09. Taylor's Theorem

9.1 Review for functions of one variable

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

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

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07 - Taylor's Theorem for a function of one variable

Taylor's Theorem for a function of one variable

Suppose $f(x)$ has derivatives on $[a,b]$ up to $(n+1)$th order
For any $x\in (a,b]$ there exists $\xi \in (a,x)$ such that

$$f(x)=f(a)+f^{\prime }(a)(x-a)+\cdots +\frac{f^{(n)}(a)}{n!}(x-a{)}^n+\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-a{)}^{n+1}$$

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9.2 Taylor’s Theorem for a function of two variables

First Order Taylor Expansion

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

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

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Second Order Taylor Expansion

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

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

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08 - Taylor's Theorem for a function of two variables

Taylor's Theorem for a function of two variables

Let $f(x,y)$ be defined on an open subset $U$, such that

• $f$ has continuous partial derivatives on $U$ up to the $(n+1)$th order

Let $(a,b)\in U$ and suppose the line segment between $(a,b)$ and $(x,y)$ lies in $U$

Then there exists $\theta \in (0,1)$ such that

$$\begin{array}{rl}f(x,y)& =f(a,b)+\sum _{k=1}^n\sum _{i+j=k,i,j\ge 0}\frac{1}{i!j!}\frac{{\partial }^{k}f(a,b)}{\partial x^i\partial y^j}(x-a{)}^i(y-b{)}^j\\ & +\sum _{i+j=n+1i,j\ge 0}\frac{1}{i!j!}\frac{{\partial }^{n+1}f(\xi )}{\partial x^i\partial y^j}(x-a{)}^i(y-b{)}^j\end{array}$$

where $\xi =\theta (a,b)+(1-\theta )(x,y)$

Note that $\xi$ is just $\theta$ of the line segment

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