Explanation:
Mean value theorem
For any
function that is continuous on [a, b] and differentiable on
(a, b) there exists some c in the interval
(a, b) such that the secant joining the
endpoints of the interval [a, b] is parallel to the tangent at c.
In calculus, the mean value theorem states, roughly, that given an arc of
a smooth differentiable (continuous) curve, there is at least
one point on that arc at which the derivative (slope) of the curve is equal
(parallel) to the "average" derivative of the arc. Briefly, a
suitable infinitesimal 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 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
(Our solved example in mathguru.com uses this concept)
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. To get at that average speed, the car
either has to go at a constant 100 miles per hour during that whole time, or,
if it goes slower at one moment, it has to go faster at another moment as well
(and vice versa), in order to still end up with an average of 100 miles per
hour. Therefore, the Mean Value Theorem tells us that at some point during the
journey, the car must have been traveling at exactly 100 miles per hour; that
is, it was traveling at its average speed.
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),
where 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 +∞ or −∞. If
finite, that limit equals f′(x).
An example where this version of the theorem applies is given by the
real-valued cube root function mapping x to x^{1/3}, whose derivative tends to infinity at the origin.
Note that the theorem is false if a differentiable function is
complex-valued instead of real-valued. For example, define f(x) = e^{ix} for all real x. Then
f (2π)
− f(0) = 0 =
0(2π − 0)
while |f′(x)|
= 1.
http://en.wikipedia.org/wiki/Mean_value_theorem
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