Math 1314
Lesson 11
Applications of the Second Derivative
Concavity
Earlier in the course, we saw that the second derivative is the rate of change of the first
derivative.
The second derivative can tell us if the rate of change of the function is increasing
or decreasing.
In business, for example, the first derivative might tell us that our profits are
increasing, but the second derivative will tell us if the pace of the increase is increasing or
decreasing.
Example 1
:
From these graphs, you can see that the shape of the curve change differs depending on
whether the slopes of tangent lines are increasing or decreasing.
This is the idea of
concavity
.
Definition
:
Let the function
f
be differentiable on an interval (
a
,
b
).
Then
f
is
concave
upward
on (
a
,
b
) if
f
′
is increasing on (
a
,
b
) and
f
is
concave downward
on (
a
,
b
) if
f
′
is
decreasing on (
a
,
b
).
Determining Where a Function is Concave Upward and Where it is Concave Downward
By Analyzing the Sign of the Second Derivative Algebraically
We can also determine concavity algebraically.
The procedure for doing this should look
pretty familiar:
1.
Find the second derivative of the function.
2.
Determine all values of
x
for which
0
)
(
=
′
′
x
f
or is undefined.
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 Fall '08
 MARKS
 Derivative, Rate Of Change, advertising campaign, Convex function, Concave function

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