taylor_polynomials

# taylor_polynomials - Taylor Polynomials Recall in the first...

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T aylor P olynomials Recall in the first quarter of Calculus, we discussed how to find the tangent line approximation of a function. That is, we said that near x = a , () ( ) f xf af a x a  Pictorially, we have the following diagram. (The true function f ( x ) appears in blue and the tangent line approximation appears in red.) a x f a f a f' a  x - a x - a Approximate value of f x True value of f x Figure 1: Tangent line approximation of f ( x ) near x = a This tangent line approximation is a special case of a Taylor Polynomial. In particular, it is defined as the Degree 1 Taylor Polynomial approximation of f ( x ) for x near a . We shall denote this by P 1 ( x ). We record this as the following: Degree 1 Taylor Polynomial approximation of f ( x ) for x near a 1 ( ) f xP x f a x a  Example 1: Find the Degree 1 Taylor Polynomial approximation of f ( x ) = e x for x near 0. Solution: Notice that f (0) = e –(0) = 1, and ( ) x f xe   , so (0) (0) 1 fe  . Thus, we have that e x º 1 – 1( x – 0) = 1 – x . 1

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The reason we use this approximation is so that not only does the approximation agree with the function at x = a , but the first derivatives agree as well.
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## This note was uploaded on 09/07/2011 for the course MATH 10C taught by Professor Hohnhold during the Spring '07 term at UCSD.

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taylor_polynomials - Taylor Polynomials Recall in the first...

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