AP CALCULUS BC 2016 SCORING GUIDELINES

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1 6 SCORING GUIDELINES Quesion A ime, he posiion of a paricle moving in he xy-plane is given by he parameric funcions ( x ( ), y ( )), where = + sin ( ). The graph of y, consising of hree line segmens, is shown in he figure above. A =, he paricle is a posiion ( 5, ). (a) Find he posiion of he paricle a =. (b) Find he slope of he line angen o he pah of he paricle a =. (c) Find he speed of he paricle a =. (d) Find he oal disance raveled by he paricle from = o =. (a) x( ) = x ( ) + x ( ) = =.77 y ( ) = The posiion of he paricle a = is (.77,.5 ). : inegral : : uses iniial condiion : answer (b) Slope y ( ).5 = = =.5 : slope x ( ) : expression for speed (c) Speed = ( x ( ) ) + ( y ( ) ) = (or 9.968) : { : answer (d) Disance = ( x ( ) ) + ( y ( ) ) = ( x ( ) ) + ( ) + ( x ( ) ) + = =.5 (or.9) : expression for disance : : inegrals : answer 6 The College Board. Visi he College Board on he Web:

2 5 SCORING GUIDELINES Quesion A ime, a paricle moving along a curve in he xy-plane has posiion ( x ( ), y ( )) wih velociy vecor.5 ( ) ( ( ) ) v = cos, e. A =, he paricle is a he poin (, 5 ). (a) Find he x-coordinae of he posiion of he paricle a ime =. (b) For < <, here is a poin on he curve a which he line angen o he curve has a slope of. A wha ime is he objec a ha poin? (c) Find he ime a which he speed of he paricle is. (d) Find he oal disance raveled by he paricle from ime = o ime =. (a) ( ) ( ) x = + cos =.557 (or.556) : inegral : : uses iniial condiion : answer (b).5 e = = cos e cos.5 ( ) = ( ) { : slope in erms of : : answer =.8 (c) Speed cos ( ) = + e ( ) cos + e = =.96 (or.95) { : speed in erms of : : answer (d) Disance = cos ( ) + e =.595 (or.59) : { : inegral : answer 5 The College Board. Visi he College Board on he Web:

3 SCORING GUIDELINES Quesion For, a paricle is moving along a curve so ha is posiion a ime is ( x(), y() ). A ime =, he + paricle is a posiion (, 5 ). I is known ha = and = sin. e (a) Is he horizonal movemen of he paricle o he lef or o he righ a ime =? Explain your answer. Find he slope of he pah of he paricle a ime =. (b) Find he x-coordinae of he paricle s posiion a ime =. (c) Find he speed of he paricle a ime =. Find he acceleraion vecor of he paricle a ime =. (d) Find he disance raveled by he paricle from ime = o =. (a) = Because = e = a ime =. >, he paricle is moving o he righ : moving o he righ wih reason : : considers : slope a = = = =.55 ( or.5) = = (b) + x( ) = + =.5 ( or.5) e : { : inegral : answer (c) Speed = ( x ( ) ) + ( y ( ) ) =.575 ( or.57) Acceleraion = x (, ) y ( ) =.,.989 : { : speed : acceleraion (d) Disance = ( x () ) + ( y () ) =.65 ( or. 65) : { : inegral : answer The College Board. Visi he College Board on he Web:

4 SCORING GUIDELINES Quesion A ime, a paricle moving in he xy-plane is a posiion ( x( ), y( )), where x( ) and y ( ) are no explicily given. For, = + and = sin ( ). A ime =, x ( ) = and y ( ) =. (a) Find he speed of he paricle a ime =, and find he acceleraion vecor of he paricle a ime =. (b) Find he slope of he line angen o he pah of he paricle a ime =. (c) Find he posiion of he paricle a ime =. (d) Find he oal disance raveled by he paricle over he ime inerval. (a) Speed = ( x ( ) ) + ( y ( ) ) =.6 or.7 Acceleraion = x (, ) y ( ) =, 5.66 or, 5.67 : { : speed : acceleraion (b) Slope y ( ) = =. x ( ) or. : answer (c) x( ) = + = y( ) = + =.6 A ime =, he paricle is a posiion (,.6 ). : : x-coordinae : inegral : answer : y-coordinae : inegral : answer (d) Disance ( ) ( ) =.9 + = : { : inegral : answer The College Board. Visi he College Board on he Web:

5 SCORING GUIDELINES (Form B) Quesion The velociy vecor of a paricle moving in he plane has componens given by cos( ) sin ( = ) and ( ) e A ime =, he posiion of he paricle is (, ). = + sin, for.5. (a) For < <.5, find all values of a which he line angen o he pah of he paricle is verical. (b) Wrie an equaion for he line angen o he pah of he paricle a =. (c) Find he speed of he paricle a =. (d) Find he acceleraion vecor of he paricle a =. (a) The angen line is verical when x () = and y (). On < <.5, his happens a =.5 and =. or.5. : ses = : : answer (b) = y () = =.867 x () x() = + x () = y() = + y () =.657 The line angen o he pah of he paricle a = has equaion y = ( x 9.5 ). : = : : x() : y() : equaion (c) Speed = ( x () ) + ( y () ) =.5 : answer (d) Acceleraion vecor: x (), y () = 8.5,.6 : x () : : y () The College Board. Visi he College Board on he Web:

6 SCORING GUIDELINES Quesion A paricle is moving along a curve so ha is posiion a ime is ( x(), y() ), where x () = + 8 and y () is no explicily given. Boh x and y are measured in meers, and is measured in seconds. I is known ha = e. (a) Find he speed of he paricle a ime = seconds. (b) Find he oal disance raveled by he paricle for seconds. (c) Find he ime,, when he line angen o he pah of he paricle is horizonal. Is he direcion of moion of he paricle oward he lef or oward he righ a ha ime? Give a reason for your answer. (d) There is a poin wih x-coordinae 5 hrough which he paricle passes wice. Find each of he following. (i) The wo values of when ha occurs (ii) The slopes of he lines angen o he paricle s pah a ha poin (iii) The y-coordinae of ha poin, given y( ) = + e (a) Speed = ( x ( ) ) + ( y ( ) ) =.88 meers per second : answer (b) x () = = + e =.587 or.588 meers Disance ( ) ( ) : { : inegral : answer (c) = = when e = and This occurs a =.79. Since x (.79) >, he paricle is moving oward he righ a ime =.7 or.8. : : considers = : =.7 or.8 : direcion of moion wih reason (d) x () = 5 a = and = A ime =, he slope is A ime =, he slope is y() = y() = + e + = =.. = = = =. = = = : : = and = : slopes : y-coordinae The College Board. Visi he College Board on he Web:

7 9 SCORING GUIDELINES Quesion A diver leaps from he edge of a diving plaform ino a pool below. The figure above shows he iniial posiion of he diver and her posiion a a laer ime. A ime seconds afer she leaps, he horizonal disance from he fron edge of he plaform o he diver s shoulders is given by x(), and he verical disance from he waer surface o her shoulders is given by y (), where x() and y () are measured in meers. Suppose ha he diver s shoulders are. meers above he waer when she makes her leap and ha =.8 and =.6 9.8, for A, where A is he ime ha he diver s shoulders ener he waer. (a) Find he maximum verical disance from he waer surface o he diver s shoulders. (b) Find A, he ime ha he diver s shoulders ener he waer. (c) Find he oal disance raveled by he diver s shoulders from he ime she leaps from he plaform unil he ime her shoulders ener he waer. π (d) Find he angle θ, < θ <, beween he pah of he diver and he waer a he insan he diver s shoulders ener he waer. (a) = only when =.675. Le b =.675. The maximum verical disance from he waer surface o he diver s shoulders is ( ). b yb = + =.6 meers. : : considers = : inegral or y () : answer Alernaively, y () = , so yb ( ) =.6 meers. (b) ( ). A y A = + =. +.6A.9A = A =.96 seconds. when : { : equaion : answer (c) ( ) ( ) A + =.96 meers : { : inegral : answer (d) A ime A, = = 9.9. = A The angle beween he pah of he diver and he waer is an ( 9.9) =.58 or.59. : : a ime : answer A 9 The College Board. All righs reserved. Visi he College Board on he Web:

8 8 SCORING GUIDELINES (Form B) Quesion A paricle moving along a curve in he xy-plane has posiion ( x(), y() ) a ime wih = and = cos. The paricle is a posiion (, 5 ) a ime =. (a) Find he acceleraion vecor a ime =. (b) Find he y-coordinae of he posiion of he paricle a ime =. (c) On he inerval, a wha ime does he speed of he paricle firs reach.5? (d) Find he oal disance raveled by he paricle over he ime inerval. (a) a( ) = x ( ), y ( ) =.,.87 : answer (b) y( ) = 5 + cos =.6 : inegrand or.6 : : uses y( ) = 5 : answer (c) Speed = ( x () ) + ( y () ) = + 9cos =.5 : : expression for speed : equaion : answer The paricle firs reaches his speed when =.5 or.6. (d) + 9cos =.8 : { : inegral : answer 8 The College Board. All righs reserved. Visi he College Board on he Web:

9 7 SCORING GUIDELINES (Form B) Quesion An objec moving along a curve in he xy-plane is a posiion ( x( ), y( )) a ime wih ( ) = arcan + and = ln ( + ) for. A ime =, he objec is a posiion (, ). (Noe: an (a) Find he speed of he objec a ime =. x = arcan x ) (b) Find he oal disance raveled by he objec over he ime inerval. (c) Find x (. ) (d) For >, here is a poin on he curve where he line angen o he curve has slope. A wha ime is he objec a his poin? Find he acceleraion vecor a his poin. (a) Speed = x ( ) + y ( ) =.9 : speed a = (b) Disance = + = 6. : { : inegral : answer (c) x( ) = x( ) + x ( ) = +.79 =.89 : inegrand : : : uses x( ) = : answer (d) The slope is, so, = or ( + ) = ( ) ln arcan. + Since >, = A his ime, he acceleraion is x ( ), y ( ) =.5766 =.5,.955. : = : : -value : values for x and y 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for sudens and parens).

10 6 SCORING GUIDELINES (Form B) Quesion An objec moving along a curve in he xy-plane is a posiion ( x(), y() ) a ime, where an( e ) for. A ime =, he objec is a posiion (, ). = and = sec( e ) (a) Wrie an equaion for he line angen o he curve a posiion (, ). (b) Find he acceleraion vecor and he speed of he objec a ime =. (c) Find he oal disance raveled by he objec over he ime inerval. (d) Is here a ime a which he objec is on he y-axis? Explain why or why no. (a) sec( e ) = = = an ( e ) sin( e ) = =.78 or.78 sin (, ) ( e ) y + = sin ( ) ( x ) e : : (, ) : equaion of angen line (b) x () =.5, y () =.596 a () =.,.5 or.,.5. : acceleraion vecor : { : speed speed ( ( e )) ( ( e )) = sec + an =.8 or.9 (c) ( x () ) + ( y () ) =.59 : { : inegral : answer (d) x( ) = x() x () = > The paricle sars o he righ of he y-axis. Since x () > for all, he objec is always moving o he righ and hus is never on he y-axis. : x( ) expression : : x () > : conclusion and reason 6 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens).

11 6 SCORING GUIDELINES Quesion An objec moving along a curve in he xy-plane is a posiion ( x(), y() ) a ime, where sin = and ( e ) for. A ime =, he objec is a he poin ( 6, ). (Noe: = + sin x = arcsin x ) (a) Find he acceleraion vecor and he speed of he objec a ime =. (b) The curve has a verical angen line a one poin. A wha ime is he objec a his poin? (c) Le m () denoe he slope of he line angen o he curve a he poin ( x(), y() ). Wrie an expression for m () in erms of and use i o evaluae lim m (). (d) The graph of he curve has a horizonal asympoe y = c. Wrie, bu do no evaluae, an expression involving an improper inegral ha represens his value c. (a) a ( ) =.95 or.96,.7 or.7 Speed = x ( ) + y ( ) =.7 or.8 : { : acceleraion : speed (b) ( e ) sin = e = = ln =.69 and when = ln : x () = : : answer (c) m () = + sin e ( ) lim () lim m = sin + e = = sin () ( ) : m () : : limi value (d) Since lim x () =, c = lim y() = + : + : inegrand : limis : iniial value consisen wih lower limi 6 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens).

12 5 SCORING GUIDELINES (Form B) Quesion An objec moving along a curve in he xy-plane has posiion ( x(), y() ) a ime wih ( ) = and = ln + ( ). A ime =, he objec is a posiion (, 5 ). A ime =, he objec is a poin P wih x-coordinae. (a) Find he acceleraion vecor a ime = and he speed a ime =. (b) Find he y-coordinae of P. (c) Wrie an equaion for he line angen o he curve a P. (d) For wha value of, if any, is he objec a res? Explain your reasoning. (a) x ( ) =, y ( ) = =.88 7 a ( ) =,.88 Speed = + ( ln ( 7) ) =.9 or. : acceleraion vecor : : speed ( ) ( ) (b) y () = y( ) + ln + ( u ) du y( ) = 5 + ln + ( u ) du =.67 : : ln ( + ( u ) ) du : handles iniial condiion : answer ln ( 7) (c) A =, slope = = =.6 y.67 =.6( x ) : slope : : equaion (d) x () = if =, y () = if = = : reason : : answer Copyrigh 5 by College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens).

13 SCORING GUIDELINES (Form B) Quesion A paricle moving along a curve in he plane has posiion ( x(), y() ) a ime, where = + 9 and = e + 5e for all real values of. A ime =, he paricle is a he poin (, ). (a) Find he speed of he paricle and is acceleraion vecor a ime =. (b) Find an equaion of he line angen o he pah of he paricle a ime =. (c) Find he oal disance raveled by he paricle over he ime inerval. (d) Find he x-coordinae of he posiion of he paricle a ime =. (a) A ime = : Speed = x () + y () = + 7 = 58 : speed : : acceleraion vecor Acceleraion vecor = x (, ) y ( ) =, (b) y ( ) 7 = = x ( ) 7 Tangen line is y = ( x ) + : slope : : angen line (c) Disance = ( + 9) + ( + 5 ) e e = 5.6 or 5.7 : : disance inegral each inegrand error error in limis : answer (d) x( ) = : inegral : : answer = 7.9 or 7.9 Copyrigh by College Enrance Examinaion Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens).

14 SCORING GUIDELINES Quesion An objec moving along a curve in he xy-plane has posiion ( x(), y() ) a ime wih = + cos ( ). The derivaive is no explicily given. A ime =, he objec is a posiion (, 8 ). (a) Find he x-coordinae of he posiion of he objec a ime =. (b) A ime =, he value of is 7. Wrie an equaion for he line angen o he curve a he poin ( x(, ) y ( )). (c) Find he speed of he objec a ime =. (d) For, he line angen o he curve a ( x(), y() ) has a slope of +. Find he acceleraion vecor of he objec a ime =. (a) ( ) = ( ) + ( + cos( )) ( ( )) x x = + + cos = 7. or 7. : ( ( )) : + cos : handles iniial condiion : answer (b) = 7 = = =.98 + cos = y 8 =.98( x ) : finds : = : equaion (c) The speed of he objec a ime = is ( x ( ) ) + ( y ( ) ) = 7.8 or 7.8. : answer (d) x ( ) =. y = = = + + cos y ( ) =.8 or.8 The acceleraion vecor a = is.,.8 or.,.8. () ( )( ( )) : : x ( ) : : answer Copyrigh by College Enrance Examinaion Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens).

15 SCORING GUIDELINES (Form B) Quesion A paricle moves in he xy-plane so ha he posiion of he paricle a any ime is given by ( ) 7 x e e = + and ( ) y = e e. (a) Find he velociy vecor for he paricle in erms of, and find he speed of he paricle a ime =. (b) Find in erms of, and find lim. (c) Find each value a which he line angen o he pah of he paricle is horizonal, or explain why none exiss. (d) Find each value a which he line angen o he pah of he paricle is verical, or explain why none exiss. 7 (a) x () = 6e 7e y () = 9e + e 7 Velociy vecor is < 6e 7 e, 9e + e > : : x( ) : y( ) : speed Speed = x() + y() = ( ) + = (b) 9e + e = = 6e 7e 7 : : in erms of : limi 9e + e 9 lim = lim = = 7 6e 7e 6 (c) Need y () =, bu 9e + e > for all, so none exiss. : : considers y( ) = : explains why none exiss (d) Need x () = and y(). 7 e = e e = 6 7 = ln 6 ( ) : considers x( ) = : : soluion Copyrigh by College Enrance Examinaion Board. All righs reserved. Available a apcenral.collegeboard.com. 5

16 SCORING GUIDELINES Quesion A paricle sars a poin A on he posiive x-axis a ime = and ravels along he curve from A o B o C o D, as shown above. The coordinaes of ( ) he paricle s posiion x( ), y( ) are differeniable funcions of, where x ( ) + = = 9cos sin 6 and y ( ) = is no explicily given. A ime = 9, he paricle reaches is final posiion a poin D on he posiive x-axis. (a) A poin C, is posiive? A poin C, is posiive? Give a reason for each answer. (b) The slope of he curve is undefined a poin B. A wha ime is he paricle a poin B? 5 (c) The line angen o he curve a he poin ( x( 8 ), y ( 8) ) has equaion y = x. Find he 9 velociy vecor and he speed of he paricle a his poin. (d) How far apar are poins A and D, he iniial and final posiions, respecively, of he paricle? (a) A poin C, is no posiive because y () is decreasing along he arc BD as increases. A poin C, is no posiive because x () is decreasing along he arc BD as increases. : : no posiive wih reason : no posiive wih reason (b) = ; ( ) cos = or 6 sin + = + = or = ; = for boh. 6 Paricle is a poin B a =. : : ses = : = (c) x (8) 9 cos ( ) sin ( ) 9 = = y(8) 5 = = x (8) y(8) = x(8) = 9 The velociy vecor is <.5,.5 >. : : x(8) : y(8) : speed Speed = = 5.7 or 5.8 (d) x(9) x() = x( ) 9 = 9.55 The iniial and final posiions are 9.55 apar. : inegral : : answer Copyrigh by College Enrance Examinaion Board. All righs reserved. Available a apcenral.collegeboard.com.

17 SCORING GUIDELINES (Form B) Quesion )F=HJE?ALAIEJDANOF=AIJD=JEJIFIEJE=J=OJEAJBHГ > J > EICELA>O NJ IE! J=@OJ J = 5AJ?DJDAF=JDBJDAF=HJE?AEJDANOF=AFHLE@A@@E?=JAJDA@EHA?JEBJE =CJDAF=JD >.E@JDAH=CABNJ =@JDAH=CABOJ?.E@JDAI=AIJFIEJELAL=KABJBHMDE?DJDAN?H@E=JABJDAF=HJE?AEI=?= IJDA@EIJ=?AJH=LAA@>OJDAF=HJE?ABHJ ГJJ CHA=JAHJD= # KIJEBO OKH=IMAH = CH=FD JDHAA?O?AIBIEA N >AJMAA Г =@ O >AJMAA Г > Г > NJ > Г > OJ >? N= J!?I! J!J J $ 5FAA@ '?I! J " )JJ $ 5FAA@ '?I '?I! J "@J Г %'%! #?IA@EJAHL=BHNJ?IA@EJAHL=BHOJ N= J!?I! J! ILAIBHJ IFAA@=JIJK@AJ\IJEA EJACH=BH@EIJ=?A??KIEMEJDKIJEBE?=JE Copyrigh by College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board.

18 AP CALCULUS BC SCORING GUIDELINES Quesion > J > & IA?@I6DAFIEJE BJDA?=H=JJEA J IA?@I?=>A@AA@F=H=AJHE?=O >O N J J " IEJ O J ГJ Г?IJ MDAHA N =@ O =HAA=IKHA@EAJAHI6DA@AHEL=JELAIBJDAIABK?JEI=HACELA>O = "?I = Г IE?I Г N J J O J J J J =.E@JDAIFABJDAF=JD=JJEA J 5DMJDA?FKJ=JEIJD=JA=@JOKH=IMAH >.E@JDA=??AAH=JELA?JHBJDA?=H=JJDAJEAMDAJDA?=H\IDHEJ=FIEJEEIN"?.E@JDAJEA J =JMDE?DJDA?=HEI=JEJI=NEKDAECDJ=@BE@JDAIFAA@EIA?BJDA?=H=J J &JDAHA=HAJMJEAI=JMDE?DJDA?=HEI=JCHK@ALA O.E@JDAIAJMJEAI =@MHEJA=ANFHAIIEJD=JCELAIJDA=LAH=CAIFAA@EIA?BJDA?=H>AJMAAJDAIAJMJEAI, O= & IE?I Г = J N = "?I %'!H%'" > NJ J " IE J " J!$"%&! N== J Г!# ' O== J #H $ )??AAH=JELA?JHEI Г!# ' # H Г!# ' $? O= J ГJIE J?IJ Г J!!H! "=J=NEKDAECDJ 5FAA@ N= J O= J N= J $ %H$ OJ MDA J =@ J " " )LAH=CAIFAA@ N= J O= " "?I J ГJIE E@AJEBEAI=??AAH=JELA?JH =I@AHEL=JELABLA?EJOLA?JH?FKJAI=??AAH=JELA?JH MDAN " IAJIO= J! IAA?JIBEHIJJ IFAA@ J J "! EEJI=@?IJ=J EJACH=@ Copyrigh by College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board.

19 AP CALCULUS BC SCORING GUIDELINES Quesion An objec moving along a curve in he xy-plane has posiion x (), y () a ime wih cos and sin for > >. A ime =, he objec is a posiion (,5). (a) Wrie an equaion for he line angen o he curve a (,5). (b) Find he speed of he objec a ime =. (c) Find he oal disance raveled by he objec over he ime inerval > >. (d) Find he posiion of he objec a ime =. (a) sin cos sin cos y Г5 5.6( x Г ) 5.6 : angen line (b) Speed = cos (8) 9 sin () =.75 : answer (c) Disance = cos 9 sin =.58 : : disance inegral < Г each inegrand error < Г error in limis : answer (d) x() cos =.95 or.95 y() 5 sin =.96 : : definie inegral for x : answer for x() : definie inegral for y : answer for y() Copyrigh by College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board.

20 !Å#ALCULUSÅ"#n!ÅMOVINGÅPARTICLEÅHASÅPOSITIONÅ XT YT ÅATÅTIMEÅTÅHEÅPOSITIONÅOFÅTHEÅPARTICLEÅATÅTIMEÅTÅÅÅISÅ ÅAND THEÅVELOCITYÅVECTORÅATÅANYÅTIMEÅTÅÅÅISÅGIVENÅBYÅ T T A &INDÅTHEÅACCELERATIONÅVECTORÅATÅTIMEÅTÅÅ B &INDÅTHEÅPOSITIONÅOFÅTHEÅPARTICLEÅATÅTIMEÅTÅÅ C &ORÅWHATÅTIMEÅTÅÅÅDOESÅTHEÅLINEÅTANGENTÅTOÅTHEÅPATHÅOFÅTHEÅPARTICLEÅATÅ XT YT ÅHAVEÅAÅSLOPEÅOFÅ D HEÅPARTICLEÅAPPROACHESÅAÅLINEÅASÅT ldå&indåtheåslopeåofåthisålineåhowåtheåworkåthatåleadsåtoåyour CONCLUSION A ACCELERATIONÅVECTORÅÅ Xaa T Yaa T T T Xaa Yaa B XT YT T # T # T T X Y # # # # X Y ÅCOMPONENTSÅOFÅACCELERATION Å ÅÅÅÅÅÅVECTORÅASÅAÅFUNCTIONÅOFÅT ÅACCELERATIONÅVECTORÅATÅT ÅANTIDIFFERENTIATION Å ÅUSESÅINITIALÅCONDITIONÅATÅT ÅPOSITIONÅATÅT.OTEÅMAXÅÅÅ;nn=ÅIFÅNOÅCONSTANTSÅOF INTEGRATION DY C ÅÅ DX T T ÅÅÅÅÅÅ T T T ÅÅÅÅÅÅT Å DY Å ÅASÅEQUATIONÅINÅT DX ÅSOLUTIONÅFORÅT D DY LIM LIM T Tld DX Tld T nåorån INCEÅXT ldåasåt ldåtheåslopeåofåtheålineåis YT T LIM LIM T Tld XT Tld T T DY YT ÅCONSIDERSÅLIMITÅOFÅ ÅORÅ Å DX X T ÅANSWER.OTEÅÅIFÅNOÅCONSIDERATIONÅOFÅLIMIT Copyrigh by College Enrance Examinaion Board and Educaional Tesing Service. All righs reserved. AP is a regisered rademark of he College Enrance Examinaion Board.

21 BC{ 999. A paricle moves in he xy{plane so ha is posiion a any ime,, isgiven by x() = ; ln( + ) an() = sin. (a) Skech he pah of he paricle in he xy{plane below. Indicae he direcion of moion along he pah. (b) A wha ime,, doesx() aain is minimum value? Wha is he posiion (x()y()) of he paricle a his ime? (c) A wha ime, <<, is he paricle on he y{axis? Find he speed and he acceleraion vecor of he paricle a his ime. (a) y ( : graph : direcion O x (b) x () = ; + = + ; = = ;+p 5 or =:68 in [] x(:68) = ;:9 y(:68) = :78 (c) x() = ; ln( + ) = =:85 or :86 x () = ; + y () = cos 8 >< >: 8 >< >: : x () = : soluion for : posiion : x() = : soluion for : speed : acceleraion vecor speed = p (x (:86)) +(y (:86)) =:96 x () =+ ( + ) y () =; sin acceleraion vecor = <x (:86)y (:86) > = <:9;:879 >

22 998 Calculus BC Scoring Guidelines Copyrigh 998 by College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board.

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