Search results
Results from the WOW.Com Content Network
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 the zeroth-order example above, the quantity "a few" was given, but in the first-order example, the number "4" is given. A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a linear approximation, straight line with a slope: a polynomial of degree 1. For example:
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 ]
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 example, the second-order equation y′′ = −y can be rewritten as two first-order equations: y′ = z and z′ = −y. In this section, we describe numerical methods for IVPs, and remark that boundary value problems (BVPs) require a different set of tools. In a BVP, one defines values, or components of the solution y at more than one ...
6.1 Example of first-order ... such as the variational method and the WKB approximation. ... in the vicinity region can be calculated in power series (like Taylor ...
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]