Here are some simple animations which might help to understand what we were doing in class today. In each case the function which we are approximating is in red, and the polynomial approximation is in blue.

The following are the different approximations to the function starting at a constant, then a straight line with non-zero gradient, then a quadratic etc. Each one matching the higher and higher derivatives of the function at . You see that by the time we get to a fifth order polynomial, the match is exact. We know this because we know that there is an exact expansion of this function which is a fifth order polynomial.

This is the same plot, but now with the function . This time the polynomial approximation doesn’t stop at the fifth order, but can keep going to any power of .

This is the same function approximated but now approximated about the point .

And this is the function approximated by different order polynomials about the point . Here we go up to the 14th order polynomial, and see that in fact here it looks like a pretty good match to the original function even quite a way away from the point about which we are expanding.