MTH 253 Calculus (Other Topics)The Mean Value TheoremTaylor’s FormulaRemainder EstimateApplication ExampleGenerating Taylor SeriesSlide 7Slide 8Some important Taylor & Maclaurin SeriesMTH 253Calculus (Other Topics)Chapter 11 – Infinite Sequences and SeriesSection 11.9 – Convergence of Taylor Series; Error EstimatesCopyright © 2009 by Ron Wallace, all rights reserved.The Mean Value TheoremFrom section 4.2( ) ( )'( )f b f af cb a-=-for some c between a and b( ) ( ) '( )( )f b f a f c b a= + -OR …Letting b = x …( ) ( ) '( )( )f x f a f c x a= + -for some c between a and xTaylor’s FormulaA generalization of the Mean Value Theorem( ) ( ) ( )n nf x P x R x= +( )0( )( ) ( )!knknkf aP x x ak== -�( 1)1( )( ) ( )( 1)!nnnf cR x x an++= -+where …( )0( )If lim ( ) 0 , then ( ) ( )!kknnkf aR x x I f x x ak���== " � = -�Remainder of order n.for some c between a and xRemainder Estimate( 1)1( )( ) ( )( 1)!nnnf cR x x an++= -+Remainder of order n.( 1)If ( ) between and ,nM f t t x a+� "1then ( )( 1)!nnx aR x Mn+-�+Application ExampleHow many terms of its Maclaurin series are needed to approximate cos x with an error less than 0.001?11[cos ]nndM xdx++� 1M� =112[ ]then ( ) 1 0.001( 1)! ( 1)!nnnxR xn np++� < <+ +gn = 6 gives 0.00468n = 7 gives 0.000921( )( 1)!nnx aR x Mn+-�+Generating Taylor SeriesGiven known Taylor Series, other series can be obtained using the following operations term by term …SubstitutionAddition & SubtractionMultiplicationDifferentiationIntegrationGenerating Taylor SeriesExample 1 2 3111x x xx= + + + +-ggglet x = -x22 4 62111x x xx= - + - ++gggintegrate3 5 71tan3 5 7x x xx x-= - + - +gggMuch easier than finding a general formula for the nth derivative of the tan-1x function.Generating Taylor SeriesExample 2 2 3 412 6 24xx x xe x= + + + + ggg2 4cos 12 24x xx = - + - gggMultiply3 4cos 13 6xx xe x x= + - - +gggMuch easier than finding derivatives excosx … try it?Some important Taylor & Maclaurin SeriesSee the list on page 815.Some things to note:• Don’t forget to consider the interval of convergence.• Some converge quickly (esp. w/ n! involved).• Some converge slowly (e.g.
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