### The Taylor Polynomial

The Taylor polynomial of degree *n* in a point *a* contained in the domain of a function f that is *n* times differentiable and continuous in *a*, is given by

Applied to f(x) = sin x, we find as Taylor polynomial of degree 3 around 0: 0+1*x-0,166666666666667*x^3

Graphically, this gives:

In particular the Taylor polynomial of degree 1 around *a* is the equation of the straight line tangent to the graph of f in *a*. This illustrates clearly that the straight line tangent to a curve in *a* is a linear approximation of the curve. As the degree of the Taylor polynomial increases, the graph of the Taylor polynomial will approximate the graph of f around *a* better, at least if there is convergence. Below, we see the straight line tangent to the graph of the sine function in the origin, found using the Taylor expansion of degree 1 around 0: