How do you find the largest interval $(c-r,c+r)$ on which the Taylor Polynomial $p_n(x,c)$ approximates a function $y=f(x)$ to within a given error?

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Answer 1 · 2014-10-06T23:44:53

Let us assume that there is $M>0$ such that

$|f^{(n+1)}(x)| le M$ for all $x$ .

If we want the error to be less than $epsilon>0$ , then

$|R_n(x;c)|=|{f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}|$ ,

where $z$ is between $x$ and $c$ .

by replacing $|f^{(n+1)}(z)|$ by $M$ ,

$le M/{(n+1)!}|x-c|^{n+1} < epsilon$

By solving for $|x-c|$ ,

$Rightarrow |x-c| < root(n+1){{epsilon (n+1)!}/M}$

Hence,

$r=root(n+1){{epsilon (n+1)!}/M}$

I hope that this was helpful.

How do you find the largest interval (c-r,c+r)(c-r,c+r) on which the Taylor Polynomial p_n(x,c)p_n(x,c) approximates a function y=f(x)y=f(x) to within a given error?