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Chap11_Sec11

Chap11_Sec11 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES

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11.11 Applications of Taylor Polynomials INFINITE SEQUENCES AND SERIES In this section, we will learn about: Two types of applications of Taylor polynomials.
APPLICATIONS IN APPROXIMATING FUNCTIONS First, we look at how they are used to approximate functions. Computer scientists like them because polynomials are the simplest of functions.

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APPLICATIONS IN PHYSICS AND ENGINEERING Then, we investigate how physicists and engineers use them in such fields as: Relativity Optics Blackbody radiation Electric dipoles Velocity of water waves Building highways across a desert
APPROXIMATING FUNCTIONS Suppose that f ( x ) is equal to the sum of its Taylor series at a : ( ) 0 ( ) ( ) ( ) ! n n n f a f x x a n = = -

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In Section 11.10, we introduced the notation T n ( x ) for the n th partial sum of this series. We called it the n th-degree Taylor polynomial of f at a . NOTATION T n ( x )
Thus, ( ) 0 2 ( ) ( ) ( ) ( ) ! '( ) ''( ) ( ) ( ) ( ) 1! 2! ( ) ... ( ) ! i n i n i n n f a T x x a i f a f a f a x a x a f a x a n = = - = + - + - + + - APPROXIMATING FUNCTIONS

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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 to f : f ( x ) ≈ T n ( x ) APPROXIMATING FUNCTIONS
Notice that the first-degree Taylor polynomial T 1 ( x ) = f ( a ) + f’ ( a )( x a ) is the same as the linearization of f at a that we discussed in Section 3.10 APPROXIMATING FUNCTIONS

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Notice also that T 1 and its derivative have the same values at a that f and f’ have. In general, it can be shown that the derivatives of T n at a agree with those of f up to and including derivatives of order n. See Exercise 38. APPROXIMATING FUNCTIONS
To illustrate these ideas, let’s take another look at the graphs of y = e x and its first few Taylor polynomials. APPROXIMATING FUNCTIONS

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The graph of T 1 is the tangent line to y = e x at (0, 1). This tangent line is the best linear approximation to e x near (0, 1). APPROXIMATING FUNCTIONS
The graph of T 2 is the parabola y = 1 + x + x 2 /2 The graph of T 3 is the cubic curve y = 1 + x + x 2 /2 + x 3 /6 This is a closer fit to the curve y = e x than T 2 . APPROXIMATING FUNCTIONS

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The next Taylor polynomial would be an even better approximation, and so on. APPROXIMATING FUNCTIONS
The values in the table give a numerical demonstration of the convergence of the Taylor polynomials T n ( x ) to the function y = e x . APPROXIMATING FUNCTIONS

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When x = 0.2, the convergence is very rapid. When x = 3, however, it is somewhat slower. The farther x is from 0, the more slowly T n ( x ) converges to e x . APPROXIMATING FUNCTIONS
When using a Taylor polynomial T n to approximate a function f , we have to ask these questions: How good an approximation is it? How large should we take n to be to achieve a desired accuracy? APPROXIMATING FUNCTIONS

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To answer these questions, we need to look at the absolute value of the remainder: | R n ( x )| = | f ( x ) – T n ( x )| APPROXIMATING FUNCTIONS
There are three possible methods for estimating the size of the error.

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Chap11_Sec11 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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