Increasing and Decreasing Functions

1. Increasing and Decreasing Functions AP Calculus – Section 3.3

2. Increasing and Decreasing Functions On an interval in which a function f is continuous and differentiable, a function is… • increasing if f ‘(x) is positive on that interval, • decreasing if f ‘(x) is negative on that interval, and • constant if f ‘(x) = 0 on that interval.

3. Visual Example f ‘(x) < 0 on (-5,-2) f(x) is decreasing on (-5,-2) f ‘(x) > 0 on (1,3) f(x) is increasing on (1,3) f ‘(x) = 0 on (-2,1) f(x) is constant on (-2,1)

4. Finding Increasing/Decreasing Intervals for a Function To find the intervals on which a function is increasing/decreasing: • Find critical numbers. • Pick an x-value in each closed interval between critical numbers; find derivative value at each. • Test derivative value tells you whether the function is increasing/decreasing on the interval.

5. Example Find the intervals on which the function is increasing and decreasing. Critical numbers:

6. Example Test an x-value in each interval. f(x) is increasing on and . f(x) is decreasing on .

7. Assignment p.181: 1-5, 7, 9

8. The First Derivative Test AP Calculus – Section 3.3

9. The First Derivative Test If c is a critical number of a function f, then: • If f ‘(c) changes from negative to positive at c, then f(c) is a relative minimum. • If f ‘(c) changes from positive to negative at c, then f(c) is a relative maximum. • If f ‘(c) does not change sign at c, then f(c) is neither a relative minimum or maximum. • GREAT picture on page 176!

10. Find all intervals of increase/decrease and all relative extrema. Test: f is decreasing before -4 and increasing after -4; so f(-4) is a MINIMUM. Test:

11. Assignment p.181: 11-21 odd