In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the "average" derivative of the section. It 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. Image File history File links Mvt2. ... For other uses, see Calculus (disambiguation). ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... For a non-technical overview of the subject, see Calculus. ...

This theorem can be understood concretely by applying it to motion: if a car travels one hundred miles in one hour, so that its average speed during that time was 100 miles per hour, then at some time its instantaneous speed must have been exactly 100 miles per hour.

An early version of this theorem was first described by Parameshvara (1370–1460) from the Kerala school of astronomy and mathematics in his commentaries on Govindasvāmi and Bhaskara II.^[1] The mean value theorem in its modern form was later stated by Augustin Louis Cauchy (1789–1857). It is one of the most important results in differential calculus, as well as one of the most important theorems in mathematical analysis, and is essential in proving the fundamental theorem of calculus. The mean value theorem can be used to prove Taylor's theorem, of which it is a special case. Look up theorem in Wiktionary, the free dictionary. ... Parameshvara (परमेश्वर) (1360-1425) was a major mathematician of the Kerala school. ... The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India which included as its prominent members Parameshvara, Nilakantha Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. ... Govindasvāmi or Govindasvāmin (c. ... Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ... Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... For a non-technical overview of the subject, see Calculus. ... For other uses, see Calculus (disambiguation). ... Analysis has its beginnings in the rigorous formulation of calculus. ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ...

Formal statement

Let f : [a, b] → R be a continuous function on the closed interval [a, b], and differentiable on the open interval (a, b). Then, there exists some c in (a, b) such that

The mean value theorem is a generalization of Rolle's theorem, which assumes f(a) = f(b), so that the right-hand side above is zero. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... For a non-technical overview of the subject, see Calculus. ... In calculus, Rolles theorem states that if a function f is continuous on a closed interval and differentiable on the open interval , and then there is some number c in the open interval such that . Intuitively, this means that if a smooth curve is equal at two points then...

The mean value theorem is still valid in a slightly more general setting, one only needs to assume that f : [a, b] → R is continuous on [a, b], and that for every x in (a, b) the limit In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ...

exists as a finite number or equals ±∞.

Proof

An understanding of this and the point-slope formula will make it clear that the equation of a secant (which intersects (a, f(a)) and (b, f(b)) ) is: This article is about the mathematical term. ... A secant line of a curve is a line that intersects two or more points on the curve. ...

The formula ( f(b) − f(a) ) / (b − a) gives the slope 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. This article is about the mathematical term. ...

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 In calculus, Rolles theorem states that if a function f is continuous on a closed interval and differentiable on the open interval , and then there is some number c in the open interval such that . Intuitively, this means that if a smooth curve is equal at two points then...

By Rolle's theorem, since g is continuous and g(a) = g(b), there is some c in (a, b) for which g '(c) = 0, and it follows from g(x) = f(x) + rx that,

as required.

Cauchy's mean value theorem

Cauchy's mean value theorem, also known as the extended mean value theorem, is the more general form of the mean value theorem. It states: If functions f(t) and g(t) are both continuous on the closed interval [a,b], differentiable on the open interval (a,b), and g'(t) is not zero on that open interval, then there exists some c in (a,b), such that

Cauchy's mean value theorem can be used to prove l'Hôpital's rule. The mean value theorem is the special case of Cauchy's mean value when g(t) = t (or more generally when g(t) is affine and not constant, meaning g(t) = pt + q where p and q are constants and ). In calculus, lHôpitals rule (sometimes spelled as lHospitals rule) uses derivatives to help compute limits with indeterminate forms. ...

Proof of Cauchy's mean value theorem

The proof of Cauchy's mean value theorem is based on the same idea as the proof of the mean value theorem. First we define a new function h(t) and then we aim to transform this function so that it satisfies the conditions of Rolle's theorem.

where m is a constant. We choose m so that

Since h is continuous and h(a) = h(b), by Rolle's theorem, there exists some c in (a, b) such that h′(c) = 0, i.e.

as required.

Mean value theorems for integration

The first mean value theorem for integration states

If G : [a, b] → R is a continuous function and φ : [a, b] → R is an integrable positive function, then there exists a number x in (a, b) such that

In particular for φ(t) = 1, there exists x in (a, b) such that This article is about the concept of integrals in calculus. ...

There are various slightly different theorems called the second mean value theorem for integration. A commonly found version is as follows:

If G : [a, b] → R is a positive monotonically decreasing function and φ : [a, b] → R is an integrable function, then there exists a number x in (a, b] such that

Here G(a + 0) stands for lim_x↓aG(x), the existence of which follows from the conditions. Note that it is essential that the interval (a, b] contains b. A variant not having this requirement is: In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...

If G : [a, b] → R is a monotonic (not necessarily decreasing and positive) function and φ : [a, b] → R is an integrable function, then there exists a number x in (a, b) such that

This variant was proved by Hiroshi Okamura in 1947.^{[citation needed]} In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... Hiroshi Okamura is a Japanese mathematician,was a professor at Kyoto University,studied IVP of ODE. He discovered the Iff condition of IVP for the solution to be unique. ...

See also

• arithmetic mean
• Newmark-beta method
• mean value theorem (divided differences)

In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... The Newmark-beta method is a method of numerical integration used to solve differential equations. ... In mathematics divided differences is a recursive division process. ...

References

The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ...

External links

• PlanetMath: Mean-Value Theorem
• Mathworld: Mean-Value Theorem