[5] Geometrically, this difference quotient measures the slope of the secant line passing through the points with coordinates (a, f(a)) and (b, f(b)).
[10] Difference quotients are used as approximations in numerical differentiation,[8] but they have also been subject of criticism in this application.
[15] The typical notion of the difference quotient discussed above is a particular case of a more general concept.
The quintessential application of the divided difference is in the presentation of the definite integral, which is nothing more than a finite difference: Given that the mean value, derivative expression form provides all of the same information as the classical integral notation, the mean value form may be the preferable expression, such as in writing venues that only support/accept standard ASCII text, or in cases that only require the average derivative (such as when finding the average radius in an elliptic integral).
(thus requiring less averaging effort): This also becomes particularly useful when dealing with iterated and multiple integrals (ΔA = AU − AL, ΔB = BU − BL, ΔC = CU − CL): Hence, and