Introduction
The Mean Value Theorem states that a function,
f
, continuous on
[a,b]
and differentiable on
(a,b)
has a point
c
, between
a
and
b
, where the rate of change of
f
at
c
is equal to the average rate of change of
f
from
a
to
b
.
In mathematical notation, the Mean Value Theorem is:
\text{Given } f \text{, continous on } [a,b] \text{ and differentiable on } (a,b) \text{, } \exists c \in (a,b) \text{, where } f'(c) = \frac{f(b) - f(a)}{b - a}
Example
f(x)=x^3 \quad a=-1 \quad b=1
The average rate of change of
f
on
[-1,1]
is
\frac{f(b) - f(a)}{b - a} = \frac{1 - (-1)}{1 - (-1)} = 1
The Mean Value Theorem states that somewhere between
x=-1
and
x=1
,
f'(x) = 1
.
In this example, there are two:
c = \pm \sqrt{1/3}
.
f'\left(-\sqrt{\frac{1}{3}}\right) = 1 \quad f'\left(\sqrt{\frac{1}{3}}\right) = 1