A generalization of the Taylor's theorem

Abstract

In mathematics, Taylor's theorem is a formula that uses the information of a function to describe its proximity to a point. If the function is smooth enough, the conductive values can be used to construct a multiform to approximate the the function in the neighboring area of that point, which can even be extended to the convergence radius of the scale. Taylor’s theorem also gives the deviation between this multiform and the actual function value. The essay includes the proof and application of Taylor’s theorem and the Taylor’s theorem in multivariate functions. Special functions are set with unknown coefficients to deduce the Taylor’s theorem. Specific questions are solved by using the Taylor’s theorem as some examples. Taylor's theorem is frequently applied in real-world settings to solve problems. For instance, in engineering, it can be used to build a variety of numerical simulations and optimization algorithms. In physics, it can be used to analyze data from physical experiments. In finance, it can be used to manage risk and set prices for financial transactions.