The Mean Value Theorem The Mean Value Theorem

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

Graph

The tangent line of

f
at
x=\pm \sqrt{1/3}
is parallel to the line drawn from
(a,f(a))
to
(b,f(b))
.
The yellow line is the function,
f(x)=x^3
.
The grey line is the line from
(-1,-1)
to
(1,1)
.
The purple line is the slope of the grey line.
The moving line is the tangent line and the moving dot represents the value of the slope/derivative of the tangent line.
When the derivative dot is on the purple line, the slope of the tangent line is equal to the slope of the purple line.
The Mean Value Theorem states that there is guaranteed to be a point where the derivative dot is on the purple line.