Interpolation polynomial types

Question

I was wondering if both the Maclaurin and Taylor series are two types of interpolation polynomials? I was under the impression that they were not because they only go though one point in an interval where as compared to the lagrange polynomial goes through multiple points.

Answer 1

In principle, one could call them interpolation polynomials, since they are a special case of Hermite interpolation.

However, this is really a stretch. The word interpolation suggests we are looking to obtain approximate values of a function between some known points. The Taylor polynomials does not fit this picture.

It is natural to consider the Taylor/Maclaurin polynomial as a limiting case of Lagrange polynomial, when all its nodes collapse into one point. This becomes clear thanks to the Newton form of the interpolation polynomial which consists of terms such as $(x-x_0)(x-x_1)\dots (x-x_k)$ multiplied by the $k$th order divided difference of $f$. As $x_1\to x_0,\dots,x_n\to x_0$, this becomes $(x-x_0)^k$ multiplied by $f^{(k)}(x_0)/k!$.