If we use polynomials to infinitely approximate a function, it is a Taylor expansion of a function at a point. Taylor series is to expand a function and turn it into a form of adding power terms. The purpose is to use relatively simple functions to fit complex functions. At this time, the relative simplicity depends on what you need. The first-order refers to the maximum number of expansions. , the second-order means that the maximum number of expansions is 2. The geometric meaning of the Taylor formula is to use the polynomial function to approximate the original function. Since the polynomial function can be derived any number of times, it is easy to calculate, and it is convenient to solve the extreme value or judge the nature of the function. Therefore, the information of the function can be obtained through the Taylor formula. At the same time, for For this approximation, an error analysis must be provided to provide the reliability of the approximation.
Taylor formula, also known as Taylor expansion. It is a formula that uses the information of a function at a certain point to describe its nearby values. If the function is smooth enough, when the derivative values of each order of the function at a certain point are known, the Taylor formula can use these derivative values as coefficients to construct a polynomial approximation function to obtain the value in the neighborhood of this point.
So what is the Taylor formula for? To put it simply, it is to use a polynomial function to approximate a given function (that is, to make the polynomial function image fit the given function image as much as possible). Note that when approximating, it must be expanded from a certain point on the function image. If you want to find the value of a certain point of a very complex function, but it is impossible to find it directly, you can use Taylor's formula to approximate the value, which is one of the applications of Taylor's formula. Taylor's formula is mainly used in gradient iteration in machine learning.
There are two types of residuals in Taylor's formula: one is qualitative Peano residuals, and the other is quantitative Lagrangian residuals. These two types of remainders are essentially the same, but have different functions. Generally speaking, when there is no need to discuss the remainder term quantitatively, Peano remainder term can be used (such as finding the limit of the undetermined form and estimating the infinitesimal order); when it is necessary to discuss the remainder term quantitatively, the Lagrangian remainder term should be used (such as using the Taylor formula to approximate the function value)
