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CSC_349_HANDOUT#33 - COMPUTER SCIENCE 349A Handout Number...

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1 COMPUTER SCIENCE 349A Handout Number 33 NUMERICAL DIFFERENTIATION FORMULAS USING TAYLOR'S THEOREM (Chapter 4, page 85 in 5 th ed. or page 90 in 6 th ed. and Chapter 23, page 632 in 5 th ed. and page 653 in 6 th ed.) In Chapter 4, Taylor’s Theorem was used to derive a numerical differentiation formula that was used to approximate a derivative in a mathematical model that was developed to determine the terminal velocity of a free-falling body (a parachutist). Recall Taylor's Theorem (with 2 = n ): ) ( ! 3 ) ( ) ( ! 2 ) ( ) ( ) ( ) ( ) ( 1 3 0 0 2 0 0 0 0 ξ f x x x f x x x f x x x f x f + + + = or (1) ) ( 6 ) ( 2 ) ( ) ( ) ( 1 3 0 2 0 0 0 ξ f h x f h x f h x f h x f + + + = + where 0 x x h = (and thus h x x + = 0 ) and 1 ξ lies between 0 x and h x x + = 0 . Solving for ) ( 0 x f from (1) gives ) ( ) ( ) ( ) ( 0 0 0 h O h x f h x f x f + + = . That is, h x f h x f x f ) ( ) ( ) ( 0 0 0 + with a truncation error that is ) ( h O ; see page 85 of the 5 th ed. or page 90 of the 6 th ed. This is called the forward difference approximation to ) ( 0 x f .
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