Taylor polynomials are a special type of polynomial that approximate a function at a specific point. They are made up of a finite number of terms and are centered around that point. They are used to approximate more complex functions and can be used to find values of a function that may be difficult to calculate directly.
Regular polynomials have a fixed degree, meaning they have a fixed number of terms. Taylor polynomials, on the other hand, do not have a fixed degree and can have a varying number of terms depending on the desired level of accuracy. Regular polynomials also approximate a function over the entire domain, while Taylor polynomials only approximate the function at a specific point.
The Taylor series is an infinite sum of terms that represents a function at a specific point. It is made up of the coefficients of the terms in a Taylor polynomial, and as the number of terms increases, the Taylor series becomes a more accurate approximation of the function. The Taylor series can also be used to find the derivatives of a function at a specific point.
The coefficients of a Taylor polynomial can be found by taking derivatives of the function at the center point and plugging those values into the Taylor polynomial formula. The general formula for the nth term of a Taylor polynomial is f(n)(a)(x-a)n/n!, where f(n)(a) represents the nth derivative of the function at the center point a.
The main purpose of using Taylor polynomials is to approximate more complex functions and to find values of a function that may be difficult to calculate directly. They can also be used to understand the behavior of a function near a specific point and to find the derivatives of a function at that point. Taylor polynomials are an important tool in calculus and are used in many scientific and engineering applications.