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Concept
Elasticity of a function
Résumé
In mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output) at point a is defined as
or equivalently
It is thus the ratio of the relative (percentage) change in the function's output with respect to the relative change in its input , for infinitesimal changes from a point . Equivalently, it is the ratio of the infinitesimal change of the logarithm of a function with respect to the infinitesimal change of the logarithm of the argument. Generalisations to multi-input-multi-output cases also exist in the literature.
The elasticity of a function is a constant if and only if the function has the form for a constant .
The elasticity at a point is the limit of the arc elasticity between two points as the separation between those two points approaches zero.
The concept of elasticity is widely used in economics and Metabolic Control Analysis; see elasticity (economics) and Elasticity coefficient respectively for details.
Rules for finding the elasticity of products and quotients are simpler than those for derivatives. Let f, g be differentiable. Then
The derivative can be expressed in terms of elasticity as
Let a and b be constants. Then
In economics, the price elasticity of demand refers to the elasticity of a demand function Q(P), and can be expressed as (dQ/dP)/(Q(P)/P) or the ratio of the value of the marginal function (dQ/dP) to the value of the average function (Q(P)/P). This relationship provides an easy way of determining whether a demand curve is elastic or inelastic at a particular point. First, suppose one follows the usual convention in mathematics of plotting the independent variable (P) horizontally and the dependent variable (Q) vertically. Then the slope of a line tangent to the curve at that point is the value of the marginal function at that point. The slope of a ray drawn from the origin through the point is the value of the average function.
Source officielle
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