Polynomials and calculus

One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, 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. Polynomials are also frequently used to interpolate functions.
Quotients of polynomials are called rational expressions, and functions that evaluate rational expressions are called rational functions. Rational functions are the only functions that can be evaluated on a computer by a fixed sequence of instructions involving operations of addition, multiplication, division, which operations on floating point numbers are usually implemented in hardware. All the other functions that computers need to evaluate, such as trigonometric functions, logarithms and exponential functions, must then be computed in software that may use approximations to those functions on certain intervals by rational functions, and possibly iteration.
Calculating derivatives and integrals of polynomials is particularly simple. For the polynomial