CK12 Calculus - Single Variable

If and convergeson a commoninterval|x| <R ab, thentheir productpowerseries also convergesonR aband is the powerseriesfor the ...

Theorem(nth-degreeTaylorpolynomial)Givena functionf with continuousnthderivativein an openin- tervalcontainingx* 0. Thereexistsu ...

Then. ExerciseFind the Taylorseriesof the followingfunctionsat the givenx 0 with givendegreen. 1.f(x) =exatx= 0,n= 3 2.f(x) =ln ...

IsT(x) convergentatx=a? Thereis no guaranteeexceptata=x 0. The secondquestionis: IfT(x) convergesatx=a, doesit equalf(a)? The an ...

Theorem(RemainderEstimation) If |f(n+1)(x)| ≤Mfor |x-x 0 | ≤r, then we havethe followingboundforRn(x): for |x-x 0 | ≤r. Example ...

Whatis the truncationerrorof approximating by its fourth-degreeMaclaurinseriesin for |x| ≤ 0.1. CombiningSeries,EulersFormula ...

Find the Maclaurinseriesof for allx. Hint: wouldyou dividexbyex? Find the Maclaurinseriesofcos 3xandsin 3xfor allxusingEuler' ...

Letrbe a real numberandf(x) = (1 +x)r. Isf(x) equalto a seriesin the form of (BE) exceptthat theremay be an infiniteseries?The a ...

Exercise Find a powerseriesrepresentationof atx= 0. Find a powerseriesrepresentationof atx= 0. Notice. Find a powerseriesrepres ...

and . This patternrepeatsand can be checkedas in the casex 0 = 0. So the TaylorSeriesis We may also applyalgebraicmanipulationto ...

Approximate to 6 decimalplaces. EvaluatingNonelementaryIntegrals Thereare manysimple-lookingfunctionsthat haveno explicitformul ...

Find the powerseriesrepresentation(MaclaurinSeries)of and approximate to 6 decimalplaces. Find the powerseriesrepresentation(Ma ...

wherethe Binomialcoefficientsare denotedby and for As a simpleBinomialfunction,takea= 1 andb=x. Then . Letrbe a real numberandf( ...

Exercise Find a powerseriesrepresentationof atx= 0. Find a powerseriesrepresentationof atx= 0. Notice. Find a powerseriesrepres ...

Thenwe may be able to deducea usefulTaylorSeriescenteredcloseto the givenx. Example 2 Approximatesin(1.1) to 4 decimalplaces. Si ...

Approximatesin(0.8) to 6 decimalplaces. Hint:considercenter Approximate to 6 decimalplaces. EvaluatingNon-ElementaryIntegral ...

Find the powerseriesrepresentation(MaclaurinSeries)of and approximate to 6 decimalplaces. Find the powerseriesrepresentation(Ma ...