AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a line segmen wih slope 5 B) 5 C) D) 5 If y = and = +, hen a = is 8 B) 8 C) D) 8 A curve in he plane is defined paramerically by he equaions equaion of he line angen o he curve a = is: = + and y = + An y = C) y = y = 8 + B) y = 8 D) y = 5 Consider he curve in he y plane represened by = e and y= e for The slope of he line angen o he curve a he poin where = is (Calculaor) 86 B) C) 5 D) 5 If = + and y =, hen d y = B) C) D) 6 6 A paricle moves along he curve y = If = and =, wha is he value of? 5 B) 6 C) D) 5 5 6 5 7 If = e and y = sin( ), hen = e sin( ) e cos( ) B) C) cos( ) e cos( ) D) e cos( ) e
8 For wha values of does he curve given by he parameric equaions y= + 8 have a verical angen? = and B) C), D),, and No value 9 For, an objec ravels along an ellipical pah given by he parameric equaions = cos and y = sin A he poin where =, he objec leaves he pah and ravels along he line angen o he pah a ha poin Wha is he slope of he line on which he objec ravels? B) C) an D) an an The posiion of a paricle moving in he y plane is given by he parameric equaions = and y= For wha values of is he paricle a res? B) C) D),,, A curve C is defined by he parameric equaions = + and y= Which of he following is an equaion of he line angen o he graph of C a he poin (, 8)? = B) = C) y = 8 D) 7 y= ( + ) + 8 y = ( + ) + 8 (97 BC, p 58 #) A kie flies according o he parameric equaions y= ( 8) where is measured in seconds and< 9 6 = and 8 (d) How high is he kie above he ground a ime = seconds? A wha rae is he kie rising a = seconds? A wha rae is he sring being reeled ou a = seconds? A wha ime does he kie sar o lose aliude? (97 BC5) Given he parameric equaions = ( sin) and y ( cos) Find in erms of Find an equaion of he line angen o he graph a = Find an equaion of he line angen o he graph a = =
(98 BC6, p 56 #9) Poin P(, y ) moves in he y plane in such a way ha = for Find he coordinaes of P in erms of if, when =, = ln and y = Wrie an equaion epressing y in erms of Find he average rae of change of y wih respec o as varies from o (d) Find he insananeous rae of change of y wih respec o when = = + and 5 (98 BC, p 58 #) The pah of a paricle is given for ime > by he parameric equaions = + and y= Find he coordinaes of each poin on he pah where he velociy of he paricle in he direcion is zero Find when = d y Find when y = 6 (989 BC, p 58 #) Consider he curve given by he parameric equaions y = = and In erms of, find Wrie an equaion for he line angen o he curve a he poin where = Find he and y coordinaes for each criical poin on he curve and idenify each poin as having a verical or horizonal angen 7 (99 BC) The posiion of a paricle a any ime is given by Find as a funcion of = and y( ) = () Lengh of Parameric Curves 8 The lengh of he curve deermined by he equaions = and y = from = o = is B) + C) + D) + + +
9 The lengh of he pah described by he parameric equaions, is given by = cos and y = sin, for cos + sin B) C) 9cos + 9sin D) 6 6 cos + sin cos sin+ sin cos 9cos sin + 9sin cos The lengh of he pah described by he parameric equaions, is given by = and y =, where B) + C) + D) + + 6 + 9 (97 BC7) Le C be he curve defined from = o = 6 by he parameric equaions + =, y= ( 6 ) Se up bu do no evaluae an inegral epression for he lengh of C (97 BC5) Given he parameric equaions = ( sin) and y ( cos) = Se up bu do no evaluae an inegral epression represening he lengh of he curve over he inerval Epress he inegrand as a funcion of Vecors If f is a vecor valued funcion defined by f( ) ( e, cos) e sin =, hen f ( ) = + C) ( e, sin) ( e, cos) e D) ( e,cos) B) cos A paricle moves in he y plane so ha a any ime is coordinaes are y= A =, is acceleraion vecor is = and (, ) B) (, ) C) (, ) D) (, ) (, 8)
5 For any ime, if he posiion of a paricle in he y plane is given by ( ) y= ln +, hen he acceleraion vecor is B), +, ( + ) C), D), ( + ) ( + ), ( + ) = + and ( ln ), 6 If a paricle moves in he y plane so ha a ime > is posiion vecor is ( ) hen a ime =, is velociy vecor is +,,8 B), C),8 8 D), 8 5, 6 7 A paricle moves on a plane curve so ha a any ime > is coordinae is coordinae is ( ) The acceleraion vecor of he paricle a = is (, ) B) (, ) C) (, 6) D) (6, ) (6, ) and is y 8 A paricle moves in he y plane so ha is posiion a any ime is given by y = sin( ) Wha is he speed of he paricle when =? = and 99 B) 6 C) 688 D) 96 79 9 (975 BC) A paricle moves on he circle by he vecor, + + + y = so ha a ime he posiion is given Find he velociy vecor Is he paricle ever a res? Jusify your answer Give he coordinaes of he poin ha he paricle approaches as increases wihou bound (987 BC5) The posiion of a paricle moving in he y plane a any ime,, is given by he parameric equaions = sin and y = cos( ) Find he velociy vecor for he paricle a anyime, For wha values of is he paricle a res? Wrie an equaion for he pah of he paricle in erms of and y ha does no involve rigonomeric funcions 5
(99 BC) A ime,, he posiion of a paricle moving along a pah in he y plane is given by he parameric equaions = e sin and y= e cos Find he slope of he pah of he paricle a ime = Find he speed of he paricle when = Find he disance raveled by he paricle along he pah from = o = (99 BC) The posiion of a paricle a any ime is given by Find he magniude of he velociy vecor a = 5 Find he oal disance raveled by he paricle from = o = 5 = and y( ) = () (99 BC) A paricle moves along he graph of y = cos so ha componen of acceleraion is always A ime =, he paricle is a he poin (, ) and he velociy vecor of he paricle is (, ) Find he and y coordinaes of he posiion of he paricle in erms of Find he speed of he paricle when is posiion is(, cos ) (995 BC) Two paricles move in he y plane For ime, he posiion of paricle A is given by = and y= ( ), and he posiion of paricle B is given by = and y = Find he velociy vecor for each paricle a ime = Se up an inegral epression ha gives he disance raveled by paricle A from = o = Do no evaluae Deermine he eac ime a which he paricles collide; ha is, when he paricles are a he same poin a he same ime Jusify your answer 5 (997 BC) During he ime period from = o = 6 seconds, a paricle moves along he pah () cos y () = 5sin given by = ( ) and ( ) Find he posiion of he paricle when = 5 On a se of and y aes, skech he graph of he pah of he paricle from = o = 6 Indicae he direcion of he paricle along his pah How many imes does he paricle pass hrough he poin found in par? (d) Find he velociy vecor for he paricle a any ime (e) Wrie and evaluae an inegral epression, in erms of sine and cosine, ha gives he disance he paricle ravels from ime = 5 o = 75 6
6 (999 BC) A paricle moves in he y plane so ha is posiion a any ime,, is given by ( ) = ln( + ) and y( ) = sin Skech he pah of he paricle on a se of and y aes Indicae he direcion of moion along he pah A wha ime,, does ( ) aain is minimum value? Wha is he posiion ( (), y() ) of he paricle a his ime? A wha ime,, is he paricle on he y ais? Find he speed and he acceleraion vecor of he paricle a his ime 7 ( BC) A moving paricle has posiion ( (), y() ) a ime The posiion of he paricle a ime = is (, 6), and he velociy vecor a any ime > is given by,+ Find he acceleraion vecor a ime = Find he posiion of he paricle a ime = For wha ime > does he line angen o he pah of he paricle a ( (), ()) (d) y have a slope of 8? The paricle approaches a line as Find he slope of his line Show he work ha leads o your conclusion 8 ( BC) An objec moving along a curve in he y plane has posiion ( (), y() ) a ime = = for A ime =, he objec is a posiion (, wih cos( ) and sin( ) 5) Wrie an equaion for he line angen o he curve a (, 5) Find he speed of he objec a ime = Find he oal disance raveled by he objec over he ime inerval (d) Find he posiion of he objec a ime = 7
9 ( BC) The figure above shows he pah raveled by a roller coaser car over he ime inerval 8seconds The posiion of he car a ime seconds can be modeled paramerically by = + sin, y= ( )( cos ), where and y are measured in meers The derivaives of hese funcions are given by = + cos, y = ( )sin + cos (d) Find he slope of he pah a ime = Show he compuaions ha lead o your answer Find he acceleraion vecor of he car a he ime when he car s horizonal posiion is = Find he ime a which he car is a is maimum heigh, and find he speed, in m/sec, of he car a his ime For <<8, here are wo imes a which he car is a ground level (y = ) Find hese wo imes and wrie an epression ha gives he average speed, in m/sec, of he car beween hese wo imes Do no evaluae he epression (B BC) A paricle moves in he y plane so ha is posiion a any ime,, is given by ( ) = sin( ) and y ( ) = Skech he pah of he paricle on a se of and y aes Indicae he direcion of moion along he pah Find he range of ( ) and he range of y( ) Find he smalles posiive value of for which he coordinae of he paricle is a local maimum Wha is he speed of he paricle a his ime? (d) Is he disance raveled by he paricle in greaer han5? Jusify your answer 8
( BC) A paricle sars a poin A on he posiive ais 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 ( ( ), y ( )) are differeniable funcions of, where 9cos sin + = = 6 and y = is no eplicily given A ime = 9, he paricle reaches is final posiion a poin D on he posiive ais A poin C, is posiive? A poin C, is posiive? Give a reason for each answer The slope of he curve is undefined a poin B A wha ime is he paricle a B? 5 The line angen o he curve a he poin ( (8), y (8)) has equaion y = 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? (B BC) A paricle moves in he y plane so ha he posiion of he paricle a any ime 7 is given by () = e + e and y () = e e (d) Find he velociy vecor for he paricle in erms of, and find he speed of he paricle a ime = Find in erms of, and find lim Find each value a which he line angen o he pah of he paricle is horizonal, or eplain why none eiss Find each value a which he line angen o he pah of he paricle is verical, or eplain why none eiss 9
( BC) An objec moving along a curve in he y plane has posiion ( (), y() ) a ime wih = + cos( ) The derivaive is no eplicily given A ime =, he objec is a posiion (, 8) Find he coordinae of he posiion of he objec a ime = A ime =, he value of = 7 Wrie an equaion for he line angen o he curve a he poin( (), y ()) Find he speed of he objec a ime = (d) For, he line angen o he curve a ( (), y() ) has a slope of + Find he acceleraion vecor of he objec a ime = (B BC) A paricle moving along a curve in he plane has posiion ( ( ), y ( )) a ime, where 9 = + and 5 = e + e for all real values of A ime =, he paricle is a he poin (, ) Find he speed of he paricle and is acceleraion vecor a ime = Find an equaion of he line angen o he pah of he paricle a ime = Find he oal disance raveled by he paricle over he ime inerval (d) Find he coordinae of he posiion of he paricle a ime = 5 (5B BC) A paricle moving along a curve in he y plane has posiion ( ( ), y ( )) a ime, where = and = ln( + ( ) ) A ime =, he paricle is a he poin (, 5) A =, he objec is a poin P wih coordinae Find he acceleraion vecor a ime = and he speed a ime = Find he y coordinae of P Wrie an equaion for he line angen o he curve a P (d) For wha value of, if any, is he objec a res? Eplain your reasoning 6 (6 BC) An objec moving along a curve in he y plane is a posiion ((), y()) a ime, where = sin ( e ) and = for > A ime =, he objec is a he poin + (6, ) Find he acceleraion vecor and he speed of he objec a ime = The curve has a verical angen line a one poin A wha ime is he objec a his poin? Le m() denoe he slope of he line angen o he curve a he poin ((), y()) Wrie an epression 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, and epression involving an improper inegral ha represens his value c
7 (6B BC) An objec moving along a curve in he y plane is a posiion ((), y()) a ime, where = an( e ) and = sec( e ) for A ime =, he objec is a he poin (, ) Wrie an equaion for he line angen o he curve a posiion (, ) Find he acceleraion vecor and he speed of he objec a ime = 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 ais? Eplain why or why no 8 (7B BC) An objec moving along a curve in he y plane is a posiion ( ( ), y( ) ) a ime wih = arcan + (,) = + for A ime =, he objec is a posiion and ln( ) Find he speed of he objec a ime = Find he oal disance raveled by he objec over he ime inerval Find () (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 9 (8B BC) A paricle moving along a curve in he y plane has posiion ( ( ), y( ) ) a ime wih = and = cos The paricle is a posiion (, 5) a ime = Find he acceleraion vecor a ime = Find he y coordinae of he posiion of he paricle a ime = 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 Answers Parameric Curves and Derivaives A 998 BC # 9% B 985 BC # 7% C 988 BC # 68% D 99 BC #5 5% 5 A 99 BC #6 9% 6 B 99 BC # 75% 7 E 997 BC # 9% 8 C 997 BC #8 6% 9 D BC # 79% C BC #7 9% A BC #7 6% Lengh of Parameric Curves 8 D 99 BC # 8% 9 D 997 BC #5 66% C 998 BC # 7% Vecors E 998 BC #77 87% D 985 BC # 69% 5 E 988 BC #5 75% 6 A 99 BC #8 86% 7 E 998 BC # 75% 8 C BC #8 6%