Taylor formula is a formula that uses the information of a function at a certain point to describe its nearby value. If the function meets certain conditions, Taylor formula can use the derivative values of each order of the function at a certain point as coefficients to construct a polynomial to approximate the function. In recent years, multiple-choice questions are relatively large and are common in the college entrance examination. In general, we will construct a function model and substitute numerical values for comparison and operation. However, it is very difficult for students to choose the function model, so we can choose to use Taylor formula to calculate the approximate value in multiple-choice questions to compare the sizes.

If the function f(x) has n-order derivatives at point x0, these derivatives constitute:

TN(x)= f(x0)+f '(x0)(x-x0)/ 1！ +f"(x0)(x-x0)^2/2！ +…+f^(n)(x0)(x-x0)^n/n！

A Taylor polynomial with function f(x) at point x0, where the coefficient f (k) (x0)/k! ? (k= 1，2，…，，？ N) is called Taylor coefficient.

Taylor expansion of function f(x) is the sum of its corresponding Taylor polynomial and an infinitesimal higher than (x-x0) n, that is, TN (x)+o ((x-x0) n) = f (x0)+f' (x0) (x-x0)/65438+. +f"(x0)(x-x0)^2/2！ +…+f^(n)(x0)(x-x0)^n/n！ +o((x-x0)^n)。 It is the basis of all Taylor expansions, so it is counted as the first commonly used Taylor expansion.

Therefore, the key to determine the Taylor expansion of the function is to determine the coefficient of each term, and more fundamentally, to determine the derivative values of each order of the function at x0.

The other nine common Taylor expansions include:

Teaching enlightenment

(1) Taylor formula is introduced for estimation. As can be seen from the above example, teachers can? In order to introduce some knowledge of Taylor formula with the help of after-class exercises, it will be convenient for students to understand some estimation and approximation problems, and they can learn and master some calculation methods of specific values of functions.

(2) Simplifying the calculation by using Taylor formula As can be seen from the above example, Taylor formula will be very accurate when estimating in a small range, so Taylor formula can often be considered when encountering such problems.

(3) Think deeply and stimulate interest.

Teachers can use the example of "Taylor formula to simplify calculation" to stimulate students' deep thinking about mathematics and stimulate students' interest in exploring mathematics.