14 - Taylor's Theorem

Taylor Expansion

Also known as Taylor Polynomial of order $n$ for $f$ about the point $a$

We want a polynomial $p_n(x)$ such that $f^{(k)}(a)=p_n^{(k)}(a)$ for $k=0,1,\cdots ,n$

So

$$p_n(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\cdots +\frac{f^{(n)}(a)}{n!}(x-a{)}^n$$

First Order Taylor Expansion

For $f(x,y)$ about $(a,b)$

$$p_1(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$$

Second Order Taylor Expansion

For $f(x,y)$ about $(a,b)$

$$\begin{array}{rl}{p}_2(x,y)& =f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\\ & +\frac{1}{2}[f_{xx}(a,b)(x-a{)}^2+2f_{xy}(a,b)(x-a)(x-b)+f_{yy}(a,b)(y-b{)}^2]\end{array}$$