mean-value theorems

mean-value theorems

In mathematics, two theorems, one associated with differential calculus and one with integral calculus. The first proposes that any differentiable function defined on an interval has a mean value, at which a tangent line is parallel to the line connecting the endpoints of the function's graph on that interval. For example, if a car covers a mile from a dead stop in one minute, it must have been traveling exactly a mile a minute at some point along that mile. In integral calculus, the mean value of a function on an interval is, in essence, the arithmetic mean (see mean, median and mode) of its values over the interval. Because the number of values is infinite, a true arithmetic mean is not possible. The theorem shows how to find the mean value using a definite integral. See also Rolle's theorem.