The Taylor's Theorem and Its Application

Taylor's theorem is an essential concept in calculus. By constructing polynomials, Taylor's formula can simplify calculations by approximating complex functions so that a variety of functions can be analyzed in detail. This paper explores the form and proof of Taylor's theorem based on Peano and Lagrange remainder forms as well as the Polynomial interpolation. Then, examples of the application of Taylor’s theorem in mathematics fields, such as limit calculation and high-order derivatives computation, are discussed. Additionally, the applications of Taylor’s theorem in other related subjects are examined. Based on the chronological investigation, the Taylor theorem is widely used in physics, engineering, and other fields as a computational tool. The Taylor theorem has gradually been used in machine learning because of the rapid development of computer science in recent years. The purpose of this paper is to summarize the different forms and proofs of Taylor's formula and discuss the development of its application over time.