The Shape of a Graph, Part I
In the previous section we saw how to use the derivative to
determine the absolute minimum and maximum values of a function. However, there
is a lot more information about a graph that can be determined from the first
derivative of a function. We will start looking at that information in this
section. The main idea we�ll be looking at in this section we will be
identifying all the relative extrema of a function.
Let�s start this section off by revisiting a familiar topic
from the previous chapter. Let�s suppose that we have a function,
.
We know from our work in the previous chapter that the first derivative,
 ,
is the rate of change of the function. We used this idea to identify where a
function was increasing, decreasing or not changing.
Before reviewing this idea let�s first write down the
mathematical definition of increasing and decreasing. We all know what the
graph of an increasing/decreasing function looks like but sometimes it is nice
to have a mathematical definition as well. Here it is.
Definition
This definition will actually be used in the proof of the
next fact in this section.
Now, recall that in the previous chapter we constantly used
the idea that if the derivative of a function was positive at a point then the
function was increasing at that point and if the derivative was negative at a
point then the function was decreasing at that point. We also used the fact
that if the derivative of a function was zero at a point then the function was
not changing at that point. We used these ideas to identify the intervals in
which a function is increasing and decreasing.
The following fact summarizes up what we were doing in the
previous chapter.
Fact
The proof of this fact is in the
Proofs From Derivative Applications section of the Extras chapter.
|
and
from
an interval I with
if
then
is
increasing on I.
and
from
an interval I with
if
then
is
decreasing on I.
for
every x on some interval I, then
is
increasing on the interval.
for
every x on some interval I, then
is
decreasing on the interval.
for
every x on some interval I, then
is
constant on the interval.
