1. Increasing and Decreasing Functions Lesson 5.1

2. The Ups and Downs Think of a function as a roller coaster going from left to right Uphill Slope > 0 Increasing function Downhill Slope < 0 Decreasing function 2

3. Definitions Given function f defined on an interval For any two numbers x 1 and x 2 on the interval Increasing function f(x 1 ) < f(x 2 ) when x 1 < x 2 Decreasing function f(x 1 ) > f(x 2 ) when x 1 < x 2 3 X2X2 X1X1 X1X1 X2X2 f(x)

4. Increasing/Decreasing and the Derivative Assuming existence of derivative on interval If f '(x) > 0 for each x f(x) increasing on interval If f '(x) < 0 for each x f(x) decreasing on interval 4 What if f '(x) = 0 on the interval? What could you say about f(x)?

5. Check These Functions By graphing on calculator, determine the intervals where these functions are Increasing Decreasing 5

6. Critical Numbers Definition Numbers c in the domain of f where f '(c) = 0 f '(c) does not exist 6 Critical Points

7. Applying Derivative Test Given a function f(x) Determine the derivative f '(x) Find critical points … Where f '(x) = 0 or f '(x) does not exist Evaluate derivative between or on either side of the critical points 7 Try it with this function

8. Applications Digitari, the great video game manufacturer determines its cost and revenue functions C(x) = 4.8x -.0004x 2 0 ≤ x ≤ 2250 R(x) = 8.4x -.002x 2 0 ≤ x ≤ 2250 Determine the interval(s) on which the profit function is increasing 8

9. Assignment Lesson 5.1 Page 313 Exercises 1 – 57 EOO 9