First presented in a variant form by the French mathematician Michel Rolle (1652-1719) in an obscure book, the mean-value theorem is one of the fundamental principles of the discipline of calculus. Stated mathematically, the mean-value theorem asserts that if a function f(x) is both continuous and differentiable over the closed interval [a, b] (that is, the interval includes the endpoints a and b), then there exists at least one number c (and perhaps more) such that the first derivative of the function (symbolized by f^'(x)), evaluated at c, is equal to the difference of the function evaluated at b and a divided by the difference of b and a. Given the conditions of continuity and a closed interval, the mean-value theorem can be written (in symbolic form) f^'(c) = [f(b)--f(a)] / (b--a). If the quantity [f(b...