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chap11sec12

# chap11sec12 - 11.12 Applications of Taylor Polynomials...

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Unformatted text preview: 11.12 Applications of Taylor Polynomials Tables of integrals and trigonometric functions can be done with expansion of power series. Suppose that f(x) = the sum of its Taylor series at a: ) ( ) ( ) ( ) ! n n n f a f x x a n ∞ = =- ∑ then the n th partial sum is the n th degree Taylor polynomial. ( ) ( ) 2 ( ) ( ) ( ) ! '( ) ''( ) ( ) ( ) ( ) ( ) ( ) 1! 2! ! ( ) ( ) ( ) n n i n i n n n n f a T x x a n f a f a f a f a x a x a x a n T f x as n so f x T x = =- = +- +- + +- → → ∞ ≈ ∑ L Polynomials are simple functions and can be manipulated without much difficulty. In Math 1550 you found a linear approximation for a function (tangent line approximation). In fact, the tangent line approximation is the 1st degree Taylor polynomial of the function. 1 ( ) ( ) ( ) '( )( ) L x T x f a f a x a = = +- Derivatives of T n at a agree with those of f up to and including derivatives of order n. When we use a Taylor polynomial to at a agree with those of f up to and including derivatives of order n....
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chap11sec12 - 11.12 Applications of Taylor Polynomials...

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