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# PP 12.11 - Application of Taylor Polynomials Suppose that f...

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Application of Taylor Polynomials

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Suppose that ( ) 0 ( ) ( ) ( ) ! n n n f a f x x a n = = - Then, the nth partial sum of the Taylor series is called the nth Taylor polynomial T n (x) of f at a and i n i i n a x i a f x T ) ( ! ) ( ) ( 0 ) ( - = =
Since f is the sum of its Taylor series, we know that T n ( x ) → f ( x ) as n → ∞. Thus, T n can be used as an approximation of f : f ( x ) ≈ T n ( x ) The advantage of such approximation is the fact that T n ( x ) is polynomial of degree n and polynomials are easy to work with.

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Let’s look at the first few Taylor polynomials. ) )( ( ) ( ) ( 1 a x a f a f x T - + = The first Taylor polynomial of f at a is the Linearization of f at a . 2 2 ) ( ! 2 ) ( ) )( ( ) ( ) ( a x a f a x a f a f x T - + - + = This is a quadratic polynomial whose graph is a parabola.
3 2 3 ) ( ! 3 ) ( ) ( ! 2 ) ( ) )( ( ) ( ) ( a x a f a x a f a x a f a f x T - + + - + - + = Now we have a cubic polynomial, so we know how to sketch its graph and how to work with it.

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Example
When using a Taylor polynomial T n to approximate a function f , we have to ask these questions: How good is the approximation?

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PP 12.11 - Application of Taylor Polynomials Suppose that f...

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