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The asymptotes most commonly encountered in the study of calculus are of curves of the form y = ƒ(x). These can be computed using limits and classified into horizontal , vertical and oblique asymptotes depending on their orientation.
An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation =, y becomes arbitrarily small in magnitude as x increases.
The asymptotic directions are the same as the asymptotes of the hyperbola of the Dupin indicatrix through a hyperbolic point, or the unique asymptote through a parabolic point. [1] An asymptotic direction is a direction along which the normal curvature is zero: take the plane spanned by the direction and the surface's normal at that point. The ...
Determine the asymptotes of the curve. Also determine from which side the curve approaches the asymptotes and where the asymptotes intersect the curve. [1] Equate first and second derivatives to 0 to find the stationary points and inflection points respectively.
The curve was first proposed and studied by René Descartes in 1638. [1] Its claim to fame lies in an incident in the development of calculus.Descartes challenged Pierre de Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines.
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.
asymptote In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors. [3]
The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and published in his Principia in 1687, [2] which was the first problem in the field to be clearly formulated and correctly solved, and was one of the most difficult problems tackled by variational methods prior to the twentieth century.
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