Definition
Let and be vector spaces. A function is called an affine function if there exists a Linear Map and a fixed vector such that for all :
f(x) = L(x) + b
>[!definition] Good approximation(affine function) > >Suppose $l(x)$ is an affine function, $l(x)$ is a good approximation of $f$ at $a$ if >$$\lim_{x\to\ a}\ \frac{f(x)-l(x)}{x-a}\ =\ 0$$ >[!theorem] > >$f$ has a good approximation at $a$ if and only if $f'(a)$ exists. If $f$ has a good approximation at $a$ then >$$l(x)\ =\ f'(x)(x-a)\ +\ f(a)$$ --- >[!definition] Taylor Polynomial > > >Let $f$ be a function that is $n$ times differentiable at $a$. The **Taylor polynomial of degree $n$** of $f$ at $a$ is > >$$ >T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n >$$ > >It provides a polynomial approximation of $f$ near $a$. --- >[!definition] Taylor's Remainder > > - Lagrange Form >If $f$ is $(n+1)$ times differentiable, the remainder of the Taylor expansion in **Lagrange form** is > >$$ >R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} (x-a)^{n+1}, \quad \text{for some } \xi \text{ between } a \text{ and } x. >$$ > - The **Cauchy form** of the remainder is > >$$ >R_n(x) = \frac{f^{(n+1)}(\xi)}{n!} (x-\xi)^n (x-a), \quad \text{for some } \xi \text{ between } a \text{ and } x. >$$ > - The **integral form** of the remainder is > >$$ >R_n(x) = \frac{1}{n!} \int_a^x f^{(n+1)}(t) (x-t)^n \, dt >$$ --- >[!remark] > >In all forms, the Taylor polynomial $T_n(x)$ plus the remainder $R_n(x)$ gives the exact value: > >$$ >f(x) = T_n(x) + R_n(x) >$$