Taylor's series expansion

An expansion of an exact function can be conveniently used to approximate that function by a finite number of terms (usually one or two) in a polynomial form. The general form of an expansion of a function f(x) about a point x0 is

The term f(n)(x0) means the nth derivative of the function evaluated at the point x0. Often times we are interested in small values of x, i.e. x near zero. Then we will expand the function around the point x0 = 0.

All of the examples given below will be of this special case.

Several examples of common expansions are given below:

  1. Exponential

  2. Sine

    3.  

  3. Square root of 1 + x