Mean value theorem
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 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.
Encyclopedia
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 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.
This theorem was first developed by
Lagrange . 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 . Then, there exists some c in such that
The mean value theorem is a generalization of
Rolle's theorem, which assumes
f =
f, 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 the
limitexists 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 is:
y = +
f.
The formula / gives the
slope of the line joining the points and , which we call a chord of the curve, while
f ' gives the slope of the tangent to the curve at the point . 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 =
f +
rx, where
r is a constant. Since
f is continuous on [
a,
b] and differentiable on , 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 =
g, there is some
c in for which
g ' = 0, and it follows
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 and are both continuous on the closed interval , differentiable on the open interval , and is not zero on that open interval, then there exists some in , 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 .
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 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 =
h, by Rolle's theorem, there exists some
c in such that
h′ = 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 such that
In particular for φ = 1, there exists
x in such that
The
second mean value theorem for integration is stated 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
- If G : [a, b] → R is a monotonically decreasing function and φ : [a, b] → R is an integrable function, then there exists a number x in such that
The latter statement was proved by Hiroshi Okamura in 1947.
See also
External links
