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The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin. The Taylor polynomials for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. For x > 1, Taylor polynomials of higher degree provide worse approximations.
In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function , the Taylor polynomial is the truncation at the order k {\textstyle k} of the Taylor series of the function.
A difference engine is an automatic mechanical calculator designed to tabulate polynomial functions. It was designed in the 1820s, and was first created by Charles Babbage . The name difference engine is derived from the method of divided differences , a way to interpolate or tabulate functions by using a small set of polynomial co-efficients.
Instead, truncated Taylor polynomial algebra can be used. The resulting arithmetic, defined on generalized dual numbers, allows efficient computation using functions as if they were a data type. Once the Taylor polynomial of a function is known, the derivatives are easily extracted.
Horner's method evaluates a polynomial using repeated bracketing: + + + + + = + (+ (+ (+ + (+)))). This method reduces the number of multiplications and additions to just Horner's method is so common that a computer instruction "multiply–accumulate operation" has been added to many computer processors, which allow doing the addition and multiplication operations in one combined step.
For a n-times differentiable function, by Taylor's theorem the Taylor series expansion is given as (+) = + ′ ()! + ()! + + ()! + (),. Where n! denotes the factorial of n, and R n (x) is a remainder term, denoting the difference between the Taylor polynomial of degree n and the original function.
In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function: [1] = + + = = + +. This series converges in the complex disk | |, except for = (where =).
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.