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The slope of a ray drawn from the origin through the point is the value of the average function. If the absolute value of the slope of the tangent is greater than the slope of the ray then the function is elastic at the point; if the slope of the secant is greater than the absolute value of the slope of the tangent then the curve is inelastic ...
Specifically, a straight line on a log–log plot containing points (x 0, F 0) and (x 1, F 1) will have the function: = (/) (/), Of course, the inverse is true too: any function of the form = will have a straight line as its log–log graph representation, where the slope of the line is m.
Castigliano's method for calculating displacements is an application of his second theorem, which states: If the strain energy of a linearly elastic structure can be expressed as a function of generalised force Q i then the partial derivative of the strain energy with respect to generalised force gives the generalised displacement q i in the direction of Q i.
Given the definition of the elasticity coefficient in terms of a partial derivative, it is possible, for example, to determine the elasticity of an arbitrary rate law by differentiating the rate law by the independent variable and scaling. For example, the elasticity coefficient for a mass-action rate law such as:
An example in microeconomics is the constant elasticity demand function, in which p is the price of a product and D(p) is the resulting quantity demanded by consumers.For most goods the elasticity r (the responsiveness of quantity demanded to price) is negative, so it can be convenient to write the constant elasticity demand function with a negative sign on the exponent, in order for the ...
The SI unit for elasticity and the elastic modulus is the pascal (Pa). This unit is defined as force per unit area, generally a measurement of pressure, which in mechanics corresponds to stress. The pascal and therefore elasticity have the dimension L −1 ⋅M⋅T −2.
The compatibility conditions in linear elasticity are obtained by observing that there are six strain-displacement relations that are functions of only three unknown displacements. This suggests that the three displacements may be removed from the system of equations without loss of information.
Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.