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Lecture-3-1-Taylor series-slides

Lecture-3-1-Taylor series-slides - BME 5020 Computer and...

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BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University Taylor Series BME 5020 Computer and Mathematical Application in Bioengineering © 2006 Jingwen Hu September 19, 2006
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Taylor’s Theorem Taylor’s Theorem First Theorem of Mean for Integrals Second Theorem of Mean for Integrals BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University n n n R a x n a f a x a f a x a f a f x f + + + + + = ) ( ! ) ( ) ( ! 2 ) ( ) )( ( ) ( ) ( ) ( 2 K + = x a n n n dt t f n t x R ) ( ! ) ( ) 1 ( ) ( ) ( ) ( a x g dt t g x a = ξ = x a x a dt t h g dt t h t g ) ( ) ( ) ( ) ( ξ ) 1 ( ) 1 ( ) ( )! 1 ( ) ( + + + = n n n a x n f R ξ Box 4.1
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Taylor’s Theorem BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University n n i i i n i i i i i i i i R x x n x f x x x f x x x f x f x f + + + + + = + + + + ) ( ! ) ( ) ( ! 2 ) ( ) )( ( ) ( ) ( 1 ) ( 2 1 1 1 K ) 1 ( ) 1 ( )! 1 ( ) ( + + + = n n n h n f R ε (x i+1 -x i )= h step size n n i n i i i i R h n x f h x f h x f x f x f + + + + + = + ! ) ( ! 2 ) ( ) ( ) ( ) ( ) ( 2 1 K
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Taylor Series Example – Example 4.1 Use zero- through fourth-order Taylor series expansions to approximate the function From x i =0 with h=1. That is, predict f(x i+1 =1) BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University 2 . 1 25 . 0 5 . 0 15 . 0 1 . 0 ) ( 2 3 4 + = x x x x x f n n n f f f f f 1 ! ) 0 ( 1 ! 2 ) 0 ( 1 ) 0 ( ) 0 ( ) 1 ( ) ( 2 + + + + K n i n i i i i h n x f h x f h x f x f x f ! ) ( ! 2 ) ( ) ( ) ( ) ( ) ( 2 1 + + + + + K
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From x i =0 with h=1. That is, predict f(x i+1 =1) BME5020 2006, Fall Semester |Biomedical Engineering, Wayne State University 2 . 1 25 . 0 5 . 0 15 . 0 1 . 0 ) ( 2 3 4 + = x x x x x f n n n f f f f f 1 ! ) 0 ( 1 ! 2 ) 0 ( 1 ) 0 ( ) 0 ( ) 1 ( ) ( 2 + + + + K Taylor Series Example – Example 4.1 n=0 n=1 n=2 n=3 n=4 2 . 1 ) 0 ( ) 1 ( = f f 95 . 0 1 * 25 . 0 2 . 1 1 ) 0 ( ) 0 ( ) 1 ( = = + f f f 45 . 0 1 * 5 . 0 95 . 0 1 ! 2 ) 0 ( 1 ) 0 ( ) 0 ( ) 1 ( 2 2 = = + + f f f f 3 . 0 1 * 15 . 0 45 . 0 1 ! 3 ) 0 ( 1 ! 2 ) 0 ( 1 ) 0 ( ) 0 ( ) 1 ( 3 3 3 2 = = + + + f f f f f 2 . 0 1 * 1 . 0 3 . 0 1 ! 4 ) 0 ( 1 ! 3 ) 0 ( 1 ! 2 ) 0 ( 1 ) 0 ( ) 0 ( ) 1 ( 4 4 4 3 3 2 = = + + + + f f f f f f
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The n th-order Taylor series expansion will be exact for an n th-order polynomial.
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