BC Calculus Review Sheet

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Nickolas Holland
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1 BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re of differet ubouded regio uder f() from (,b], where lim f( ) or, where the re is represeted by b f ( ),(sometimes clled verticl improper). Give f(), fid rc legth of the fuctio o the itervl (, f()) d (b,f(b)). 4. Give curve i prmetric form where f t, y g t, fid the rc legth of the curve o the itervl t1, t. 5. Give F(, y) y d iitil poit (, y ) ( 11, ), fid pproimte vlue for o o f (.) 1 d Give the differetil equtio of the form dp P kp 1 for P s fuctio of t, where k d L L re costts. Rh 010

2 7. Give the differetil equtio dp 1P 4P where P is mesurig the umber iml preset o 0. Fid the vlue of P whe the umber of these imls is icresig the fstest. 8. Give the differetil equtio dp 100P 400P where P is mesurig the umber iml preset o 0. Determie the lim Pt ( ). 9. Give tht lie segmet hs edpoits of (1,) d (5,10), write set of prmetric equtios for the lie tht psses through these two poits. 10. Give the positio fuctio of two prticles i prmetric form, () t f(), t y () t g() t d 1 1 () t h(), t y () t k() t prticles itersect or collide., determie if the 11. Give set of prmetric equtios where f t, y gt, fid or the slope of the tget lie. Rh 010

3 1. A pth of prticle is described with set of prmetric equtios f t, y g t. Fid the equtio of the tget lie whe t = to. 1. A pth of prticle is described with set of prmetric equtios f t y g t,.. Fid ll vlues of t where the prticle s pth is verticl. b. Fid ll vlues of t where the prticle s pth is horizotl. 14. Give set of prmetric equtios where f t, y gt, fid 15. Give the positio vector of prticle movig i the ple is rt () t (), yt (). Fid the velocity vector. 16. The positio vector of prticle movig i the ple is rt () t (), yt (). Fid the ccelertio vector. 17. The positio vector of prticle movig i the ple is rt () t (), yt (). Fid the speed of the prticle t momet t time t =. 18. Give the velocity vector vt ( ) '( t), y'( t) d positio vector t t = 0 s vector t time t =. (), 0 y() 0, fid the positio Rh 010

4 19. Give vt ( ) '( t), y'( t) determie whe the prticle is stopped. 0. Give vt ( ) '( t), y'( t) fid the slope of the tget lie to the vector t t Give prticle moves log fuctio y = +1, the rte of chge of or t for t>0 d (0)=1. Fid the prticle s positio t time t =.. Fid the slope of the tget lie to the polr curve. r f( ). Give polr curve r f, fid horizotl tgets to curve. 4. Fid verticl tgets to polr curve r f. 5. Fid the re iside oe of the petls o the flower described by r cos( ). Rh 010

5 r ( ) 1 6. Fid the re outside but iside r ( ). r ( ) 1 7. Fid the rc legth of fuctio from d b Fid the sum Determie if the series 4 1 coverges or diverges 0. Determie if the series coverges or diverges 1 1 Rh 010

6 1. Determie if the series coverges or diverges si. Determie if the series coverges or 1 diverges.. Determie if the series coverges or diverges Determie if the series 1 coverges or 1 diverges. 5. Write series for cos where is iteger 6. Write series for l( 1 ) cetered t = 0. Rh 010

7 If f( )... represets T 4 f Tylor Polyomils bout = 0, fid 8. Write the th degree Tylor Polyomil for f() t = c. 9. Give Tylor series, fid the Lgrge form of the remider for the 4 th term. 40 Let S 4 be the sum of the first 4 terms of ltertig series for f(). Approimte f( ) S Give the polyomils f 1..., wht is f()?!! 4. Give the polyomil f......, wht is! 5! 7! 1! f()? 4. Give the polyomil f wht is f()? ! 4! 6!!, Rh 010

8 44. Fid the itervl of covergece of series. f( ) 45. Fid lim g ( ) 46. Fid 1 Rh 010

9 BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl).. Fid the re of differet ubouded regio uder f() from (,b], where lim f( ) or, where the re is represeted by b f ( ),(sometimes clled verticl improper). Give f(), fid rc legth of the fuctio o the itervl (, f()) d (b,f(b)). This is wht you thik of doig... b Set up lim f( ) b 1 coverges. Set up lim b ( ) coverges. f Use the itegrl: L 1 f. b to see if the re diverges or to see if the re diverges or 4. Give curve i prmetric form where f t, y g t, fid the rc legth of the curve o the itervl t1, t. Use the itegrl: L t t1 5. Give F(, y) y d iitil poit (, y ) ( 11, ), fid pproimte vlue for o o f (.) 1 d 01. Thik bout Euler s Method to drw tget lies d pproimte log the tget lies. First clculte the slope t (1,1) d write equtio of tget lie to f t (1,1). Use this lie to pproimte ew poit t =1.1 usig 01.. This gives you secod poit to repet the procedure gi. Write other tget lie with ew slope d pproimte the vlue of f(1.) by movig log this secod tget lie to the poit = Give the differetil equtio of the form dp P kp 1 for P s fuctio of t, where k d L L re costts. Seprte the differetils, use prtil frctios, itegrte, use iitil coditio to solve for the costt d ed up with equtio of the form: L P 1 Mkt Ae Rh 010

10 7. Give the differetil equtio dp 1P 4P where P is mesurig the umber iml preset o 0. Fid the vlue of P whe the umber of these imls is icresig the fstest. dp 7. First otice tht 1P 4P is prbol, so dp P rewritig it i the form kp 1 or L dp P dp 1P 1 tells us tht the v whe P=0 or 0 P=. The umber of imls is icresig fstest t the midpoit of 0 d or Give the differetil equtio dp 100P 400P where P is mesurig the umber iml preset o 0. Determie the lim Pt ( ). 9. Give tht lie segmet hs edpoits of (1,) d (5,10), write set of prmetric equtios for the lie tht psses through these two poits. dp P Fctorig 100P 4P 100P1 we c see 00 dp tht 0 whe P=0 d P=00. Therefore, P=00 is the lim Pt ( ) sice the grow icreses betwee P = 0 d P = 00 but stops t P = 00. Determie the slope (m) of the lie segmet (m=), write equtio for the lie segmet usig poit slope form (y=(-1)+), d the rewrite this equtio s prmetric equtios where (t)=t d y(t)=(t-1)+ or y(t)=t. Select vlues for t from kowig tht or t strts t 1 d goes to 5 so 1 t Give the positio fuctio of two prticles i prmetric form, () t f(), t y () t g() t d 1 1 () t h(), t y () t k() t prticles itersect or collide., determie if the For the pths to itersect ( t ) ( t ) d 1 1 y ( t ) y ( t ). Solve these equtios simulteously 1 1 to fid the time whe the pths itersect. For the prticles to collide they must be t poit t the sme time. Determie the times whe ech prticle is t the give poit. If the times mtch, the prticles collide, otherwise their pths oly cross. 11. Give set of prmetric equtios where f t, y gt, fid or the slope of the tget lie. Recll tht Rh 010

11 1. A pth of prticle is described with set of prmetric equtios f t, y g t. Fid the equtio of the tget lie whe t = to. 1. A pth of prticle is described with set of prmetric equtios f t y g t,.. Fid ll vlues of t where the prticle s pth is verticl. b. Fid ll vlues of t where the prticle s pth is horizotl. Determie the poit where the prticle is t ( ), yt ( ). The fid the slope of the grph t the time o t=t o by clcultig o tt0. The write the equtio of the lie i poit-slope form.. Determie the times whe 0 d 0. b. Determie the times whe 0 d Give set of prmetric equtios where f t, y gt, fid d First fid d the clculte. 15. Give the positio vector of prticle movig i the ple is rt () t (), yt (). Fid the velocity vector. Recll tht the velocity vector is vt ( ) '( t), y'( t) which mes tht you must differetil (t) d y(t) respect to t d the write vector. 16. The positio vector of prticle movig i the ple is rt () t (), yt (). Fid the ccelertio vector. Recll tht the ccelertio vector is t ( ) ''( t), y''( t) which mes tht you must differetil (t) d y (t) respect to t d the write vector. 17. The positio vector of prticle movig i the ple is rt () t (), yt (). Fid the speed of the prticle t momet t time t =. Recll tht speed is the mgitude of the velocity vector d is foud by clcultig '( ), '( ) '( ) '( ) v y y 18. Give the velocity vector vt ( ) '( t), y'( t) d positio vector t t = 0 s vector t time t =. (), 0 y() 0, fid the positio Recll tht the positio vector is ( 0 ) '( t ), y ( 0 ) y '( t ) 0 0 Rh 010

12 19. Give vt ( ) '( t), y'( t) determie whe the prticle is stopped. You must cosider both (t) d y (t). You eed to determie whe both (t) d y (t) equl zero. 0. Give vt ( ) '( t), y'( t) fid the slope of the tget lie to the vector t t 1. y '( t) You must clcultor d evlute this epressio t '( t) t Give prticle moves log fuctio y = +1, the rte of chge of or t for t>0 d (0)=1. Fid the prticle s positio t time t =. Fid the chge i the directio or () '() t Determie o 0 0 the y coordite usig the fuctio y = f() ( 1 ) Write the coordite: (1, 884). Fid the slope of the tget lie to the polr curve. r f( ) Recll tht r cos, y r si d d. Give polr curve r f, fid horizotl tgets to curve. Recll tht r cos, y r si d the fid where r si 0 d where r cos 0 4. Fid verticl tgets to polr curve r f. Recll tht r cos, y r si d the fid where r cos 0 d where r si 0 5. Fid the re iside oe of the petls o the flower described by r cos( ). Recll tht oe petl c be trced by d the re 6 6 c be foud by clcultig the itegrl Rh 010

13 r ( ) First fid the poits of itersectio d bd the 1 r ( b ). 1 itegrte r ( ) r1( ) d 6. Fid the re outside but iside r ( ) 1 7. Fid the rc legth of fuctio from d b. rcos y rsi Recll the d to covert from polr form to prmetric form. The use the itegrl for rc legth with prmetric equtios. Perform the itegrl b d d d Fid the sum. Notice tht the sum is geometric series where 1 r r 1 1 d so the sum is give by 1 9. Determie if the series 4 1 coverges or diverges Thik bout the th term i this series: lim test, sice the series diverges.. By the th term 0. Determie if the series coverges or diverges 1 1 Thik bout usig the itegrl test. lim 1 1 b 1 1 b 1 lim l b sice the itegrl diverges. b Therefore, the series diverges Rh 010

14 1. Determie if the series coverges or diverges. 1 1 Oe test you might thik of usig is the p-series test. Sice p 0 coverges. 1 si. Determie if the series coverges or 1 diverges. 1 si 0 1 Sice the series coverges d the series coverges by the compriso test.. Determie if the series coverges or diverges Usig the rtio test, 1 1 lim So the series coverges Determie if the series 1 coverges or 1 diverges. 1 Usig the ltertig series test, sice ech term decreses s pproches ifiity d coverges to 0, the the ltertig series coverges. 5. Write series for cos where is iteger 4 6 Recll tht 1 d the multiply cos !!! through by. 6. Write series for l( 1 ) cetered t = 0. 1 Recll tht l( 1 ) ( 1) Substitute for. Rh 010

15 If f( )... represets T 4 f Tylor Polyomils bout = 0, fid Notice tht f() is geometric series with = d r = the sum of the series c be writte s or 1 r so 8. Write the th degree Tylor Polyomil for f() t = c. T ( ) f( c) f '( c) c f "( c) ( ) c f () c! c! 9. Give Tylor series, fid the Lgrge form of the remider for the 4 th term. This error is o greter th the vlue of the 5 th term t some 1 f 1 vlue of betwee d c. R c 1! 40 Let S 4 be the sum of the first 4 terms of ltertig series for f(). Approimte f( ) S. 4 You should recogize this s the error for the 4 th term of ltertig series which is o greter th the bsolute vlue of the 5 th term. 41. Give the polyomils f 1..., wht is f()?!! f( ) e 4. Give the polyomil f......, wht is! 5! 7! 1! f()? 4. Give the polyomil f wht is f()? ! 4! 6!!, f() = si () f() = cos () Rh 010

16 44. Fid the itervl of covergece of series. Apply the rtio test to fid the itervl d the test covergece t the edpoits. f( ) 45. Fid lim g ( ) Check to see if you c use L Hopitl s Rule. Check to see if f( ) g( ) 0 or f( ) g( ). If this is true, the f( ) f '( ) lim. You my hve to repet these steps. g ( ) g '( ) 46. Fid 1 Use prtil frctio to set up two itegrls: A B. Solve for A d B d the complete 4 the itegrtio l C 7 4 Rh 010

BC Calculus Review Sheet. converges. Use the integral: L 1

BC Calculus Review Sheet. converges. Use the integral: L 1 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