BC Clculus Review Sheet Whe yu see the wrds.. Fid the re f the uuded regi represeted y the itegrl (smetimes f ( ) clled hriztl imprper itegrl).. Fid the re f differet uuded regi uder f() frm (,], where lim f( ) r, where the re is represeted y f ( ),(smetimes clled verticl imprper). Give f(), fid rc legth f the fucti the itervl (, f()) d (,f()). This is wht yu thik f dig... Set up lim f( ) cverges. Set up lim ( ) cverges. f Use the itegrl: L f. t see if the re diverges r t see if the re diverges r 4. Give curve i prmetric frm where f t, y g t, fid the rc legth f the curve the itervl t, t. Use the itegrl: L t t 5. Give F(, y) y d iitil pit (, y ) (, ), fid pprimte vlue fr f (.) d 0. Thik ut Euler s Methd t drw tget lies d pprimte lg the tget lies. First clculte the slpe t (,) d write equti f tget lie t f t (,). Use this lie t pprimte ew pit t =. usig 0.. This gives yu secd pit t repet the prcedure gi. Write ther tget lie with ew slpe d pprimte the vlue f f(.) y mvig lg this secd tget lie t the pit =.. 6. Give the differetil equti f the frm P kp fr P s fucti f t, where k d L L re cstts. Seprte the differetils, use prtil frctis, itegrte, use iitil cditi t slve fr the cstt d ed up with equti f the frm: L P Mkt Ae Rh 00
7. Give the differetil equti P 4P where P is mesurig the umer iml preset dy 0. Fid the vlue f P whe the umer f these imls is icresig the fstest. 7. First tice tht P 4P is prl, s P rewritig it i the frm kp r L P P tells us tht the v whe P=0 r 0 P=. The umer f imls is icresig fstest t the midpit f 0 d r.5. 8. Give the differetil equti 00P 400P where P is mesurig the umer iml preset dy 0. Determie the lim Pt ( ). 9. Give tht lie segmet hs edpits f (,) d (5,0), write set f prmetric equtis fr the lie tht psses thrugh these tw pits. P Fctrig 00P 4P 00P we c see 00 tht 0 whe P=0 d P=00. Therefre, P=00 is the lim Pt ( ) sice the grw icreses etwee P = 0 d P = 00 ut stps t P = 00. Determie the slpe (m) f the lie segmet (m=), write equti fr the lie segmet usig pit slpe frm (y=(-)+), d the rewrite this equti s prmetric equtis where (t)=t d y(t)=(t-)+ r y(t)=t. Select vlues fr t frm kwig tht r t strts t d ges t 5 s t 5 0. Give the psiti fucti f tw prticles i prmetric frm, () t f(), t y () t g() t d () t h(), t y () t k() t prticles itersect r cllide., determie if the Fr the pths t itersect ( t ) ( t ) d y ( t ) y ( t ). Slve these equtis simulteusly t fid the time whe the pths itersect. Fr the prticles t cllide they must e t pit t the sme time. Determie the times whe ech prticle is t the give pit. If the times mtch, the prticles cllide, therwise their pths ly crss.. Give set f prmetric equtis where f t, y gt, fid r the slpe f the tget lie. Recll tht Rh 00
. A pth f prticle is descried with set f prmetric equtis f t, y g t. Fid the equti f the tget lie whe t = t.. A pth f prticle is descried with set f prmetric equtis f t y g t,.. Fid ll vlues f t where the prticle s pth is verticl.. Fid ll vlues f t where the prticle s pth is hriztl. Determie the pit where the prticle is t ( ), yt ( ). The fid the slpe f the grph t the time t=t y clcultig tt0. The write the equti f the lie i pit-slpe frm.. Determie the times whe 0 d 0.. Determie the times whe 0 d 0. 4. Give set f prmetric equtis where f t, y gt, fid d First fid d the clculte. 5. Give the psiti vectr f prticle mvig i the ple is rt () t (), yt (). Fid the velcity vectr. Recll tht the velcity vectr is vt ( ) '( t), y'( t) which mes tht yu must differetil (t) d y(t) respect t t d the write vectr. 6. The psiti vectr f prticle mvig i the ple is rt () t (), yt (). Fid the ccelerti vectr. Recll tht the ccelerti vectr is t ( ) ''( t), y''( t) which mes tht yu must differetil (t) d y (t) respect t t d the write vectr. 7. The psiti vectr f prticle mvig i the ple is rt () t (), yt (). Fid the speed f the prticle t mmet t time t =. Recll tht speed is the mgitude f the velcity vectr d is fud y clcultig '( ), '( ) '( ) '( ) v y y 8. Give the velcity vectr vt ( ) '( t), y'( t) d psiti vectr t t = 0 s vectr t time t =. (), 0 y() 0, fid the psiti Recll tht the psiti vectr is ( 0 ) '( t ), y ( 0 ) y '( t ) 0 0 Rh 00
9. Give vt ( ) '( t), y'( t) determie whe the prticle is stpped. Yu must csider th (t) d y (t). Yu eed t determie whe th (t) d y (t) equl zer. 0. Give vt ( ) '( t), y'( t) fid the slpe f the tget lie t the vectr t t. y '( t) Yu must clcultr d evlute this epressi t '( t) t.. Give prticle mves lg fucti y = +, the rte f chge f r t fr t>0 d (0)=. Fid the prticle s psiti t time t =. Fid the chge i the directi r () '() t. Determie 0 0 the y crdite usig the fucti y = f() ( ) 884. Write the crdite: (, 884). Fid the slpe f the tget lie t the plr curve. r f( ) Recll tht r cs, y r si d d. Give plr curve r f, fid hriztl tgets t curve. Recll tht r cs, y r si d the fid where r si 0 d where r cs 0 4. Fid verticl tgets t plr curve r f. Recll tht r cs, y r si d the fid where r cs 0 d where r si 0 5. Fid the re iside e f the petls the flwer descried y r cs( ). Recll tht e petl c e trced y d the re 6 6 c e fud y clcultig the itegrl Rh 00
r ( ) First fid the pits f itersecti d d the r ( ). itegrte r ( ) r( ) d 6. Fid the re utside ut iside r ( ) 7. Fid the rc legth f fucti frm d. rcs y rsi Recll the d t cvert frm plr frm t prmetric frm. The use the itegrl fr rc legth with prmetric equtis. Perfrm the itegrl d d d 6 6 8. Fid the sum. Ntice tht the sum is gemetric series where r r d s the sum is give y 9. Determie if the series 4 cverges r diverges Thik ut the th term i this series: lim 0 test, sice the series diverges.. By the th term 0. Determie if the series cverges r diverges Thik ut usig the itegrl test. lim lim l sice the itegrl diverges. Therefre, the series diverges Rh 00
. Determie if the series cverges r diverges. Oe test yu might thik f usig is the p-series test. Sice p 0 cverges. si. Determie if the series cverges r diverges. si 0 Sice the series cverges d the series cverges y the cmpris test.. Determie if the series cverges r diverges. 4 Usig the rti test, lim 4 4 4 4 4 4 4 S the series cverges. 4. Determie if the series cverges r diverges. Usig the ltertig series test, sice ech term decreses s pprches ifiity d cverges t 0, the the ltertig series cverges. 5. Write series fr cs where is iteger 4 6 Recll tht d the multiply cs 4 6...!!! thrugh y. 6. Write series fr l( ) cetered t = 0. Recll tht l( ) ( ) Sustitute fr. Rh 00
7. If f( )... represets T 4 f Tylr Plymils ut = 0, fid Ntice tht f() is gemetric series with = d r = the sum f the series c e writte s r r s 8. Write the th degree Tylr Plymil fr f() t = c. T ( ) f( c) f '( c) c f "( c) ( ) c f () c! c! 9. Give Tylr series, fid the Lgrge frm f the remider fr the 4 th term. This errr is greter th the vlue f the 5 th term t sme f vlue f etwee d c. R c! 40 Let S 4 e the sum f the first 4 terms f ltertig series fr f(). Apprimte f( ) S. 4 Yu shuld recgize this s the errr fr the 4 th term f ltertig series which is greter th the slute vlue f the 5 th term. 4. Give the plymils f..., wht is f()?!! f( ) e 4. Give the plymil 5 7 f......, wht is! 5! 7!! f()? 4. Give the plymil f wht is f()? 4 6......! 4! 6!!, f() = si () f() = cs () Rh 00
44. Fid the itervl f cvergece f series. Apply the rti test t fid the itervl d the test cvergece t the edpits. f( ) 45. Fid lim g ( ) Check t see if yu c use L Hpitl s Rule. Check t see if f( ) g( ) 0 r f( ) g( ). If this is true, the f( ) f '( ) lim. Yu my hve t repet these steps. g ( ) g '( ) 46. Fid Use prtil frcti t set up tw itegrls: A B. Slve fr A d B d the cmplete 4 the itegrti. 7 7 4 l C 7 4 Rh 00