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 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.
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.
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.
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

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:

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.
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, 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'Hopital'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 p≠0).
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

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 limx↓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:
- 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]
See also
References
External links
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