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The polynomials, exponential function e x, and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if x is far from b.
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.
In mathematics, a cubic function is a function of the form () = + + +, that is, a polynomial function of degree three. In many texts, the coefficients a , b , c , and d are supposed to be real numbers , and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to ...
An important example in calculus is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, and the Stone–Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by ...
This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots. As a cubic polynomial has three roots (not necessarily distinct) by the fundamental theorem of algebra, at least one root must be real.
The sine function and all of its Taylor polynomials are odd functions. The cosine function and all of its Taylor polynomials are even functions. In mathematics , an even function is a real function such that f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} for every x {\displaystyle x} in its domain .
For example, if a quantity is constant within the whole interval, approximating it with a second-order Taylor series will not increase the accuracy. In the case of a smooth function, the nth-order approximation is a polynomial of degree n, which is obtained by truncating the Taylor series
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.