## Newton-Raphson by Taylor polynomials

Hi,

this is from my book:

Suppose that $f\in C^2[a,b]$. Let $\overline{x}\in [a,b]$ be an approximation to $p$ such that $f'(\overline{x})\neq 0$ and $|p-\overline{x}|$ is "small". Consider the first Taylor polynomial for $f(x)$ expanded about $\overline{x}$,

$f(x) = f(\overline{x})+(x-\overline{x})f'(\overline{x})+\frac{(x-\overline{x})^2}{2}f''(\xi(x))$,

where $\xi(x)$ lies between $x$ and $\overline{x}$.

Why do i need to use $\xi(x)$? As far as I know, Newton-Raphsons assumes that $|p-\overline{x}|$ is so small that we can ignore $(x-\overline{x})^2$ and thus also the second derivative.. Someone care to explain?

Thanks.