How do you use the Taylor Remainder term to estimate the error in approximating a function $y=f(x)$ on a given interval $(c-r,c+r)$ ?

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Answer 1 · 2014-10-16T21:39:42

Assume that there exists a finite $M>0$ such that

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

for all $x$ in $(c-r,c+r)$ .

The error of approximating $f(x)$ by the Taylor polynomial $p_n(x;c)$ can be estimated by

$|f(x)-p_n(x;c)|$

$=|R_n(x;c)|$

$=|{f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}|$ , where $z$ is between $x$ and $c$

$le M/{(n+1)!}r^{n+1}$

I hope that this was helpful.

How do you use the Taylor Remainder term to estimate the error in approximating a function y=f(x)y=f(x) on a given interval (c-r,c+r)(c-r,c+r)?