CSC_349_HANDOUT#5 - COMPUTER SCIENCE 349A Handout Number 5...

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1 COMPUTER SCIENCE 349A Handout Number 5 Taylor’s Theorem is the fundamental tool for deriving and analyzing numerical approximation formulas in this course. It states that any “smooth” function (one with a sufficient number of derivatives) can be approximated by a polynomial, and it includes an error (remainder) term that indicates how accurate the polynomial approximation is. Taylor’s Theorem also provides a means to estimate the value of a function ) ( x f at some point 1 + i x using the values of ) ( x f and its derivatives at some nearby point i x . Taylor’s Theorem (page 74 of the 5 th edition; page 79 of the 6 th ) Let 0 n and let a be any constant. If ) ( x f and its first 1 + n derivatives are continuous on some interval containing x and a , then n n n R a x n a f a x a f a x a f a x a f a f x f + + + + + + = ) ( ! ) ( ) ( ! 3 ) ( ) ( ! 2 ) ( ) )( ( ) ( ) ( ) ( 3 2 L where the remainder (or error) term is 1 ) 1 ( ) ( )! 1 ( ) ( + + + = n n n a x n f R ξ and is some value between x and a . Note that n n n a x n a f a x a f a x a f a x a f a f x P ) ( ! ) ( ) ( ! 3
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CSC_349_HANDOUT#5 - COMPUTER SCIENCE 349A Handout Number 5...

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