MTH 253 Calculus (Other Topics)Back to the goal of the chapter …Local Linear ApproximationsLocal Quadratic ApproximationsLocal Cubic ApproximationsSlide 6ExampleSlide 8Generalization … nth degree Polynomial ApproximationTaylor SeriesMaclaurin SeriesQuestion …Example …Three Important Maclaurin SeriesOne more … ln t ?Slide 16Slide 17Slide 18MTH 253Calculus (Other Topics)Chapter 11 – Infinite Sequences and SeriesSection 11.8 –Taylor and Maclaurin SeriesCopyright © 2009 by Ron Wallace, all rights reserved.Back to the goal of the chapter …Approximating transcendental functions through the use of algebraic functions.Solution ..Transcendental function must be differentiable of ALL orders.Find a power series equal to the transcendental function.Consider the interval of convergence.Use a portion of the series to approximate the function.Local Linear ApproximationsReview of Section 3.10Find a linear equation that approximates a function around a point given the derivative of the function at that point.Tangent Linef(x)a( )f aSlope: '( )m f a=( )Point: , ( )a f a( ) '( )( )y f a f a x a= + -Local Quadratic ApproximationsFind a quadratic equation that approximates a function around a point given the first & second derivatives of the function at that point.2''( )( ) '( )( ) ( )2f ay f a f a x a x a= + - + -20 1 2( ) ( )y u u x a u x a= + - + -1 2' 2 ( )y u u x a= + -2'' 2y u=( ) ( )y a f a='( ) '( )y a f a=''( ) ''( )y a f a=0u=1u=22u=f(x)a( )( )y a f a=y(x)Local Cubic ApproximationsFind a Cubic equation that approximates a function around a point given the first, second, and third derivatives of the function at that point.2 3''( ) '''( )( ) '( )( ) ( ) ( )2 6f a f ay f a f a x a x a x a= + - + - + -( ) ( )y a f a='( ) '( )y a f a=''( ) ''( )y a f a='''( ) '''( )y a f a=f(x)a( )( )y a f a=y(x)Local Cubic ApproximationsFind a Cubic equation that approximates a function around a point given the first, second, and third derivatives of the function at that point.2 3( ) '( ) ''( ) '''( )( ) ( ) ( )0! 1! 2! 3!f a f a f a f ay x a x a x a= + - + - + -( ) ( )y a f a='( ) '( )y a f a=''( ) ''( )y a f a='''( ) '''( )y a f a=f(x)a( )( )y a f a=y(x)ExampleFind the Linear, Quadratic, and Cubic equations that approximate the above function around a = 1.1( ) tanf x x-=(1) 4f p=21'( )1f xx=+1'(1)2f =2 22''( )(1 )xf xx-=+1''(1)2f =-22 36 2'''( )(1 )xf xx-=+1'''( )2f x =1( 1)4 2y xp= + -21 1( 1) ( 1)4 2 4y x xp= + - - -2 31 1 1( 1) ( 1) ( 1)4 2 4 12y x x xp= + - - - + -1( ) tanf x x-=ExampleFind the Linear, Quadratic, and Cubic equations that approximate the above function around x0 = 1.1( 1)4 2y xp= + -21 1( 1) ( 1)4 2 4y x xp= + - - -2 31 1 1( 1) ( 1) ( 1)4 2 4 12y x x xp= + - - - + -1( ) tanf x x-=Generalization … nth degree Polynomial ApproximationIf f(x) can be differentiated n times at x = a, then the nth Taylor Polynomial for f(x) about x = a is …( )0( )( ) ( )!knknkf ap x x ak== -�2( )3( ) '( ) ''( )( ) ( )0! 1! 2!'''( ) ( ) ( ) ( )3! !nnf a f a f ax a x af a f ax a x an= + - + -+ - + + -gggTaylor Series( )0( )( ) ( )!kkkf ap x x ak�== -�2(3) ( )3''( )( ) '( )( ) ( )2( ) ( ) ( ) ( )6 !nnf af a f a x a x af a f ax a x an= + - + -+ - + + - +ggg gggNOTE: This assumes that f (n)(a) exists for all n.Maclaurin Series( )0(0)( )!kkkfp x xk�==�2(3) ( )3''(0)(0) '(0)2(0) (0) 6 !nnff f x xf fx xn= + ++ + + +ggg gggNOTE: This assumes that f (n)(0) exists for all n.Taylor Serieswith a = 0Question …For a function f(x) that is differentiable for all orders at x = a, will the Taylor Series converge to f(x) for each value of the domain of the function?Maybe!For some … yesFor some … noFor some … for part of the domainFor all … yes, when x = aExample …Find the Taylor Series for f(x) = ln x at x = 1.Determine the interval of convergence.Three Important Maclaurin Series2 3 401 ! 2! 3! 4!kxkx x x xe xk�== = + + + + +�ggg( )2 2 4 60cos 1 1 2 ! 2! 4! 6!kkkx x x xxk�== - = - + - +�ggg( )2 1 3 5 70sin 1 (2 1)! 3! 5! 7!kkkx x x xx xk+�== - = - + - ++�gggAll three of these converge for all values of x.One more … ln t ?( ) ln(1 ) at 0f x x x= + =Begin with …1'( ) (1 )f x x-= +2''( ) (1 )f x x-=- +3'''( ) 2(1 )f x x-= +(4) 4( ) 3!(1 )f x x-=- +( ) 1( ) ( 1) ( 1)!(1 )n n nf x n x+ -= - - +@ x=01-12-3!(-1)n+1(n-1)!11( 1)ln(1 )n nnxxn+�=-+ =�1 1x- < �One more … ln t ?1 2 31( 1)ln(1 ) 2 3n nnx x xx xn+�=-+ = = - + -�ggg1 1x- < �1 2 31( 1) ( )ln(1 ) 2 3n nnx x xx xn+�=- -- = =- - - -�ggg1 1x- � <3 5ln(1 ) ln(1 ) 2 3 5x xx x x� �+ - - = + + +� �� �ggg1 1x- < <SubtractingOne more … ln t ?3 5ln(1 ) ln(1 ) 2 3 5x xx x x� �+ - - = + + +� �� �ggg1 1x- < <1ln(1 ) ln(1 ) ln1xx xx+� �+ - - =� �-� �1 1 1 1x tt xx t+ -= � =- +0 1< 1t x> � - <One more … ln t ?3 51ln 2 1 3 5x x xxx� �+= + + +� �-� �ggg1 1x- < <0t >( ) ( )3 51 111 1111111ln ln 2 1 3 5t ttt ttttttt- --+ ++-+-+� �� �+= = + + +� �� �-� �� �� �gggExample …( ) ( )3 52 243 3623461ln 5 ln 2 1.58351 3 5� �� �+= = + + + �� �� �-� �� �� �gggln 5