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.
Taylor formula is named after British mathematician Brook Taylor, who first described it in a letter 17 12. Taylor formula is one of the commonly used approximate methods to study the properties of complex functions, and it is also an important application content of function differential calculus.
Taylor formula is a very important content in advanced mathematics. It approximately represents some complex functions as simple polynomial functions. This function of Taylor formula makes it a powerful tool to analyze and study many mathematical problems.
Brook Taylor, a mathematician, was one of the most outstanding representatives of the British Newton School in the early18th century. His main work was The Method of Positive and Negative Increments published in 17 15, in which he stated the famous theorem-Taylor in a letter to his teacher Meiqin in July12. 17 17 Taylor uses Taylor's definition to understand numerical equations.
Taylor formula is developed from Gregory-Newton interpolation formula, which uses the information of a function at a certain point to describe the value near it. If the function is smooth enough, Taylor formula can construct a polynomial to approximate the function value in the neighborhood of the point with these derivative values as coefficients on the premise of knowing the derivatives of each order.
In 1772, Lagrange emphasized the importance of Taylor formula and called it the basic theorem of differential calculus, but the convergence of series was not considered in the proof of Taylor theorem. This work was not completed by Cauchy until the 1920s of 19. Taylor theorem initiated the finite difference theory, which made any unary function expand into a power series. Therefore, people call Taylor the founder of finite difference theory.
Taylor formula is an important content in mathematical analysis, and it is also an indispensable mathematical tool to study the function limit and estimation error. Taylor formula embodies the essence of "approximate method" of calculus and has unique advantages in approximate calculation.
Taylor formula can transform nonlinear problems into linear problems with high accuracy, so it has important applications in all aspects of calculus. Taylor formula can be used to find the limit, judge the extreme value of function, find the value of higher derivative at a certain point, judge the convergence of generalized integral, approximate calculation, inequality proof and so on.