http___en.wikipedia.org_wiki_Inflection_point_3.xml
1.1 2.1 2.2 3.1 4.1 5.1 5.2 6.1 7.1 8.1 9.1 9.2 10.1 10.2
Инфлексна точка В областта на математиката и в частност на диференциалното смятане (математически анализ) инфлексна точка или точка на инфлексия на една функция е точка от кривата, описваща тази функция, при преминаване през която функцията от вдлъбната се променя в изпъкнала, или обратното. Ако си представим каране на превозно средство по криволичещ път, в инфлексната точка кормилото е изправено, преди да бъде завъртяно в другата посока. Инфлексната точка може да се определи още като: - точка от крива, в която тангентата пресича кривата; - точка от крива, в която втората производна на функцията, описваща кривата, си сменя знака. Това определение е близко до предното, защото знакът на кривината винаги съвпада със знака на втората производна; - точка от кривата, при която стойността на втората производна на функцията е равна на нула, тоест f′′(x)=0, или f′′(x) не съществува. - точка от кривата на графиката на дадена функция, за която първата ѝ производна има локален екстремум. Необходими, но недостатъчни условия[редактиране | edit source] Условието f′′(x)=0 е изпълнено и при някакъв локален екстремум. Тъй като f′(x)=0, то и втората производна е нула, но в тази точка функцията може да не се изменя от вдлъбната в изпъкнала или обратното. Също така една функция може да се променя от вдлъбната в изпъкнала (или обратното) в точка, която не е от дефиниционното множество на функцията. При този случай тя няма инфлексна точка.
1.1 2.1 3.1 4.1 5.1 6.1 7.1 7.2 7.3 7.4 8.1 9.1 10.1 11.1 12.1 12.2 12.3 12.4 12.5 13.1 13.2 14.1 15.1 16.1 17.1 18.1 18.2 19.1 19.2 20.1 21.1 21.2 21.3 21.4 21.5 22.1 23.1 24.1 25.1 26.1 27.1 28.1 29.1 30.1 31.1 31.2
Inflection point In differential calculus, an inflection point, point of inflection, flex, or inflection (inflexion) is a point on a curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa. A point where the curvature vanishes but does not change sign is sometimes called a point of undulation or undulation point. In algebraic geometry an inflection point is defined slightly more generally, as a point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4. Contents Equivalent forms[edit] A differentiable function has an inflection point at (x, f(x)) if and only if its first derivative, f′, has an isolated extremum at x. (This is not the same as saying that f has an extremum). That is, in some neighborhood, x is the one and only point at which f′ has a (local) minimum or maximum. If all extrema of f′ are isolated, then an inflection point is a point on the graph of f at which the tangent crosses the curve. For an algebraic curve, a non singular point is an inflection point if and only if the multiplicity of the intersection of the tangent line and the curve (at the point of tangency) is odd and greater than 2.[1] For a curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.[2] For a twice differentiable function, an inflection point is a point on the graph at which the second derivative changes sign. A necessary but not sufficient condition[edit] If x is an inflection point for f then the second derivative, f″(x), is equal to zero if it exists, but this condition does not provide a sufficient definition of a point of inflection. One also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of such an undulation point is x = 0 for the function f given by f(x) = x4. This definition assumes that f has some higher-order non-zero derivative at x, which is not necessarily the case, But if it has one, it follows from the definition that the sign of f′(x) is the same on either side of x in a neighborhood of x. If this is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection. Categorization of points of inflection[edit] Points of inflection can also be categorized according to whether f′(x) is zero or not zero. - if f′(x) is zero, the point is a stationary point of inflection - if f′(x) is not zero, the point is a non-stationary point of inflection An example of a saddle point is the point (0,0) on the graph y = x3. The tangent is the x-axis, which cuts the graph at this point. A non-stationary point of inflection can be visualised if the graph y = x3 is rotated slightly about the origin. The tangent at the origin still cuts the graph in two, but its gradient is non-zero. Functions with discontinuities[edit] Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. Take, for example, the function 2x2/(x2 – 1). It is concave when |x| > 1 and convex when |x| < 1. However, it has no points of inflection because 1 and -1 are not in the domain of the function. See also[edit] - Critical point (mathematics) - Ogee, an architectural form with an inflection point - Vertex (curve), a local minimum or maximum of curvature - Hesse configuration formed by the nine inflection points of an elliptic curve References and Sources[edit] References[edit] Sources[edit] - Weisstein, Eric W., "Inflection Point", MathWorld. - Hazewinkel, Michiel, ed. (2001), "Point of inflection", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4