1. 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

2. AP Exam Practice

3. I. Increasing & decreasing functions
A function, f, is increasing on an interval for any
in the interval, then
implies that
A function is decreasing on an interval for

4. Testing for 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].

5. Ex 1
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.

6. Ex 2: Strictly Monotonic
Find the intervals on which f(x) = x³ is increasing or decreasing.

7. Ex 3: Not Strictly Monotonic
Find the intervals on which
is increasing or decreasing.

8. AP Exam Practice

9. II. The 1st derivative test
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.

10. EX 1:
Find the relative extrema of in the interval (0, 2π).

11. Ex 2:
Find the relative extrema of

12. homework
Pg 181 #1-31 odds