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Asymptote. The graph of a function with a horizontal ( y = 0), vertical ( x = 0), and oblique asymptote (purple line, given by y = 2 x ). A curve intersecting an asymptote infinitely many times. In analytic geometry, an asymptote ( / ˈæsɪmptoʊt /) of a curve is a line such that the distance between the curve and the line approaches zero as ...
A sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve . A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: [ 1] Other standard sigmoid functions are given in the Examples section. In some fields, most notably in the context of ...
The folium of Descartes is related to the trisectrix of Maclaurin by affine transformation. To see this, start with the equation and change variables to find the equation in a coordinate system rotated 45 degrees. This amounts to setting. In the plane the equation is. If we stretch the curve in the direction by a factor of this becomes which is ...
Asymptotic analysis. In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function f (n) as n becomes very large. If f(n) = n2 + 3n, then as n becomes very large, the term 3n becomes insignificant compared ...
Truncus (mathematics) In analytic geometry, a truncus is a curve in the Cartesian plane consisting of all points ( x, y) satisfying an equation of the form. A mathematical graph of the basic truncus formula, marked in blue, with domain and range both restricted to [-5, 5]. where a , b, and c are given constants.
e. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function. Formal definitions, first devised in the early 19th century, are given below.
Newton's diagram. Newton's diagram (also known as Newton's parallelogram, after Isaac Newton) is a technique for determining the shape of an algebraic curve close to and far away from the origin. It consists of plotting (α, β) for each term Axαyβ in the equation of the curve. The resulting diagram is then analyzed to produce information ...
The generalized logistic function or curve is an extension of the logistic or sigmoid functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named Richards's curve after F. J. Richards, who proposed the general form for the family of models in 1959.