Chapter Taylor Theorem Revisited

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1 Captr 0.07 Taylor Torm Rvisitd Atr radig tis captr, you sould b abl to. udrstad t basics o Taylor s torm,. writ trascdtal ad trigoomtric uctios as Taylor s polyomial,. us Taylor s torm to id t valus o a uctio at ay poit, giv t valus o t uctio ad all its drivativs at a particular poit,. calculat rrors ad rror bouds o approimatig a uctio by Taylor sris, ad 5. rvisit t captr wvr Taylor s torm is usd to driv or plai umrical mtods or various matmatical procdurs. T us o Taylor sris ists i so may aspcts o umrical mtods tat it is imprativ to dvot a sparat captr to its rviw ad applicatios. For ampl, you must av com across prssios suc as 6 cos( ) + + ()!! 6! 5 7 si( ) + + ()! 5! 7! () All t abov prssios ar actually a spcial cas o Taylor sris calld t Maclauri sris. Wy ar ts applicatios o Taylor s torm importat or umrical mtods? Eprssios suc as giv i Equatios (), () ad () giv you a way to id t approimat valus o ts uctios by usig t basic aritmtic opratios o additio, subtractio, divisio, ad multiplicatio. Eampl Fid t valu o 0.5 usig t irst iv trms o t Maclauri sris. T irst iv trms o t Maclauri sris or ! is 0.07.

2 0.07. Captr ! T act valu o up to 5 sigiicat digits is also.80. But t abov discussio ad ampl do ot aswr our qustio o wat a Taylor sris is. Hr it is, or a uctio () providd all drivativs o ist ad ar cotiuous btw ad +. Wat dos tis ma i plai Eglis? As Arcimds would av said (witout t i prit), Giv m t valu o t uctio at a sigl poit, ad t valu o all (irst, scod, ad so o) its drivativs, ad I ca giv you t valu o t uctio at ay otr poit. It is vry importat to ot tat t Taylor sris is ot askig or t prssio o t uctio ad its drivativs, just t valu o t uctio ad its drivativs at a sigl poit. Now t i prit: Ys, all t drivativs av to ist ad b cotiuous btw (t poit wr you ar) to t poit, + wr you ar watig to calculat t uctio t at. Howvr, i you wat to calculat t uctio approimatly by usig t ordr st d t Taylor polyomial, t,,..., drivativs d to ist ad b cotiuous i t t closd itrval [, + ], wil t ( +) drivativ ds to ist ad b cotiuous i t op itrval (, + ). Eampl Tak si, w all kow t valu o si. W also kow t cos ad cos 0. Similarly si() ad si. I a way, w kow t valu o si ad all its drivativs at. W do ot d to us ay calculators, just plai dirtial calculus ad trigoomtry would do. Ca you us Taylor sris ad tis iormatio to id t valu o si?

3 Taylor Torm Rvisitd So Hc!! ( + ) ( ) si, si cos, 0 si, cos(), 0 si(), + + +! +!! + +! ( 0.90) ( 0.90) ( 0.90) ( 0.90) !!! T valu o si I gt rom my calculator is wic is vry clos to t valu I just obtaid. Now you ca gt a bttr valu by usig mor trms o t sris. I additio, you ca ow us t valu calculatd or si coupld wit t valu o cos (wic ca b calculatd by Taylor sris just lik tis ampl or by usig t si + cos idtity) to id valu o si at som otr poit. I tis way, w ca id t valu o si or ay valu rom 0 to ad t ca us t priodicity o si, tat is si si +,,, si at ay otr poit. ( ) to calculat t valu o Eampl Driv t Maclauri sris o si( ) + + I t prvious ampl, w wrot t Taylor sris or si aroud t poit Maclauri sris is simply a Taylor sris or t poit 0. si, ( 0 ) 0! 5 5! 7 7!.

4 0.07. Captr 0.07 cos, si, cos si, cos() 0 ( 0 ) 0, 0 ( 0 ) 0, 0 Usig t Taylor sris ow, ( ) ! 5! So 5 +! 5! 5 si( ) +! 5! ( + ) Eampl Fid t valu o ( 6) giv tat ( ) 5 otr igr drivativs o at ar zro. Sic ourt ad igr drivativs o, 7, ( ) 0 ( + ) ar zro at. ( + ) !! , ad all

5 Taylor Torm Rvisitd Not tat to id 6 actly, w oly dd t valu o t uctio ad all its drivativs at som otr poit, i tis cas,. W did ot d t prssio or t uctio ad all its drivativs. Taylor sris applicatio would b rdudat i w dd to kow t prssio or t uctio, as w could just substitut 6 i it to gt t valu o ( 6). Actually t problm posd abov was obtaid rom a kow uctio wr ( ) 5, ( ) 7, ( ) 0, ( ) 6, ad all otr igr drivativs ar zro. Error i Taylor Sris As you av oticd, t Taylor sris as iiit trms. Oly i spcial cass suc as a iit polyomial dos it av a iit umbr o trms. So wvr you ar usig a Taylor sris to calculat t valu o a uctio, it is big calculatd approimatly. T Taylor polyomial o ordr o a uctio () wit ( +) cotiuous drivativs i t domai [, + ] is giv by ( ) '' R!! wr t rmaidr is giv by + ( ) ( + R ) () c. ( + )! wr < c < +, +. tat is, c is som poit i t domai Eampl 5 T Taylor sris or at poit 0 is giv by ! 5! a) Wat is t trucatio (tru) rror i t rprstatio o i oly our trms o t sris ar usd? b) Us t rmaidr torm to id t bouds o t trucatio rror. a) I oly our trms o t sris ar usd, t T trucatio (tru) rror would b t uusd trms o t Taylor sris, wic t ar

6 Captr E t + +! 5! 5 + +! 5! b) But is tr ay way to kow t bouds o tis rror otr ta calculatig it dirctly? Ys, ( ) R! wr + ( ) ( + R ) () c, < c < +, ad ( + )! c is som poit i t domai (, + ). So i tis cas, i w ar usig our trms o t 0, Taylor sris, t rmaidr is giv by + ( 0 ) ( + R ) () c ( + )! ( ) () c! c Sic < c < + 0 < c < < c < T rror is boud btw 0 < R () < < R () < < R () < 0. 6 So t boud o t rror is lss ta o Eampl wic dos cocur wit t calculatd rror T Taylor sris or at poit 0 is giv by ! 5! As you ca s i t prvious ampl tat by takig mor trms, t rror bouds dcras ad c you av a bttr stimat o. How may trms it would rquir to gt a approimatio o witi a magitud o tru rror o lss ta 0 6?

7 Taylor Torm Rvisitd Usig ( + ) trms o t Taylor sris givs a rror boud o + ( ) ( + R ) () c ( + )! 0,, + ( 0 ) ( + R ) 0 () c ( + )! + ( ) c ( + )! Sic < c < + 0 < c < < c < < R ( 0) < ( + )! ( + )! So i w wat to id out ow may trms it would rquir to gt a approimatio o witi a magitud o tru rror o lss ta 0 6, 6 < 0 ( + )! 6 ( + )! > 0 ( + )! > 0 6 (as w do ot kow t valu o but it is lss ta ). 9 So 9 trms or mor will gt witi a rror o 0 6 i its valu. W ca do calculatios suc as t os giv abov oly or simpl uctios. To do a similar aalysis o ow may trms o t sris ar dd or a spciid accuracy or ay gral uctio, w ca do tat basd o t cocpt o absolut rlativ approimat rrors discussd i Captr 0.0 as ollows. W us t cocpt o absolut rlativ approimat rror (s Captr 0.0 or dtails), wic is calculatd atr ac trm i t sris is addd. T maimum valu o m m, or wic t absolut rlativ approimat rror is lss ta % is t last umbr o sigiicat digits corrct i t aswr. It stabliss t accuracy o t approimat valu o a uctio witout t kowldg o rmaidr o Taylor sris or t tru rror.

8 Captr 0.07 INTRODUCTION TO NUMERICAL METHODS Topic Taylor Torm Rvisitd Summary Ts ar ttbook ots o Taylor Sris Major All girig majors Autors Autar Kaw Dat April, 009 Wb Sit ttp://umricalmtods.g.us.du

Worksheet: Taylor Series, Lagrange Error Bound ilearnmath.net

Worksheet: Taylor Series, Lagrange Error Bound ilearnmath.net Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.