h dv dh = 144π d d dh 1 dv.95 d = 144π d = or.96 f hr = 8 = 8 1 : V ( 8) dv 1 : equaion relaing and d 4 : dh 1 : d = 8 1 : unis of f hr and f hr dh d 1 The College Board.

5 1 SCORING GUIDELINES (Form B) Quesion 4 A squirrel sars a building A a ime = and ravels along a sraigh wire conneced o building B. For 18, he squirrel s velociy is modeled by he piecewise-linear funcion defined by he graph above. (a) A wha imes in he inerval < < 18, if any, does he squirrel change direcion? Give a reason for your answer. (b) A wha ime in he inerval 18 is he squirrel farhes from building A? How far from building A is he squirrel a his ime? (c) Find he oal disance he squirrel ravels during he ime inerval 18. (d) Wrie expressions for he squirrel s acceleraion a (), velociy v (), and disance x() from building A ha are valid for he ime inerval 7 < < 1. (a) The squirrel changes direcion whenever is velociy changes sign. This occurs a = 9 and = 15. (b) Velociy is a =, = 9, and = 15. posiion a ime = = = 115 : { 1 : -values 1 : explanaion 1 : idenifies candidaes : { s The squirrel is farhes from building A a ime = 9; is greaes disance from he building is (c) The oal disance raveled is v () d= = 15. ( 1) (d) For 7 < < 1, a () = = v () = 1( 7) = x( 7) = = 1 7 ( u u) x() = x( 7) + ( 1u + 9) du = = u= u = 7 4 : 1 : 1 : : a () v () x() 1 The College Board.

6 1 SCORING GUIDELINES (Form B) Consider he differenial equaion dy = x + 1. y Quesion 5 (a) On he axes provided, skech a slope field for he given differenial equaion a he welve poins indicaed, and for 1 < x < 1, skech he soluion curve ha passes hrough he poin (, 1 ). (Noe: Use he axes provided in he exam bookle.) (b) While he slope field in par (a) is drawn a only welve poins, i is defined a every poin in he xy-plane for which y. Describe all poins in he xy-plane, y, for dy which 1. = (c) Find he paricular soluion y = f( x) o he given differenial equaion wih he iniial condiion f ( ) =. (a) : 1 : zero slopes 1 : nonzero slopes 1 : soluion curve hrough (, 1) (b) x = y = x 1 y dy = 1 for all ( x, y ) wih y = x 1 and y 1 : descripion (c) ydy= ( x+ 1) y x = + x + C ( ) = + + C C = y = x + x + 4 Since he soluion goes hrough (, ), y mus be negaive. Therefore y = x + x : 1 : separaes variables 1 : aniderivaives 1 : consan of inegraion 1 : uses iniial condiion 1 : solves for y Noe: max 5 [1-1---] if no consan of inegraion Noe: 5 if no separaion of variables 1 The College Board.

7 1 SCORING GUIDELINES (Form B) Quesion 6 Two paricles move along he x-axis. For 6, he posiion of paricle P a ime is given by π p() = cos ( ), while he posiion of paricle R a ime is given by r () = (a) For 6, find all imes during which paricle R is moving o he righ. (b) For 6, find all imes during which he wo paricles ravel in opposie direcions. (c) Find he acceleraion of paricle P a ime =. Is paricle P speeding up, slowing down, or doing neiher a ime =? Explain your reasoning. (d) Wrie, bu do no evaluae, an expression for he average disance beween he wo paricles on he inerval 1. (a) r () = = ( 1)( ) r () = when = 1 and = r () > for < < 1 and < < 6 r () < for 1< < 1 : r () : Therefore R is moving o he righ for < < 1 and < < 6. (b) p π π () = sin( ) 4 4 p () = when = and = 4 p () < for < < 4 p () > for 4 < < 6 Therefore he paricles ravel in opposie direcions for < < 1 and < < 4. : 1 : p () 1 : sign analysis for p () (c) p π π π () = cos 4 4 ( 4 ) p ( ) ( ) ( ) π π π = cos = > p ( ) < Therefore paricle P is slowing down a ime =. 1 : p ( ) : wih reason 1 (d) p() r() d 1 : { 1 : inegrand 1 : limis and consan 1 The College Board.