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The Shape of a Graph, Part II
In the previous section we saw how we could use the first derivative of a function to
get some information about the graph of a function. In this section we are going to
look at the information that the second derivative of a function can give us a about the
graph of a function.
Before we do this we will need a couple of definitions out of the way. The main
concept that we’ll be discussing in this section is concavity. Concavity is easiest to
see with a graph (we’ll give the mathematical definition in a bit).
So a function is
concave up
if it “opens” up and the function is
concave down
if it
“opens” down. Notice as well that concavity has nothing to do with increasing or
decreasing. A function can be concave up and either increasing or decreasing.
Similarly, a function can be concave down and either increasing or decreasing.
It’s probably not the best way to define concavity by saying which way it “opens”
since this is a somewhat nebulous definition. Here is the mathematical definition of
concavity.
Definition 1
Given the function
then
1.
is
concave up
on an interval
I
if all of the tangents to the curve on
I
are
below the graph of
.
2.
is
concave down
on an interval
I
if all of the tangents to the curve
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I
are above the graph of
.
To show that the graphs above do in fact have concavity claimed above here is the
graph again (blown up a little to make things clearer).
So, as you can see, in the two upper graphs all of the tangent lines sketched in are all
below the graph of the function and these are concave up. In the lower two graphs all
the tangent lines are above the graph of the function and these are concave down.
Again, notice that concavity and the increasing/decreasing aspect of the function is
completely separate and do not have anything to do with the other. This is important
to note because students often mix these two up and use information about one to get
information about the other.
There’s one more definition that we need to get out of the way.
Definition 2
A point
is called an
inflection point
if the function is continuous at the point and
the concavity of the graph changes at that point.
Now that we have all the concavity definitions out of the way we need to bring the
second derivative into the mix. We did after all start off this section saying we were
going to be using the second derivative to get information about the graph. The
following fact relates the second derivative of a function to its concavity. The proof
of this fact is in the
Proofs From Derivative Applications
section of the Extras
chapter.
Fact
Given the function
then,
1.
If
for all
x
in some interval
I
then
is
concave up on
I
.
2.
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This note was uploaded on 11/06/2011 for the course MATH 151 taught by Professor Sc during the Fall '08 term at Rutgers.
 Fall '08
 sc
 Derivative

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