Taylor Series

Abstract

It is known that, zooming-in close enough to a curve, it will start to look like a straight line. This can be tested easily by using any graphic software to draw a curve, and then zooming into a smaller and smaller region of it. It is also the reason why the Earth appears flat to us; it is of course spherical, but humans on its surface see a small portion up close so that it appears like a plane. This leads to the intuition for the third mathematical and modelling tool in this book: it is possible to represent a high-order polynomial (such as a curve or a sphere) with a lower-order polynomial (such as a line or a plain), at least over a small region. The mathematical tool that allows this is called the Taylor series. And, since the straight line mentioned in the first intuitive example is actually the tangent (or first derivative) of the curve, it should come as no surprise that this Taylor series will make heavy use of derivatives of the functions being modelled.