The Taylor Expansion - a Useful Tool in Numerical Differentiation

Here we will discuss a few basic ideas in the construction of a representation of a function at a point nearby a given point. For the present we focus on a function of a single variable f(x). The generalization to functions of several variables is straightforward and will be given when needed. We focus on using the so-called Taylor series expansion, since we shall see later that with it one can reduce the problem of generating approximations to derivatives to algebraic manipulations. Further, with the Taylor series method one automatically gets an indication of the order of accuracy of the particular approximation.