Feb 7 - Taylor Series

Today was an introduction to the idea of a Taylor Series. The general idea of a Taylor Series is that we are going to take functions and represent them as "infinite" polynomials. The "Why?" question will be answered eventually but for now we're going to focus on the "How?" question.

Most of the Taylor Series we will work with will have the form:

a0 + a1*x + a2*x^2 + a3*x^3 + ... + an*x^n + ...

where each of the coefficients has the form an = f^n(0) / n! (where f^n(0) means "the nth derivative of f(x) evaluated at x = 0).

Generally, a Taylor Series has the form:

a0 + a1*(x - c) + a2*(x - c)^2 + a3*(x - c)^3 + ... + an*(x - c)^n + ...

where each of the coefficients has the form an = f^n(c) / n!

You will eventually be responsible for the Taylor series for e^x, sin x and cos x.

Here is the animation I used in class to illustrate the Taylor series for sin x.



Notes from today are below.

p1

p2