Search results
Results from the WOW.Com Content Network
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.
For example, the exponential function is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. However, one may equally well define an analytic function by its Taylor series. Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in ...
Taylor's theorem; Rules and identities ... from elementary calculus to the corresponding multi-variable case. Below are some examples. ... formula is used for the ...
Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case = states that = + ′ () + where is the remainder term. The linear approximation is obtained by dropping the remainder: f ( x ) ≈ f ( a ) + f ′ ( a ) ( x − a ) . {\displaystyle f(x)\approx f(a)+f'(a)(x-a).}
In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.
Taylor's formula has an effect of dividing a function by variables, which can be illustrated by the next typical theoretical use of the formula. Example: [8] Let : be a linear map between the vector space of smooth functions on with rapidly decreasing derivatives; i.e., | | < for any multi-index ,.
For the second-order approximations of the third central moment as well as for the derivation of all higher-order approximations see Appendix D of Ref. [3] Taking into account the quadratic terms of the Taylor series and the third moments of the input variables is referred to as second-order third-moment method. [4]
1.6.2 Using the Taylor series and Newton's method for the inverse ... for example, once defined the ... are independent random variables, then two distinct linear ...