[미적분]다항식의 추정값(polynomial appoximation to functions)

7. Polynomial Approximations to Functions.

Theorem 7.1. Let f be a function with derivatives of order n at the point x=0. Then there exists one and only one polynomial P of degree ≤ n which satisfies the n+1 conditions p(0) = f(0), P`(0) = f`(0), ....., P(n)(0) = f(n)(0). This polynomial is given by the formula P(x) = Tn f(x).

Point. point x = a, P(x) = Tn f(x;a).

Point. T2n+1(sinx) = x - x3/3! + x5/5! - x7/7! + ... + (-1)n x2n+1/(2n+1)!

Point. T2n(cosx) = 1 - x2/2! + x4/4! - x6/6! + .... + (-1)n x2n/(2n)!

Theorem. 7.2. The Taylor operator Tn has the following properties:
(a) Linearity property. If c1 and c2 are constants, then Tn(c1f + c2g) = c1Tn(f) + c2Tn(g)
(b) Differentiation property. The derivative of a Taylor polynomial of f is a Taylor polynomial of f`; in fact, we have (Tnf)` = Tn-1(f`).
(c) Integration property. An indefinite integral of a Taylor polynomial of f is a Taylor polynomial of an indefinite integral of f. More precisely, if g(x) = , t…