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Derivative test. In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is ...
For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous. [1] In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem.
This is called the second derivative test. An alternative approach, called the first derivative test, involves considering the sign of the f' on each side of the critical point. Taking derivatives and solving for critical points is therefore often a simple way to find local minima or maxima, which can be useful in optimization.
A function of class or -function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a -function. However, it may also mean "sufficiently differentiable" for the ...
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals ", it has two major branches, differential calculus and integral calculus.
Because of this, different methods need to be used to solve BVPs. For example, the shooting method (and its variants) or global methods like finite differences, [3] Galerkin methods, [4] or collocation methods are appropriate for that class of problems. The Picard–Lindelöf theorem states that there is a unique solution, provided f is ...
t. e. In numerical analysis, the Runge–Kutta methods (English: / ˈrʊŋəˈkʊtɑː / ⓘ RUUNG-ə-KUUT-tah[1]) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. [2]
Mathematical optimization. Graph of a surface given by z = f (x, y) = − (x ² + y ²) + 4. The global maximum at (x, y, z) = (0, 0, 4) is indicated by a blue dot. Nelder-Mead minimum search of Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value. Mathematical optimization (alternatively ...