《高數(shù)雙語》課件section 3-3

Section3.3Taylor’sTheoremAndItsApplications2Overviewvariablexin,where

isafixedpoint,canbecalculatedsimplyThisapproximationissimpleandiseasilyused,butithaslowprecision,since

andtheapproximationis,andalsothisInChapter2,wehadseenthatthevalueofasmoothfunctionforsomebyapproximation.thedifferencebetweentheaccuratevalueofjustainfinitesimalofhigherorderwithrespectto

isverysmall.approximationcanonlybeusedinthecaseofInthislecture,wewillformanewtechniquetoapproximatethevalueofafunctionwithhigherorderofcomputationerror.3Taylor’sTheoremisusingaline,,toapproximateacurve.Itcanbefindasuitablepolynomialsofdegreetoapproximateagivencurveoffunction,suchthattheapproximateerroris.Thatis Ifwecandoso,whatarethecoefficientsof,andhowcanweobtainItiseasytoseethattheapproximation,,infact,imagedthatifweuseasuitablecurvetoapproximatecurve,theapplicablewillbewiderandtheprecisionwillbeimproved.Itseemsnaturethatwechosethepolynomialsasthesuitablecurve.aninfinitesimalofhigherorderwithrespectto.them?Thenthequestionbecomesto,wetryto4Taylor’sTheoremTaylorPolynomialPeanoremainderTaylorcoefficientsTaylor,

Brook

(1685-1731)EnglishmathematicianTheorem

(Taylor’stheoremwithPeanoremainder)Supposethatthefunctionfisdifferentiableofordernatthepointx0.Then5Taylor’sTheoremTaylor,

Brook

(1685-1731)Englishmathematician

LagrangeremainderLagrangeFormula6Taylor’sTheoremComparetothePeanoremainder,theLagrangeremaindercanbeusedtoestimatetheerrormoreprecisely.

suchthat,,Infact,ifthefunctionfisdifferentiableofordern+1on[a,b],andthereisaconstantthenWecanseefromtheinequalitythat

as.Thismeansthat

toapproximateadifferentiablefunction,theerrormaybemakearbitrarilyifweutilizethepolynomialofanyorderinthewholeintervalsmall.

istakenlargeenough.

while7Taylor’sTheoremColinMaclaurin(1698-1746)Scottishmathematician

Ifwelet,thentheLagrangeformulabecomesandthisformulaiscalledtheMaclaurinformula.8MaclaurinFormulaeforSomeElementaryFunctionsMaclaurinformulafortheexponentialfunctionSincewehavewhere

and.

and,9MaclaurinFormulaeforSomeElementaryFunctionsMaclaurinformulafor

andLet,since

or,thenwhere

and.wehave10MaclaurinFormulaeforSomeElementaryFunctionsSimilarly,wehaveMaclaurinformulafor

andwhere

and.11MaclaurinFormulaeforSomeElementaryFunctionsMaclaurinformulaforWehadknownthatsoThusWhere

and.12MaclaurinFormulaeforSomeElementaryFunctionsMaclaurinformulaforWehadknownthatsoThusWhere

and.13SomeApplicationsofTaylor’sTheorem(1)Approximatecalculations14SomeApplicationsofTaylor’sTheorem(1)Approximatecalculations15SomeApplicationsofTaylor’sTheorem(1)Approximatecalculations16SomeApplicationsofTaylor’sTheorem(1)Approximatecalculations17SomeApplicationsofTaylor’sTheorem(1)Approximatecalculations18SomeApplicationsofTaylor’sTheorem(1)Approximatecalculations,wehavesince,soItmaybeseenthat,as,whichmeansthat

isapproximated

andestimationoftheerror.Example

ApproximatecalculationofthevalueofSolution

inthemaclaurinformulaeforLetbytheapproximationformula:TheerrorcanbemadearbitrarilysmallaslongifFinish.

istakenlargeenough.19SomeApplicationsofTaylor’sTheorem(1)ApproximatecalculationsSincetheerrorofapproximationofTayloriswhen

liesbetweenthepoints

and.Since

and

forall,weknowthattheerrorintheapproximationisnomorethan.,weonlyneed.InordertohaveSolution:20SomeApplicationsofTaylor’sTheorem(1)ApproximatecalculationsExample

Findanapproximatevalueofarealrootfortheequationontheparameter;wedenotethisrootby.Thisfunctioncanbeseen

isdifferentiabletoanyorder.Thus,byTaylor,weobtainwhere

isaverysmallparameter.Itiseasytoprovethatthisequationhasarealrootanditisdependentbythegivenfunction.asanimplicitfunctiondeterminedItcanbeprovethattheimplicitfunctionformulaandtakingforinstance,.Solution21SomeApplicationsofTaylor’sTheorem(1)ApproximatecalculationsExample

Findanapproximatevalueofarealrootfortheequationwhere

isaverysmallparameter.Solution(continued),wehave,thatisWetakedifferenttoeachsideofthegiven,wehaveWewilldeterminethecoefficientsinthelastequation.TakeequationwithrespecttoDifferentiatingbothsidesofthegivenequationagain,wehave sothat.22SomeApplicationsofTaylor’sTheoremSothat(1)ApproximatecalculationsExample

Findanapproximatevalueofarealrootfortheequationwhere

isaverysmallparameter.Solution(continued)therefore,theseconddegreeapproximateoftherealrootis

.Finish.23SomeApplicationsofTaylor’sTheorem(2)FindinglimitsSolution,wehaveFinish.Example

Find

ByMaclaurinformulawithPeanoremainderto24SomeApplicationsofTaylor’sTheorem(3)Provinginequalities

ProofBytheMaclaurinformulawehave,liesbet