MEAN-VALUE THEOREM

theorem in mathematical analysis dealing with a type of average useful for approximations and for establishing other theorems, such as the fundamental theorem of calculus. The theorem states that the slope of a line connecting any two points on a smooth curve is the same as the slope of some line tangent to the curve between the two points. In symbols, if g(x) represents the function, x0 and x1 the two given points, and c1 the point between, then [g(x1) - g(x0)]/(x1 - x0) = g(c1), in which g(c1) represents the slope of the tangent line at c1, as given by the derivative. Although the mean-value theorem seems obvious geometrically, proving the result without reference to diagrams involves deep properties of real numbers and continuous functions. Other mean-value theorems can be obtained from this basic one by letting g(x) be some special function.