Section 3: Increasing & Decreasing functions & the first derivative test

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# Section 3: Increasing & Decreasing functions & the first derivative test - PowerPoint PPT Presentation

Learning Target: I will determine if a function is increasing or decreasing and find extrema using the first derivative. Section 3: Increasing & Decreasing functions & the first derivative test. AP Exam Practice. A function, f, is increasing on an interval for any

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Learning Target: I will determine if a function is increasing or decreasing and find extrema using the first derivative.

### Section 3: Increasing & Decreasing functions & the first derivative test

A function, f, is increasing on an interval for any

in the interval, then

implies that

A function is decreasing on an interval for

in the interval, then

implies that

I. Increasing & decreasing functions

Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b).

1. If f’(x) > 0 for all x in (a, b) then f is increasing on [a, b].

2. If f’(x) < 0 for all x in (a, b), then f is decreasing on [a, b].

3. If f’(x) = 0 for all x in (a, b), then f is constant on [a, b].

Testing for increasing & decreasing functions

Find the open intervals on which

is increasing or decreasing.

1. Find the critical points and any gaps in the graph.

2. Test the intervals

3. Write the intervals where increasing and decreasing.

Ex 1

Let c be a critical number of a function, f, that is continuous on an open interval, I, containing c. If f is differentiable on the interval, except (possibly) at c, then f(c) can be classified as:

1. If f’(x) changes from + to – at c, then f(c) is a relative maximum.

2. If f’(x) changes from – to + at c, then f(c) is a relative minimum.

3. If f’(x) does NOT change at c, then f’(c) is not a maximum or minimum.

II. The 1st derivative test