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Lesson 10 – Applications of the First Derivative
1
Lesson 10
Applications of the First Derivative
Determining the Intervals on Which a Function is Increasing or Decreasing
From the Graph of
f
A function is
increasing
on an interval (
a, b
) if, for any two numbers
1
x
and
2
x
in (
a, b
),
)
(
)
(
2
1
x
f
x
f
<
, whenever
2
1
x
x
<
.
A function is
decreasing
on an interval (
a, b
) if, for
any two numbers
1
x
and
2
x
in (
a, b
),
)
(
)
(
2
1
x
f
x
f
, whenever
2
1
x
x
<
.
In other words, if the
y
values are getting bigger as we move from left to right across the
graph of the function, the function is increasing. If they are getting smaller, then the
function is decreasing.
Example 1:
You are given the graph of
f
(
x
).
State the interval(s) on which
f
is increasing
and the interval(s) on which
f
is decreasing.
The rate of change of a function at a point is given by the derivative of the function at
that point.
So we can use the derivative to determine where a function is increasing and
where a function is decreasing.
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2
Let’s look at the slopes of some lines that are tangent to this graph at various points:
Here are some generalizations:
At a point where the derivative is positive, a function is increasing.
At a point where the derivative is negative, a function is decreasing.
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 Fall '08
 CONSTANTE
 Derivative

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