The Further Mathematics Support Programme
Degree Topics in Mathematics
Mean Value Theorem and Rolle’s Theorem
Before looking at these two theorems, we need to consider what it means for a function to be
differentiable.
If you have studied differentiation from first principles at A level, you will recognise the
following definition of a derivative:
(
( )
)
( )
If this limit exists, we say f is differentiable at the point
For example, consider the function ( )
obtain
.
. If we look at the limit for this function we
(
( )
Simplifying the numerator and cancelling the
obtain
( )
)
term from numerator and denominator we
(
)
and so as
the limit is ( )
and this exists for all values of . Hence the function
is differentiable for all values. We would normally write this result as ( )
.
What types of functions are not differentiable?
This is the function
This function could not be differentiable at
as it is
not actually defined there. In fact, there is an asymptote
at
– we say the function is not continuous.
A function cannot be differentiable at a given point if it is
not defined at that point.
This function is
(
) . We can see that
the gradient is infinite at the point
. The
derivative of this function is ( )
(
)
It is easy to see that this derivative is not defined
at the point
.
A function is not differentiable at a given point if
the graph has infinite gradient at that point.
The Further Mathematics Support Programme
Task 1
Is the function ( )
continuous for all real numbers? Is it differentiable?
Task 2
Which of the following would be a suitable set of values for
( )
is continuous and differentiable?
(a) {
}
(b) {
}
to ensure that the function
(c) {
}
Task 3
Determine for what set of values of
differentiable.
the function ( )
√
is defined, continuous and
This line is called a
secant
Rolle’s Theorem
Rolle’s theorem states that if a function
( ):
-
is continuous on the closed
interval [a, b]
is differentiable on the open
interval (a, b)
has two values and such that
( )
( )
then there exists at least one number
the interval (a, b) such that ( )
.
Image taken from http://en.wikibooks.org/wiki/Calculus/Rolle%27s_Theorem
This is basically a sophisticated way of saying that if a curve has no discontinuities and has
two values which produce the same value when substituted into the function, then there
will be a maximum or a minimum point between those two values. In your A-level studies,
you will have found the location of maxima and minima by differentiating a function and
putting the derivative equal to zero.
Task 4
Show that Rolle’s theorem holds for a range of
function ( )
values (which you should specify) on the
in
The Further Mathematics Support Programme
The Mean Value Theorem
The mean value theorem states that if
f is continuous on the interval [a, b] and
is differentiable on the interval (a, b)
then there exists a point c in the interval
(a, b) such that
( )
( )
( )
In other words, there is at least one
point between a and b at which the
gradient is equal to the gradient of the
secant between a and b.
Image taken from
http://www.digplanet.com/wiki/Mean_value_theorem
Task 5
Show that the mean value theorem holds for the function ( )
where
and
. Find the value of which satisfies the theorem.
on the interval
Task 6
There is a connection between the mean value theorem and Rolle’s theorem: in fact, one is
a special case of the other.
Explain the link between the two theorems.
The Further Mathematics Support Programme
Solutions
Task 1
The function ( )
is continuous and has no
discontinuities, as can be seen from the graph of the
function.
There are no points where the gradient is infinite (no
vertical sections) and so the function is differentiable
for all values of .
From first principles, or using the standard laws of
differentiation, we know ( )
Task 2
This graph has a discontinuity at
so any set of
values which includes 0 would be unsuitable. Therefore
the only possible set of values that would be suitable
} i.e. just consider the right hand branch of
is {
the graph.
From first principles or using the standard laws of
differentiation, we know that for this section of the
graph, ( )
Task 3
The graph of the function ( ) √
is shown
here. It is not defined for
(as we cannot find
the square root of a negative number).
For
the gradient of the graph is infinite.
Hence the set of
{
}
values required is
The Further Mathematics Support Programme
Task 4
Check the conditions for Rolle’s theorem:
The function is clearly continuous from the graph
(we know quadratic functions have no
discontinuities).
From first principles we can see:
(
( )
(
)
)
(
( )
)
(
)
Hence the function is differentiable for all
values
We now need to identify two values of for which the function has the same value. A
( )
natural choice is ( )
. This means that by Rolle’s Theorem there must be a
value in the interval (0, 4) for which the derivative is zero.
This is true, as we can clearly see that
( )
Task 5
The function ( )
is
continuous on the interval [0, 3]
It is also differentiable on the interval
(0 ,3) (this can be checked by
differentiation from first principles if
required).
Draw in vertical lines at
and at
and the secant between the yvalues.
Calculating ( ) and ( ) gives
( )
( )
So
( )
( )
( )
This means there is a point in the interval (0,3) which has a gradient of 4. We can check
this by differentiating: ( )
and equating this to 4 gives
and so
.
Looking
at
the
graph
around
the
point
it
is
easy
to
see
that
at
a
√
tangent at that point would be parallel to the secant.
The Further Mathematics Support Programme
Task 6
The mean value theorem states that if f is continuous on the interval [a, b] and is
differentiable on the interval (a, b) then there exists a point c in the interval (a, b) such that
( )
( )
( )
Rolle’s theorem states that if a function ( ) is continuous on the closed interval [a, b], is
differentiable on the open interval (a, b), and has two values and such that ( )
( ),
then there exists at least one number in the interval (a, b) such that ( )
.
Therefore Rolle’s theorem is a special case of the mean value theorem. If we choose two
points a and b for which ( )
( ), then the formula for the mean value theorem becomes
( )
( )
( )
which is the condition for Rolle’s theorem.