Taylor's Formula - Taylor formula

Taylor's Formula - Taylor formula

Taylor formula is applied mathematics, physics, is described by a function formula of a value in the vicinity thereof in a point information. If the smoothing function is enough, a known function in the case of a point below the respective first derivative values, as coefficients of the Taylor formula can be used to construct a polynomial approximation function value at this point in the neighborhood of these derivative values. Taylor formula is also given polynomial and the deviation between the actual function value.

Taylor is one of the formula $x = x_{0}$ Department has $n$ a function of the derivative $f(x)$ about the use $(x - x_{0})$ Of $n$ -order polynomial approximation function method.

If the function $f(x)$ comprising $x_{0}$ Of a closed interval $[a, b]$ having the $n$ order derivative, and the open interval $(a, b)$ having the $(n + 1)$ The first derivative of the closed interval $[a, b]$ an arbitrary point $x$ , the following equation:

among them, $f^{(n)}(x_{0})$ Represents $f(x)$ is $n$ order derivative, a polynomial function is called after the equal sign $f(x)$ in $x_{0}$ At the Taylor expansion, remaining $R_{n}(x)$ is the remainder of Taylor's formula is $(x - x_{0})^{n}$ high order infinitesimal.

The Lagrange form of the remainder term states that there exists a number $c$ between $a$ and $x$ SUCH that
Lagrange remaining items are shown in the form of $a$ and $x$ exists between the number $c$ , such that

The Cauchy form of the remainder term states that there exists a number c between a and x such that
Cauchy remaining items are shown in the form of $a$ and $x$ exists between the number $c$ , such that

The integral form of the remainder term is
integer remainder term is

1. Taylor formula of commonly used functions