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For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. [1]
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:
This is the quadratic function whose first and second derivatives are the same as those of f at a given point. The formula for the best quadratic approximation to a function f around the point x = a is () + ′ () + ″ (). This quadratic approximation is the second-order Taylor polynomial for the function centered at x = a.
The second-order Taylor expansion of f around ... If the second derivative is positive, the quadratic approximation is a convex function of , and its minimum can be ...
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.
If all second-order partial derivatives of exist, then the Hessian matrix of is a square matrix, usually defined and arranged as = []. That is, the entry of the i th row and the j th column is ( H f ) i , j = ∂ 2 f ∂ x i ∂ x j . {\displaystyle (\mathbf {H} _{f})_{i,j}={\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}.}
"Sur l'ordre de la meilleure approximation des fonctions continues par les polynômes de degré donné" [On the order of the best approximation of continuous functions by polynomials of a given degree]. Mem. Acad. Roy. Belg. (in French). 4: 1– 104. Faber, Georg (1914). "Über die interpolatorische Darstellung stetiger Funktionen" [On the ...
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.