SCING GUIDELINES (Form B) Quesion 4 A paricle moves along he x-axis wih velociy a ime given by v( ) = 1 + e1. (a) Find he acceleraion of he paricle a ime =. (b) Is he speed of he paricle increasing a ime =? Give a reason for your answer. (c) Find all values of a which he paricle changes direcion. Jusify your answer. (d) Find he oal disance raveled by he paricle over he ime inerval. 1 (a) a () = v() = e a() = e : 1 : v( ) 1 : a() (b) a () < v() = 1 + e < Speed is increasing since v () < and a () <. 1 : answer wih reason 1 (c) v () = when 1 = e, so = 1. v () > for < 1 and v () < for > 1. Therefore, he paricle changes direcion a = 1. : 1 : solves v ( ) = o ge = 1 1 : jusifies change in direcion a = 1 (d) Disance = v () d 1 1 1+ + 1 1 1 1 1 1 e e 1 1 1 e e 1 1 = ( ) + ( + ) = ( ) ( ) e d e d = ( + ) + ( + ) 4 : 1 : limis 1 : inegrand 1 : anidiffereniaion 1 : evaluaion = e + 1 e 1 () = x e x() = e x (1) = x() = e Disance = ( x(1) x() ) + ( x(1) x() ) ( + e) + 1+ e = ( ) = e + 1 e 1 : any aniderivaive 1 : evaluaes x ( ) when =, 1, 4 : 1 : evaluaes disance beween poins 1 : evaluaes oal disance Copyrigh by College Enrance Examinaion Board. All righs reserved. Available a apcenral.collegeboard.com. 5
5 SCING GUIDELINES (Form B) Quesion A waer ank a Camp Newon holds 1 gallons of waer a ime =. During he ime inerval hours, waer is pumped ino he ank a he rae () 95 sin ( ) W = gallons per hour. 6 During he same ime inerval, waer is removed from he ank a he rae () 75sin ( ) R = gallons per hour. (a) Is he amoun of waer in he ank increasing a ime = 15? Why or why no? (b) To he neares whole number, how many gallons of waer are in he ank a ime =? (c) A wha ime, for, is he amoun of waer in he ank a an absolue minimum? Show he work ha leads o your conclusion. (d) For >, no waer is pumped ino he ank, bu waer coninues o be removed a he rae R() unil he ank becomes empy. Le k be he ime a which he ank becomes empy. Wrie, bu do no solve, an equaion involving an inegral expression ha can be used o find he value of k. (a) No; he amoun of waer is no increasing a = 15 since W( 15) R( 15) = 11.9 <. 1 : answer wih reason (b) 1 + ( W() R() ) d = 19.788 11 gallons : 1 : limis 1 : inegrand (c) W() R() = =, 6.4948, 1.9748 (hours) gallons of waer 1 6.495 55 1.975 1697 11 1 : inerior criical poins 1 : amoun of waer is leas a : = 6.494 or 6.495 1 : analysis for absolue minimum The values a he endpoins and he criical poins show ha he absolue minimum occurs when = 6.494 or 6.495. k (d) R () d= 11 1 : limis : 1 : equaion Copyrigh 5 by College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and www.collegeboard.com/apsudens (for AP sudens and parens).
7 SCING GUIDELINES (Form B) Quesion A paricle moves along he x-axis so ha is velociy v a ime is given by v () = sin ( ). The graph of v is shown above for 5 π. The posiion of he paricle a ime is x( ) and is posiion a ime = is x ( ) = 5. (a) Find he acceleraion of he paricle a ime =. (b) Find he oal disance raveled by he paricle from ime = o =. (c) Find he posiion of he paricle a ime =. (d) For 5 π, find he ime a which he paricle is farhes o he righ. Explain your answer. (a) a( ) = v ( ) = 6cos9 = 5.466 or 5.467 1 : a ( ) (b) Disance = v () d= 1.7 For < <, v ( ) = when = π = 1.7745 and = π =.566 x ( ) = 5 x x ( ) () π = 5 + v d = 5.8948 π π ( ) () π = 5 + v d = 5.441 x( ) = 5 + v( ) d = 5.7756 ( π ) ( ) ( π ) ( π ) ( ) ( π ) x x + x x + x x = 1. 7 : { 1 : seup 1 : answer 1 : inegrand (c) x( ) = 5 + v( ) d = 5.77 or 5.774 : 1 : uses x( ) = 5 1 : answer (d) The paricle s righmos posiion occurs a ime = π = 1.77. The paricle changes from moving righ o moving lef a hose imes for which v ( ) = wih v ( ) changing from posiive o negaive, namely a = π, π, 5π ( = 1.77,.7,.96 ). T Using x( T) = 5 + v( ) d, he paricle s posiions a he imes i changes from righward o lefward movemen are: T: π π 5π xt ( ): 5 5.895 5.788 5.75 The paricle is farhes o he righ when T = π. : 1 : ses v () = 1 : answer 1 : reason 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and www.collegeboard.com/apsudens (for sudens and parens).
1 SCING 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 () = 6 + 9+. 4 (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 + 9 = ( 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 = > 4 4 8 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. Visi he College Board on he Web: www.collegeboard.com.
1 SCING GUIDELINES (Form B) Quesion 4 A squirrel sars a building A a ime = and ravels along a sraigh wire conneced o building B. For, he squirrel s velociy is modeled by he piecewise-linear funcion defined by he graph above. (a) A wha imes in he inerval < <, if any, does he squirrel change direcion? Give a reason for your answer. (b) A wha ime in he inerval 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. (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 9 + 5 9 = 14 6 + 4 15 14 1 = 9 + 9 + 1 = 115 : { 1 : -values 1 : explanaion 1 : idenifies candidaes : { 1 : answers The squirrel is farhes from building A a ime = 9; is greaes disance from he building is 14. (c) The oal disance raveled is v () d= 14 + 5 + 5 = 15. 1 : answer ( 1) (d) For 7 < < 1, a () = = 1 7 1 v () = 1( 7) = 1 + 9 7 + 5 x( 7) = = 1 7 ( u u) x() = x( 7) + ( 1u + 9) du = 1 + 5 + 9 = 5 + 9 65 u= u = 7 4 : 1 : 1 : : a () v () x() 1 The College Board. Visi he College Board on he Web: www.collegeboard.com.