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Ee1031dS10 - Taylor's 0th Order Theorem(Mean Value Theorem...

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Taylor’s 0 th Order Theorem (Mean Value Theorem) f ( ) ( ) f x f x x x ( ) ( ) ( ) ( ) ( ) f x f x f x f x x x x x x x , x x , , 0 ( ) ( ) '( )( ) ' ( ) ( ) '( ) th x x x x f x f x f x x Taylor s Order Theorem f x f x f Mean Value Theorem x x EE103 Slides 1D (SEJ) 1 Taylor’s Theorem for Functions of One Variable Taylor’s theorem is an important analysis tool as opposed to a computational tool Taylor’s theorem is an important analysis tool, as opposed to a computational tool, and an understanding of this theorem and the resulting approximations are important. We’ve already seen examples of Taylor’s 0 th Order theorem (Mean Value Theorem). th th , Given , Order Theorem: Taylor's 0 ( ) ( ) '( )( ) x x x x f x f x f x x Order Approximation: ( ) ( ) Taylor's 0 f x f x 2 1 2 , 1 Order Theorem: Taylor's ( ) ( ) '( )( ) ''( )( ) l ' '( )( ) st t x x f x f x f x x x f x x f 1 Order Approximation: ( ) ( ) Taylor's , st f x f x x Linear Approximation abou x x t x 2 3 1 1 Order Theorem: Taylor's 2 ( ) ( ) '( )( ) ''( )( ) '''( )( ) nd f x f x f x x x f x x x f x x 2! 3! 2 1 2! , Order Approximat , ion: ( ) ( ) Taylor s 2 ( )( ( )( ( Taylor's 2 '( )( ) ''( )( ) nd x x Quadratic Approximation a f x f x bout x f x x x f x x x EE103 Slides 1D (SEJ) 2
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