Section 4.7 Taylor Polynomials
If \(g\) is a function of one variable then the Taylor polynomial of order \(n\) about \(a\) is defined by
\begin{equation*}
T_n(a+h)=g(a)+g'(a)h+\frac{g''(a)}{2!}h^2+\dots +\frac{g^{(n)}(a)}{n!}h^n\text{.}
\end{equation*}
As you know from first year calculus \(T_n\) is the best approximation of \(f\) near \(a\) by an \(n\)-th order polynomial. From the above formula we now want to derive the Taylor polynomials of order one and two for a function, \(f\text{,}\) defined on \(D\subset\mathbb R^N\text{.}\) Given an interior point of \(\vect a\in D\) and \(h\in\mathbb R^N\) we consider the function
\begin{equation*}
g_{\vect h}(t):=f(\vect a+t\vect h),
\end{equation*}
and write down its Taylor polynomial of order \(2\) about \(t=0\text{:}\)
\begin{equation*}
g_{\vect h}(0)+g_{\vect h}'(0)t+\frac{g_{\vect h}''(0)}{2!}t^2\text{.}
\end{equation*}
\begin{equation*}
g_{\vect h}'(0) =\bigl(\grad f(\vect a)\bigr)\cdot \vect h\text{.}
\end{equation*}
If
\(\vect h\neq 0\) then
\(\vect v:=\vect h/\|\vect h\|\) is a unit vector, and by the formula
(4.3) the above becomes
\begin{equation*}
g_{\vect h}'(0)
=\|h\|\bigl(\grad f(\vect a)\bigr)\cdot \vect v
=\|h\|\frac{\partial f}{\partial\vect v}(\vect a)\text{.}
\end{equation*}
Applying the same arguments to \(g_{\vect h}'(t)\) we get
\begin{equation*}
g_{\vect h}''(0)
=\|h\|^2\frac{\partial^2 f}{\partial\vect v^2}(\vect a)\text{.}
\end{equation*}
Using the formula for the second directional derivative from
Proposition 4.43 and the definition of
\(\vect v\) we get
\begin{align*}
g_{\vect h}''(0)\amp =\|h\|^2\vect v^TH_f(\vect a)\vect v\\
\amp=\|h\|^2\frac{\vect h^T}{\|h\|}H_f(\vect a)\frac{\vect h}{\|h\|}\\
\amp=\vect h^TH_f(\vect a)\vect h\text{,}
\end{align*}
where
\(H_f(\vect a)\) is the Hessian matrix as defined in
Definition 4.40. The above motivates the following definition.
Definition 4.44. Taylor polynomials.
The functions
\begin{align*}
T_1(\vect a+\vect h)\amp :=f(\vect a)+\bigl(\grad f(\vect a)\bigr)\cdot\vect h\\
T_2(\vect a+\vect h)\amp :=f(\vect a)+\bigl(\grad f(\vect a)\bigr)\cdot\vect h +\frac{1}{2}\vect h^TH_f(\vect a)\vect h
\end{align*}
are called the Taylor polynomial of \(f\) of order one and two, respectively, centred at \(\vect a\text{.}\)
By construction
\(T_1\) and
\(T_2\) are the Taylor polynomials of
\(g_{\vect h}\) at
\(t=1\text{,}\) so we can apply Taylor's theorem to
\(g_{\vect h}\) at
\(t=1\text{.}\) Doing this, with some effort involved in dealing with the remainder term, we get the following version of Taylor's Theorem for a function of several variables. For a rigorous proof we refer to Apostol
[2](pages 308-310).
Theorem 4.45. Taylor's Theorem.
Suppose that all first and second order partial derivatives of \(f\) are continuous at \(\vect a\text{.}\) Then for \(n=1,2\)
\begin{equation*}
f(\vect a+\vect h)=T_n(\vect h)+R_n(\vect h),
\end{equation*}
where the remainder term \(R_n(\vect h)\) has the property that
\begin{equation*}
\lim_{\vect h\to\vect 0}\frac{R_n(\vect h)}{\|\vect h\|^n}
=0,
\end{equation*}
We conclude this section by giving one example.
Example 4.46.
Suppose that
\(f(x,y):=x^3-3x^2y\) for all
\((x,y)\in\mathbb R^2\) is as in
Example 4.37. Then we saw that
\begin{equation*}
\grad f(x,y)=(3x^2-6xy,-3x^2)
\end{equation*}
and
\begin{equation*}
H_f(x,y)=
\begin{bmatrix}
6x-6y \amp -6x \\
-6x \amp 0
\end{bmatrix}\text{.}
\end{equation*}
Hence the Taylor polynomial of order one centred at \(\vect a=(1,1)\) is
\begin{align*}
T_1(1+h_1,1+h_2)\amp=-2+(-3,-3)\cdot(h_1,h_2)\\
\amp=-2-3(h_1+h_2)
\end{align*}
for all \((h_1,h_2)\in\mathbb R^2\text{.}\) The Taylor polynomial of order two centred at \(\vect a=(1,1)\) is
\begin{align*}
T_2(1+h_1,1+h_2)\amp =T_1(1+h_1,1+h_2)+\frac{1}{2}
\begin{bmatrix}
h_1 \amp h_2
\end{bmatrix}
\begin{bmatrix}
0 \amp -6 \\
-6 \amp 0
\end{bmatrix}
\begin{bmatrix}
h_1 \\h_2
\end{bmatrix}\\
\amp =-2-3(h_1+h_2)-6h_1h_2\text{.}
\end{align*}
