Ee1031dS10

Ee1031dS10 - Taylors 0 th Order Theorem (Mean Value...

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Unformatted text preview: Taylors 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 , , ( ) ( ) '( )( ) ' ( ) ( ) '( ) 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 Taylors Theorem for Functions of One Variable Taylors theorem is an important analysis tool as opposed to a computational tool Taylors theorem is an important analysis tool, as opposed to a computational tool, and an understanding of this theorem and the resulting approximations are important. Weve already seen examples of Taylors 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 th Order Approximation: ( ) ( ) Taylor's 0 f x f x 2 1 2 , 1 Order Theorem: Taylor's ( ) ( ) '( )( ) ''( )( ) st t x x f x f x f x x x f x x 1 Order Approximation: ( ) ( ) Taylor's '( )( ) , st f x f x f 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 Theorem: Order Approximat , ion: ( ) ( ) Taylor s 2 ( ) ( ) ( )( ) ( )( ) ( )( ) Taylor's 2 '( )( ) ''( )( ) nd x x Quadratic Approximation a f x f x bout x f x f x f x x x f x x x f x x f x x x f x x x EE103 Slides 1D (SEJ) 2 Taylors Theorem for Functions of One Variable 1 ( ) ( 1) ( ) ( ) k n n k n th x x x x n Order Theo m re ( ) ( 1) , 1 1 ( ) ( 1) ( ) ( ) ( ) ( ) ! ( 1 ) ! ( ) ( ) ( ) ( ) k n x x k k n n k n f x f x f x f k n x x x x f f ( ) ( 1) , , ( ) ( ) ! ( 1 ) ! where is some point between . k n x x k x x f x f k n x a x nd ( ) k t h f isthek derivative of f (0) , ! 1 f f and EE103 Slides 1D (SEJ) 3 The Error Term 1 ( ) ( 1) ( ) ( 1)! n x x n x x f , ( 1)! x x n When we drop the error term we have Taylor's When we drop the error term, we have Ta order approx ylors . th n ( ) 1 ( ) ( ) ( ) ( ) ! k n k k x x f x f x f x k EE103 Slides 1D (SEJ) 4 Linear Approximation: Taylors 1 st Order Approx 3 ( ) ' ( ; ) ( ) ( )( ) 1 f x x l x x f x f x x x EE103 Slides 1D (SEJ) 5 1 x Quadratic Approximation (Taylors 2 nd Order Approximation) 3 ( ) ( ) '( )( ( ) , 1 ( ; ) ) f f x f x f x x x x x x x l x EE103 Slides 1D (SEJ) 6 2 1 2!...
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This note was uploaded on 06/01/2010 for the course EE EE 103 taught by Professor Jacobsen during the Spring '09 term at UCLA.

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Ee1031dS10 - Taylors 0 th Order Theorem (Mean Value...

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