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Next: Accuracy and Tolerance Up: Taylor Polynomials Previous: Getting Started

Background

The idea of the Taylor polynomial approximation of order $n$ at $x=a$, written $P_n(x,a)$, to a smooth function $f(x)$ is to require that $f(x)$ and $P_n(x,a)$ have the same value at $x=a$. Furthermore, their derivatives at $x=a$ must match up to order $n$. For example the Taylor polynomial of order three for $\sin(x)$ at $x=0$ would have to satisfy the conditions

\begin{displaymath}\begin{array}{ccccc}
P_3(0,0) & = & \sin(0) & = & 0\\
P_3'...
... & = & 0 \\
P_3'''(0,0) & = & -\cos(0) & = & -1
\end{array}\end{displaymath}

You should check for yourself that the cubic polynomial satisfying these four conditions is

\begin{displaymath}P_3(x,0) = x - \frac{1}{6} x^3.\end{displaymath}

The general form of the Taylor polynomial approximation of order $n$ to $f(x)$ is given by the following

Theorem 1   Suppose that $f(x)$ is a smooth function in some open interval containing $x=a$. Then the $n$th degree Taylor polynomial of the function $f(x)$ at the point $x=a$ is given by

\begin{displaymath}P_n(x,a) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k \end{displaymath}


\begin{displaymath}= f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 + \cdots
+ \frac{f^{(n)}(a)}{n!} (x-a)^n\end{displaymath}

We will be seeing this formula a lot, so it would be good for you to memorize it now! The notation $f^{(k)}(a)$ is used in the definition to stand for the value of the $k$-th derivative of $f$ at $x=a$. That is, $f^{(1)}(a) = f'(a)$, $f^{(3)}(a) = f'''(a)$, and so on. By convention, $f^{(0)}(a) =
f(a)$. Note that $a$ is fixed and so the derivatives $f^{(k)}(a)$ are just numbers. That is, a Taylor polynomial has the form

\begin{displaymath}\sum_{k=0}^{n} a_k (x-a)^k \end{displaymath}

which you should recognize as a power series that has been truncated.
next up previous
Next: Accuracy and Tolerance Up: Taylor Polynomials Previous: Getting Started
William W. Farr
2006-04-03