Reply:
f '(xo) is the exact value, f' '(ξ) that is a first-order Taylor's remainder. So, he was launched into the first order.
Taylor formula is a function at x = x0 f (x) having n first derivative method using a polynomial approximation function with respect to (x-x0) n times.
If the function f (x) with n-order derivative on the containing x0 in a closed interval [a, b], and has (n + 1) order derivative in the open interval (a, b), is of the closed interval [a , b] at any point on the x, the following equation:
Wherein represents the n-order derivative f (x), the polynomial function after the equal sign is called f (x) at x0 Taylor expanding the remainder Rn (x) is the remainder of the Taylor formula is (x-x0 ) n high order infinitesimal.
Extended Information:
Practical application, the need to cut off Taylor formula, taking only a limited term, limited term of a Taylor series function is called the Taylor expansion. Taylor Formula it can be used for items such approximation error estimate.
The importance of Taylor expansion is reflected in the following five areas:
1, the power series derivation and integration can be carried out one by one, and therefore relatively easy summation function.
2, an analytic function can be extended to an analytic function on an open sheet in the complex plane is defined, and complex analysis that this approach is feasible.
3, the Taylor series approximation may be used to calculate the value of the function, and the estimation error.
4, prove the inequality.
5, seeking undetermined type of limit.