Home > Mean value theorem

This theorem was developed by Lagrange. Some mathematicians consider this theorem to be the most important theorem of calculus (see also: the fundamental theorem of calculus). The theorem is not often used to solve mathematical problems; rather, it is more commonly used to prove other theorems. The mean value theorem can be used to prove Taylor's theorem, of which it is a special case.

More precisely, the theorem states: for some continually differentiable curve; for every secant, there is some parallel tangent. In addition, the tangent runs through a pointThe word point can refer to: a location in physical space a unit of angular measurement; see navigation point is a typographic unit of measure in typography equal inch or sometimes approximated as inch; on computer displays it should be equal to point in located between the intersection points of said secant.

Let f : [a, b] → R be continuous on the closed intervalTopology In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. Interval notation is where the permitted values for a variable are expressed as ranging over an in [a, b], and differentiable on the open interval (a, b). Then there exists some c in (a, b) such that

Generalization: The theorem is usually stated in the form above, but it is actually valid in a slightly more general setting: We only need to assume that f : [a , b] → R is continuous on [a , b], and that for every x in (a , b) the limitA limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. in the form of an inequality, such as: Chandrasekhar limit Greisen exists or is equal to ± infinity.

1 Proof

An understanding of this and the Point-Slope FormulaIn mathematics, the slope (or gradient especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the "steepness" of said line. This pages focuses on such slopes. With an understandi will make it clear that the equation of a secant (which intersects (a, f(a)) and (b, f(b)) ) is: y = {[f(b) - f(a)] / [b - a]}(x - a) - f(a).

The formula ( f(b) - f(a) ) / (b - a) gives the slopeIn mathematics, the slope (or gradient especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the "steepness" of said line. This pages focuses on such slopes. With an understandi of the line joining the points (a , f(a)) and (b , f(b)), which we call a chord of the curve, while f ' (x) gives the slope of the tangent to the curve at the point (x , f(x) ). Thus the Mean value theorem says that given any chord of a smooth curve, we can find a point lying between the end-points of the chord such that the tangent at that point is parallel to the chord. The following proof illustrates this idea.

Define g(x) = f(x) + rx , where r is a constant. Since f is continuous on [a , b] and differentiable on (a , b), the same is true of g. We choose r so that g satisfies the conditions of Rolle's theorem, which means

By Rolle's Theorem, there is some c in (a , b) for which g '(c) = 0, and it follows

as required. The mean value theorem in the following form is considered more useful.