So we’ve now looked at a couple of different functions and found polynomials which approximate the functions to different levels of accuracy. Let’s try and come up with a general method of formulating this. Let’s say that we have some function and we want to approximate it close to . We will then assume that we can write the polynomial approximation as:
Note that previously we wrote but it’s good to get used to slightly changeable notation. The context is what should tell you the meaning.
We want to have that:
We will first ask that the value of the polynomial is equal to the value of the function at . We do this by setting in both sides of the above. Note that we are being slightly ambiguous in what we mean by the approximation here because in a moment we will go from a sign to an sign.…