Calculus BC Bible. (3rd most important book in the world) (To be used in conjunction with the Calculus AB Bible)

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1 PG. Clculus BC Bile (rd most importt ook i the world) (To e used i cojuctio with the Clculus AB Bile) Topic Rectgulr d Prmetric Equtios (Arclegth, Speed) 4 Polr (Polr Are) 4 5 Itegrtio y Prts 5 Trigoometric Itegrls 7 Trigoometric Sustitutios 8 9 Itegrtio Techiques (Log Divisio, Prtil Frctios) 9 Improper Itegrls 0 Covergece/Divergece Tests Tylor/McLuri Series 4 Rdius d Itervl of Covergece 5 Writig Tylor series usig kow series 5 Seprtig Vriles L'Hopitl's Rule Work Hooke's Lw 7 Logisticl/Epoetil Growth 8 Euler's Method 8 Slope Fields 9 Lgrge Error Boud 9 Are s Limit 0 Summry of Tests for Covergece Power Series for Elemetry Fuctios Specil Itegrtio Techiques -4 After Clculus BC (Vector Clculus d Clculus III)

2 APPLICATIONS OF THE INTEGRAL Rectgulr Equtios y f () Legth of Curve (Arclegth) Surfce Are ( ) L + f d S.A. π f d + f '( ) Prmetric Equtios Prmetric equtios re set of equtios, oth fuctios of t such tht: f t y g( t) Legth of Curve ( ) + g' ( t) dt L f ' t or Surfce Are d dt + dt dt S.A. π g t ( f '( t) ) + g' ( t) dt Speed Equtio : speed dt + d dt EX# : If smooth curve c is give y the equtio, f (t) d y g(t), the the slope of c t (, y) is d dt, d dt d dt 0 d y d d d d d y d d d y d d d dt d d dt d dt d y d d dt t y 4 t 4 d dt t d dt d dt t t t 0 dt t Fid slope d cocvity t (, ) Must fid t t 4 (t 4) t 4 t 4 d (,) 8 d y d t 4 Cocve up d y d t t d y d t t t

3 Polr Coordites Polr coordites re defied s such: r cosθ r + y y r siθ θ t y Are Uder Curve A or ( f ( θ)) dθ A A r dθ A Are Betwee Two Curves ( f ( θ) ) g( θ) or ( R) r dθ dθ EX# : Fid the Are of oe lef of the three-lef rose r cos θ. (Oe lef is formed from -π to π ) π A (cos θ) dθ 9 π cos θ dθ 9 π π π 8 0 π 8 0 π π 4 π cosθ dθ π 9 si θ θ 4 8 π EX# : Fid the Ier Loop of r siθ. (siθ t π d 5π to form the ier loop) 5π 5π A π ( siθ) dθ π π π ( 4siθ + 4si θ) dθ π 5π ( 4siθ cosθ ) dθ siθ θ + cosθ π π EX# : π cosθ 4siθ + 4 dθ 5π π Fid the Totl Are of r siθ. (Hlf the figure is etwee 5π d π, so we doule tht re) π A 5π ( siθ) dθ π 5π π 5π ( 4siθ + 4 si θ) dθ ( 4siθ cosθ ) dθ θ + 4 cosθ si θ π EX#4 : π 5π π 5π cosθ 4siθ + 4 dθ 9π π + Fid the Are of the Outer Loop of r siθ. (Whole figure Ier Loop) A

4 EX#5 : Fid the Are of the eclosed regio etwee r cosθ d r cosθ. cosθ cosθ whe cosθ which is θ π d θ 5π. They form two ideticl eclosures. Fid the Are of oe eclosure d doule it. They meet t θ π i the first eclosure. You hve to figure out which equtio outlies your ecosure. r outlies the eclosure from θ 0 to θ π d r outlies the eclosure from θ π to θ π. π A ( cosθ) dθ + π 0 π ( cosθ) dθ π INTEGRATION TECHNIQUES Itegrtio y Prts Itegrtio y Prts is doe whe tkig the itegrl of product i which the terms hve othig to do with ech other. 5π u dv uv v du Refer to the Clculus AB Bile for the geerl techique. If the derivtive of oe product will evetully rech zero, use the tulr method. Tulr method : List the term tht will rech zero o the left d keep tkig the derivtive of tht term util it reches zero. List the other term o the right d tke the itegrl for s my times s it tkes for the left side to rech zero. EX : cos d + cos si 0 cos Multiply the terms s show. The first oe o the left multiplies with the secod oe o the right d the sig is positive, the secod term o the left mtches with the third oe o the right d is egtive, d cotiue o util the lst term o the right is mtched up, ltertig sigs s you go. cos d si + cos + C 4

5 Specil cses (Itegrtio y Prts) ( l must e u whe doig itegrtio y prts ) u l l d l d l + C dv d du v (Whe either fuctio goes wy, you my pick either fuctio to e u, ut you must pick the sme kid of fuctio throughout) ( i.e. If you pick the trig. fuctio to e u the you must pick the trig. fuctio to e u the et time lso. You will perform itegrtio y prts twice.) st time e si d e si e cos d u si dv e e si e cos e si d du cos d v e e si e cos e si d (Add to other side) d time e si d u cos dv e e si d du si d v e Trigoometric Itegrls sim cos d if is odd, sustitute u si if m is odd, sustitute u cos e si e cos e si e cos + C Useful Idetity: cos si Useful idetity: si cos If m d re oth eve, the reduce so tht either m or re odd, usig trigoometric (doule gle) idetities. Doule gle idetities : si si cos cos cos si Hlf - Agle Idetities : si cos cos + cos EX# : si cos 4 d si ( cos )cos 4 d d du si cos 4 cos u 4 u (u cos du -si d) u5 5 + u7 7 + C 5 cos 5 + ( cos )7 7 + C EX# : si cos 5 d cos d si si (Ech cos si ) cos d du si si 4 + si u u 4 + u (u si du cos d) u u5 5 + u7 7 + C ( si ) si ( si )7 7 + C

6 EX# : si cos 4 d cos d si cos si + cos i d 8 si d + 8 si cos d si si cos d cos + cos si 8 cos 4 d + 8 u du u si du cos si u + C si 4 4 si + + C 48 cos 4 tm sec d If is eve, sustitute u t Useful idetity: t sec If m is odd, sustitute u sec Useful idetity: t sec If m is eve d is odd, reduce powers of sec loe Useful idetity: t sec EX# : sec 4 t d sec ( + t )t d ( sec + t ) d du sec t + t 4 u + u 4 (u t du sec d) u + u5 5 + C ( t ) + ( t )5 5 + C EX# : sec 5 t d d ( t sec ) d sec t sec 4 t d sec t sec 4 sec sec t sec 4 sec (u sec du sec t d) u 4 u du 5 u5 5 u7 7 + C sec 5 ( sec )7 7 + C Trigoometric Sustitutios If the itegrd cotis the sustitute siu d cosu du The rdicd ecomes ( si u) ( si u) cos u cosu If the itegrd cotis + the sustitute tu d sec u du The rdicd ecomes ( + t u) ( + t u) sec u secu If the itegrd cotis the sustitute secu d secu tu du The rdicd ecomes sec u ( sec u ) t u tu

7 EX# : 9 d 9 siθ cosθ dθ 9 si θ siθ 9cos θ cosθ dθ 9 cos θ d d cosθ dθ 9θ + 9si θ 4 + C 9 rcsi + 9 cosθ dθ 9 + cosθ dθ 9 + C 9 siθ 9 rcsi C siθ EX# : d + 5 EX# : d ( Sustitute 4 tθ d 4sec θ dθ ) Sustitute 5secθ d 5secθ tθ dθ 4 tθ 4 sec θ dθ 5sec θ 5 + t θ 5secθ tθ dθ 5secθ 4 tθ 4sec θ dθ 5 sec θ + t θ 5secθ tθ dθ 4 tθ 4 sec θ dθ sec θ 4 tθ secθ dθ 4 secθ + C C + C 5secθ 5 t θ 5secθ tθ dθ 5secθ 5 t θdθ 5( sec θ )dθ ( 5sec θ 5)dθ 5 tθ 5θ + C rcsec 5 + C 5 5rcsec 5 + C θ 4 tθ 4 tθ θ 4 + 5secθ 5 5 secθ θ 5 7

8 Itegrtio Techiques Log Divisio (Must use log divisio if the top fuctio hs the sme or greter power s the ottom fuctio) Step - Divide the umertor y the deomitor. Seprte the itegrd ito seprte itegrls for ech term EX. + d + d l Add d sutrct terms i the umertor + d C + + d ( + ) + 4 d l rct + + C d Prtil Frctios ( Used whe the deomitor is fctorle) Before performig prtil frctios, you must use log divisio if the degree of the umertor is greter or equl to the deomitor. Step - Fctor the deomitor of the fuctio Step - Seprte ito prtil frctios Rewrite s frctiol terms - ritrry costts divided y ech fctor of the deomitor. If fctor is repeted more th oce, write tht my terms for tht fctor 4 EX# : d 4 4 d A + B 4 + d ( + ) Multiply y the commo deomitor. 4 A + + B( 4) Plug i the roots of the deomitor d solve for the costts Let 4 5 7A A 5 7 Let 7B B 7 The itegrl hs ee reduced dow to somethig much more resole d ow we re le to itegrte. Therefore, 4 d d 5 7 l l + + C 8

9 EX# : d + + d A + B + + Multiply y the commo deomitor. + A + + B( + ) + C Plug i the roots of the deomitor d solve for the costts Let 0 A Let C C Let 5 4A + B + C B C ( + ) d The itegrl hs ee reduced dow to somethig much more resole d ow we re le to itegrte. Therefore, + + d + + d l l C Simple prtil frctios ( Other method) EX# : 8 + ( 5) d A + B + 5 d d l + + l 5 + C 8 A( 5) + B( + ) 8 A 5A + B + B l C A + B 0 A + B 0 5A + B 8 5A + B 8 8A 8 A B Improper Itegrls f ( ) d lim f ( ) d d f ( ) d + f ( ) f I the first cse: If f ( )coverges the the itegrl eists If f ( )diverges the the itegrl is or EX # : e d lim e d lim c e ( ) lim e c d + e 0 + e e So e d coverges EX # : d d l lim l l So d diverges. EX # : d d lim 0 + So d coverges. EX #4 : d 0 d 0 0 lim 0 + So d diverges. 0 9

10 Geerl Series d Sequeces For the sequece { } m, if lim eists, the the sequece coverge to L. If lim does ot eist, the the sequece diverges to or. Boudess refers to whether or ot sequece hs fiite rge If If { } m { } m The th prtil sum S coverges, the diverges, the { } m { } m is ouded is uouded refers to the sum of the first terms of sequece. If d oth coverge, the + does lso d + + Covergece Tests ( PARTING CC) th term test (Used to show immedite divergece) If top power is the sme or greter th the ottom power the the series is diverget. If ottom power is greter we must proceed d use differet test, ecuse this test cot prove covergece. + ( ) EX# : d 5 These re diverget y the th term test. 5 + P -Series EX# : EX# : d d coverges if p > d diverges if p p Coverge y p-series. Diverge y p-series Itegrl Test for Covergece (for oegtive sequeces) (Used if you kow the itegrl of the series) EX# : EX# : 0 + coverges iff f ( )d coverges diverges iff f ( )d diverges 0 + d rct 0 lim rct rct 0 π 0 π d l lim l l 0 so series Diverges so series Coverges 0

11 Altertig Series Test for Covergece (for ltertig sequeces) EX# : The series ( ) coverges if { } is positive decresig sequece d lim 0. ( ) Coverges ecuse the deomitor is lrger i the ltertig series. + Altertig series c ot e used to prove divergece. Direct Compriso Test for Covergece d Divergece (for positive sequeces) EX# : EX# : If coverges d 0 the coverges If diverges d 0 the Diverges sice Coverges sice > d we kow 7 + < 7 diverges d we kow diverges y p-series. 7 coverges y geometric series. Limit Compriso Test (for oegtive sequeces) (Used o messy lgeric series) Assume lim > 0 If coverges, the EX# : 0 lim lim Diverges sice we compred with diverget series. EX# : lim lim Coverges sice we compred with coverget series. Geometric Series Test EX# : coverges. If diverges, the diverges. Limit is fiite d positive. 7 Limit is fiite d positive. A geometric series r coverges iff r <. A geometric series r diverges iff r. m If geometric series coverges, the r r m Sum : S r r m 5 Coverges ecuse r 0 5. The sum of the series is S 5 5 m EX# : 7 Diverges ecuse r 7. No sum sice series diverges. 0

12 Rtio Test (for oegtive series) (Used for relly ugly prolems) EX# : EX# : The series coverges if lim The series diverges if lim If lim + ( )! lim e ( ) 5 lim + + < >, the this test is icoclusive. ( + )! e +! e ( + ) ( + )! lim e lim + 5 e! lim Root Test for Covergece (for oegtive series) EX# : The sequece coverges if lim < The sequece diverges if lim > If lim ( ) lim e, the this test is icoclusive. e lim e e ( + ) e 5 + lim 5 so series Coverges so series Diverges so series Coverges EX# : ( ) 7 lim lim so series Diverges Telescopig Series ( Limit 0 d the terms get smller s we pproch ) If series is telescopig series the it is coverget. EX# : EX# : + is coverget. The sum is coverget. The sum

13 SERIES AND SEQUENCES Power Series : A power series is defied s c c 0 + c + c + c Tylor s Theorem d Relted Series (If the series is cetered t c 0 the it is McLuri Series) to the th term: f ( ) f ( c) + f ( c) ( c) + f ( c) ( c) + f ( c) ( c) f (c) Tylor s theorem for pproimtig f Tylor series of f ( ) this is power series!! 0! f c! ( c)! ( c) EX# : Fid the fourth degree McLuri Series cetered t c 0 f ( ) cos f ( 0) f ( ) f f f f 4 ( ) si f 0 ( ) cos f 0 ( ) si f ( 0) 0 ( ) cos f 4 ( 0) for f ( ) cos.! 0 f ( ) 0 f ( )!! + 4 4! 0! ! + 0 4! ( )! ! EX# : Fid the fourth degree Tylor Series for f f ( ) l f 0 f ( ) 0 + f f f f 4 ( ) ( ) f ( ) f l cetered t c. f f ( ) ( )!!! f ( ) ( ) ( ) f 4 4 +! +! + 4! ( )4 4! ( ) + + EX # : f 5 f 4 f f 7 rd Degree Tylor Polyomil : P ( ) 5 + 4! Usig rd Degree Tylor Polyomil to pproimte.: P (.) 5 + 4(. ) +.! + 7 (. )! ( )!

14 Rdius of covergece / Itervl of covergece EX# : Fid the rdius d itervl of covergece ( +) 0 + < + < < + < < < + Geometric Series 0 < Coverges iff r < Do't hve to check edpoits ecuse edpoits of geometric series ever work. I'll check ywy. Check edpoits Check, Check, 0 Diverges ( ) Diverges 0 EX# : Fid the rdius d itervl of covergece ( ) ( 4) Use Rtio Test 0 lim ( 4) + ( +) + ( 4) ( 4) lim + < 4 < < 4 < < < 7 Check edpoits Check, ( ) Diverges 0 ( ) Check 7, Coverges 0 Sice 7 works iclude 7 i itervl. Ceter of covergece: Rdius of covergece: Itervl of Covergece: < < Ceter of covergece: 4 Rdius of covergece: Itervl of Covergece: < 7 Whe fidig the itervl of covergece usig rtio test : If lim 0 the the series coverges from to. If lim the the series coverges t the ceter c oly. EX# : ( 4) lim! 0 so the series coverges from to. EX# : ( +)! lim 0 ( 4) + + ( +)! ( 4) + ( 4) lim + ( +)!! ( 4) lim! ( +) + ( +)! + ( +)! so the series coverges t the ceter oly. ( +) + ( +)! lim + ( 4) ( +) 0 ( +)! lim ( +) ( +) 4

15 Memorize these kow series Series for e : + +! +! +... Series for! Series for si : EX# : 0 + : ! + 5 5!... Series for cos : ( + )!! + 4 4!...! Use the series from ove to write ech of the followig series f ( ) cos f ( ) 4! + 8 f ( ) cos f ( )! + 5 f ( ) si f ( )! + 5 f ( ) e f ( ) + + e f f ( ) f ( ) 0 d f 0 4 4!... ( Replce with ( i the kow cosie series) )! 0 + 4!... (Multiply the kow cosie series y )! 0 + 5!... (Tke the itegrl of the kow cosie series) ( + )!! ( )! : (Replce with i the kow e series)! 0! + 4! : DIFFERENTIAL EQUATIONS ( Seprtig Vriles) Seprle Differetil Equtios 0 If P( ) + Q( y) d 0 the Q y EX# : Fid the geerl solutio give d y d y y d y d 0 0 (Tke the defiite itegrl of e ) (Replce with i the kow series (Multiply the foud series ) + y ) P ( )d the Q ( y ) P( )d EX# : Fid the prticulr solutio y f () for EX# give (, 5) y + C y + C y + C C 7 C y + 7 y + 7 EX# : Fid the prticulr solutio y f () give d d y y d ( 0, 5 ) d y l y + C y e +C 5 y C e 5 C y 5e

16 f () L Hôpitl s Rule : If lim g() 0 0 or, the lim EX # : EX # : e lim lim e so lim lim e so lim 9 0 Idetermite forms : 0,,, 0, 0 0 f () g() lim f '() g'() lim e Remider: I limits 0 d 0 Idetermite forms must e chged to 0 0 or EX # : lim l 0 so we must covert to lim l so lim 0 + lim EX #4 : l y lim l y lim Work lim + l + + W F d cosθ so we must rig dow the so let y lim + l y lim l y l + 0 y e or the dot product of the Force vector d the distce vector. W F d W F d ; where F is cotiuously vryig force. l + l y lim 0 0 d tke l of oth sides. l y lim Hooke s Lw : The force F required to compress or stretch sprig (withi its elstic limits) is proportiol to the distce d tht the sprig is compressed or stretched from its origil legth. Tht is, F k d W k d + where the costt of proportiolity k (the sprig costt) depeds o the specific ture of the sprig. EX : Compressig Sprig A force of 750 pouds compresses sprig iches from its turl legth of 5 iches. Fid the work doe i compressig the sprig dditiol iches. 750 k W k 50 F( ) k so F( ) d 75 ich pouds

17 Logisticl Growth Epoetil Growth dt ky y L L ; y ; L Y 0 + e kt Y 0 k costt of proportiolity L crryig cpcity 7 dt ky ; y Cekt k growth costt C iitil mout Y 0 iitil mout Logistic Growth dt ky( y L ) where L is the crryig cpcity (upper oud) d k is the costt of proportiolity. If you solve the differetil equtio usig very tricky seprtio of vriles, you get the geerl solutio: EX# : y L + e kt ; L Y 0 Y 0 A highly cotgious pikeye (scietific me: Cojuctivitus itchlikecrzius) is rvgig the locl elemetry school. The popultio of the school is 900 (icludig studets d stff), d the rte of ifectio is proportiol oth to the umer ifected d the umer of studets whose eyes re pus-free. If sevety-five people were ifected o Decemer 5 d 50 hve cotrcted pikeye y Decemer 0, how my people will hve gotte the gift of crusty eyes y Christms Dy? Solutio Becuse of the proportiolity sttemets i the prolem, logistic growth is the pproch we should tke. The upper limit for the disese will e L 900; it is impossile for more th 900 people to e ifected sice the school oly cotis 900 people. This gives us the equtio L y 900 +e kt Five dys lter, 50 people hve cotrcted pikeye, so plug tht iformtio to fid k: Filly, we hve the equtio y Decemer 5, so t e k 5 50(+e 5k ) 900 e 5k e 5k 5 55 l e 5k l 55 5k k We wt to fid the umer of ifectios o t +e 900 y +e ( )(0) y So lmost 558 studets hve cotrcted pikeye i time to ope presets.

d 0 0 d 0. 0.5 0..05 0.05 d Slope Fields Drw the slope field for d y Plug ech poit ito d d grph the tget lie t the poit At (, ).

18 Euler s Method Uses tget lies to pproimte poits o the curve. New y Old y + d d EX# : Give: f (0) Origil (0, ) d y d : chge i Use step of h 0. to fid f (0.) New y + (0.)(0) f ( 0.) At (0, ) New y + (0.)(0.5).05 f ( 0.).05 At (0., ) New y.05 + (0.)(0.05).0455 f ( 0.).0455 At (0.,.05) d Derivtive (slope) t the poit. d 0 0 d d Slope Fields Drw the slope field for d y Plug ech poit ito d d grph the tget lie t the poit At (, ). At (, ). At ( 0, ) d d d 0 0 EX : Here re the slope fields for the give differetil equtios d y ( 8 y) d + y The slope fields give ALL the solutios to the differetil equtio. 8

19 Lgrge Error Boud Error f ( ) P ( ) R ( ) where R ( ) f + ( z) ( c) + f + ( +)! ( z) the mimum of the ( +)th derivtive of the fuctio EX # : cos! cos ( 0. ) 4! ( ) f 5 ( z) ( c) 5 f 5 ( z) si z We eed si z to e s lrge s possile (which is ecuse si z ) R 4 5! 0. R 4 ( 0.) < 5 5! EX # : f f 5 f 7 f d Degree Tylor Polyomil : P ( ) ( ) Usig d Degree Tylor Polyomil to pproimte.: P. ( c) Lgrge Error Boud : R ( ) f z! + 5. (. ) 0.00 Error!! + 7 (. ).55! Are s limit Are lim i f + i height width Summtio formuls d properties i itervl ) c c ) i + ) i + i i i ( + ) 4) i + 5) ( i ± i ) i ± i ) k i k i, k is costt 4 i i i i i i EX# : Are f ( ) [ 0, ] sudivisios Fid Are uder curve. lim f i 0 i lim i i lim 4 i i + lim EX# : lim Aswer : d (You hve to recogize tht the prolem is skig for Are from 0, 0 [ ] with sudivisios) 9

20 Summry of tests for Series Test Series Coverges Diverges Commet This test cot e th-term lim 0 used to show covergece. Geometric Series r r < r Sum: S r 0 Telescopig + lim L Sum: S L p-series Altertig Series p > p p ( ) 0 < + d lim 0 Remider: R N N + Itegrl ( f is cotiuous, positive, d decresig), f 0 f ( )d coverges f ( )d diverges Remider: 0 < R N < f ( )d N Root lim < lim > Test is icoclusive if lim. Rtio lim + < lim + > Test is icoclusive if lim +. Direct compriso (, > 0) 0 < d coverges 0 < d diverges Limit Compriso (, > 0) lim L > 0 d coverges lim L > 0 d diverges 0

21 Fuctio Power Series for Elemetry Fuctios Itervl of Covergece ( ) + ( ) ( ) + ( ) 4 + ( ) ( ) + 0 < < ( ) + < < l + ( ) ( ) ( ) + 0 < e + +! +! + 4 4! !! + < < si! + 5 5! 7 7! + 9 9! + + ( + )! + < < cos! + 4 4!! + 8 8! +! + < < rct rcsi ! +! ( k ) ( + ) k + k + k k + k k + k k!! *The covergece t ± depeds o the vlue of k. ( k ) ( k ) 4 4! + < < *

22 Procedures for fittig itegrds to Bsic Rules Techique Epd (umertor) Seprte umertor Emple ( + e ) d + e + e d + e + e + C + d d rct + l + + C Complete the squre d d rcsi ( ) + C Divide improper rtiol fuctio d + + d rct + C Add d sutrct terms i the umertor d d d l C Use trigoometric Idetities t sec si cos cot csc Multiply d Divide y Pythgore cojugte d + si si cos d si + si si d d cos si cos d sec sec t t + sec + C si d si Hppy stuig d good luck! compiled y Kris Chisguthum, clss of 97 edited d recompiled y Michel Lee, clss of 98 edited d recompiled y Doug Grhm, clss of Forever

23 VECTOR CALCULUS A vector hs two compoets: mgitude d directio, epressed s ordered pir of rel umers: (, ) where 0 d y y 0 :( 0, y 0 ) d, y vector s directed lie segmet respectively. θ is the gle etwee two vectors. Legth (mgitude) of vector : + Uit vectors : i (, 0) j (0,) + ( +, + ) -, c ( c, c ) i + j c c + cosθ represet the iitil d termil poit of the If i 0, the d re perpediculr Agle etwee two vectors : cosθ Projectio (of vector oto ) i pr Vector - Vlued Fuctios If c, the d re prllel ( ) pr + pr ' where ' is perpediculr to A vector - vlued fuctio hs domi ( set of rel umers) d rule (which ssigs to ech umer i the domi vector) costi + sitj e.g. F t If F( t) f ( t)i + f ( t) j lim F( t)eists oly if lim f ( t) d f ( t) eist lim F( t) lim f ( t) )i + lim f ( t) ) j t c t c t c A vector vlued fuctio is cotiuous t t 0 if its compoet fuctios re cotiuous t t 0 f ( t) + f ( t) dt f ( t) dt + f ( t) dt F t F t Motio (defied s vector vlued fuctio of time) Positio: r( t) ( t)i + y( t) j Velocity: v( t) ( t)i + y ( t) j Speed: Accelertio: Tget vector: ( t)i + y ( t) j r ( t) / r ( t) / r ( t) T ( t) / T ( t) v t t T t Curvture: k T t Norml vector: N t

24 OTHER DIFFERENTIAL EQUATIONS (Clculus III) Lier First Order Differetil Equtios A lier first order differetil equtio hs this form: Solve usig this equtio: y e S e S Secod Order Lier Differetil Equtios Q( )d A secod order differetil equtio hs this geerl form: Homogeeous Equtios A secod order lier equtio is homogeeous if g d y d + d + cy 0 To solve, mke use of these equtios: d + P y Q where S P d d y d + d + cy g( ) 0; thus s + 4c d s 4c There re three cses depedig o the solutio to 4c If 4c > 0 y C e s + C e s where C d C re costts If 4c 0 y C e s + C e s where C d C re costts If 4c < 0 y C e u siv + C e u cosv where C d C re costts u d v 4c Nohomogeeous Equtios A secod order differetil equtio is ohomogeeous if g To solve:. Fid the solutio for d y d + d + cy 0 See Homogeeous Equtios. Fid y p y p u y + u y where: ( y g) u y y y y y u g y y y y d y d y re solutios i Step. Epress the solutio s y y p + the solutio give y Step 4 does ot equl zero

Calculus BC Bible. (3rd most important book in the world) (To be used in conjunction with the Calculus AB Bible)

Calculus BC Bible. (3rd most important book in the world) (To be used in conjunction with the Calculus AB Bible) PG. Clculus BC Bile (3rd most importt ook i the world) (To e used i cojuctio with the Clculus AB Bile) Topic Rectgulr d Prmetric Equtios (Arclegth, Speed) 3 4 Polr (Polr Are) 4 5 Itegrtio y Prts 5 Trigoometric