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Second-order Taylor series approximation (in orange) of a function f (x,y) = e x ln(1 + y) around the origin. In order to compute a second-order Taylor series expansion around point (a, b) = (0, 0) of the function (,) = (+), one first computes all the necessary partial derivatives:
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 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. A simulation-based alternative to this approximation is the application of Monte Carlo simulations.
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]
The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems , linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems . [ 1 ]
Linear approximations in this case are further improved when the second derivative of a, ″ (), is sufficiently small (close to zero) (i.e., at or near an inflection point). If f {\displaystyle f} is concave down in the interval between x {\displaystyle x} and a {\displaystyle a} , the approximation will be an overestimate (since the ...
First-order approximation is the term scientists use for a slightly better answer. [3] Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has 4 × 10 3, or four thousand, residents"). In the case of a first-order approximation, at least one number given is exact.
An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively). In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors".
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