Polynomial - Polynomials and Calculus

Polynomials and Calculus

One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomial functions. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Polynomial functions are also frequently used to interpolate functions.

Calculating derivatives and integrals of polynomial functions is particularly simple. For the polynomial function

the derivative with respect to x is

and the indefinite integral is