Search results
Results from the WOW.Com Content Network
A sigmoid function is constrained by a pair of horizontal asymptotes as . A sigmoid function is convex for values less than a particular point, and it is concave for values greater than that point: in many of the examples here, that point is 0.
The graph of a function with a horizontal (y = 0), vertical (x = 0), and oblique asymptote (purple line, given by y = 2x) A curve intersecting an asymptote infinitely many times In analytic geometry , an asymptote ( / ˈ æ s ɪ m p t oʊ t / ) of a curve is a line such that the distance between the curve and the line approaches zero as one or ...
Examples of applications are the following. In applied mathematics, asymptotic analysis is used to build numerical methods to approximate equation solutions. In mathematical statistics and probability theory, asymptotics are used in analysis of long-run or large-sample behaviour of random variables and estimators.
In physics and other fields of science, one frequently comes across problems of an asymptotic nature, such as damping, orbiting, stabilization of a perturbed motion, etc. Their solutions lend themselves to asymptotic analysis (perturbation theory), which is widely used in modern applied mathematics, mechanics and physics. But asymptotic methods ...
The vertical and horizontal lines are asymptotes. In the same way, it can be shown that the reciprocal of a continuous function r = 1 / f {\displaystyle r=1/f} (defined by r ( x ) = 1 / f ( x ) {\displaystyle r(x)=1/f(x)} for all x ∈ D {\displaystyle x\in D} such that f ( x ) ≠ 0 {\displaystyle f(x)\neq 0} ) is continuous in D ∖ { x : f ...
For example, the parent function = / has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant.
The following are usually easy to carry out and give important clues as to the shape of a curve: Determine the x and y intercepts of the curve. The x intercepts are found by setting y equal to 0 in the equation of the curve and solving for x.
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.