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 \(f(x)\) and we want to approximate it close to \(x=a\). We will then assume that we can write the polynomial approximation as:

\[\sum_{i=0}^n c_i (x-a)^i\]

Note that previously we wrote \(a_i\) 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:

\[f(x)\approx \sum_{i=0}^n c_i (x-a)^i\]

We will first ask that the value of the polynomial is equal to the value of the function at \(x=a\). We do this by setting \(x=a\) 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 \(\approx\) sign to an \(=\) sign.…