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

3. Definitions Given function f defined on an interval
For any two numbers x1 and x2 on the interval
Increasing function
f(x1) < f(x2) when x1 < x2
Decreasing function
f(x1) > f(x2) when x1< x2 X1 X2 X2 X1
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
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

6. Critical Numbers Definition Numbers c in the domain of f where
f '(c) = 0
f '(c) does not exist
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
Try it with this function

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

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