Section 3.3 Intervals of Increase and Decrease - UH

[Pages:5]Section 3.3 ? Intervals of Increase and Decrease

Let be a function whose domain includes an interval .

We say that is increasing on if for every two numbers 1, 2 in , 1 2 implies that 1 2 .

We say that is decreasing on if for every two numbers 1, 2 in , 1 2 implies that 1 2 .

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Example 1: Given the graph of a polynomial function below, give the intervals of increase and decrease.

Increasing:

Decreasing:

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One way we can find intervals of increase and decrease is to graph the function. Example 2: Given 5| 2| + 1, when is this function increasing? When is it decreasing?

Increasing:

Decreasing:

Example 3: Given decreasing?

2 + 1,

0

2, 0 5 , when is this function increasing? When is it

,

5

Increasing:

Decreasing:

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Let us use the graph below to observe the slopes of the tangent lines as the graph increases and decreases.

Over the intervals where the function is increasing, the tangent lines have positive slope. On the other hand, over the intervals of decrease, the tangent lines have negative slope.

Theorem: Suppose that is differentiable on the interior of an interval and continuous on all of .

x If 0 for all in , then increases on . x If 0 for all in , then decreases on .

Example 4: Given 65 403 + 10, when is this function increasing? When is it decreasing?

Increasing:

Decreasing:

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Example 5: Given

2-8 -4 2

2 , when is this function increasing? When is it decreasing?

-4

Increasing:

Decreasing:

Example 6: Given 8 cos4 , when is this function increasing on 0, ? When is it decreasing on 0, ?

32 cos3 sin

Increasing:

Decreasing:

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Page 5 of 5 Example 7: Given 4 2/3, when is this function increasing? When is it decreasing?

Increasing:

Decreasing:

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