In calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, the mean value theorem states, roughly, that given an arc of a differentiable

Differentiable function

In calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain...

 (continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

) curve, there is at least one point on that arc at which the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 (slope) of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal

Infinitesimal

Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

 element of the arc is parallel to the secant chord connecting the endpoints of the arc. The theorem is used to prove theorems that make global conclusions about a function on an interval starting from local hypotheses about derivatives at points of the interval.

More precisely, if a function f(x) is continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

 on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) such that

This theorem can be understood intuitively by applying it to motion: If a car travels one hundred miles in one hour, then its average speed during that time was 100 miles per hour.