Section 4.3
Two special cases


Our first special case concerns n-th-order approximations of functions. The map f -> f mod Xn+1 for polynomials f is the truncation map T:

T: R[X] -> R[X]<n + 1,
a0 + a1X + ··· + anXn + ··· + asXs -> a0 + a1X + ··· + anXn.

In terms of polynomial functions, the truncation is an approximation of f about 0 to order n. We will transfer this principle to arbitrary, sufficiently often differentiable functions. Taylor

Let f be a real-valued function defined on an interval containing 0 R and sufficiently often differentiable. Then the polynomial a = a0 + a1X + ··· + anXn is called the n-th-order approximation of f about 0 if

f(x) = a(x) + O(xn+1)    if x -> 0.

Such an n-th-order approximation is unique; in fact it consists of the first n + 1 terms of the Taylor series of f about 0.

Theorem

Let f be a continuous n-times differentiable real-valued function. Then the polynomial

F = f(0) + f(1)(0)/1! X + ··· + f(n)(0)/n! Xn in R[X]

is the n-th-order approximation of f about 0.

Furthermore, if G is an n-th order approximation of a function g, then (FG) mod Xn + 1 and (F+G) mod Xn + 1 are the n-th-order approximations of fg and f + g, respectively.