1. 3. Increasing, Decreasing, and the 1st derivative test

2. Graphs
We will be spending the next several days talking about curve sketching
We’ll use information about the function, the 1st derivative, and the 2nd derivative in order to make a sketch of the function

3. If a graph exists on an interval, it is doing one of three things
Increasing (y-values increase as x increases)
Decreasing (y-values decrease as x increases)
Staying constant (y values stay the same as x increases)
The derivatives tell us this information because it tells us the slope.

5. Where does a graph change its increasing/decreasing/constant status??
It only changes at a critical value or at a discontinuity!!
Without a graph, we must find any critical values or discontinuities, then test the intervals in between them.

6. 1st derivative test for local extrema
At a critical point c
If f’ changes sign from positive to negative, then f has a local max
If f’ changes sign from negative to positive, then f has a local min
If f’ doesn’t change sign, then no local extrema
When using the 1st derivative test to justify relative extrema, you MUST write a concluding statement clearly communicating the type of sign change of f’ at each x = c. You may not use the word “IT”

7. Example 1 Inc Dec Inc -2 2 Plug in values in each interval to f’
Local max Local min

8. Example 2

9. Example 3
Find the x-coordinates of the local extrema of the function on the interval

10. Example 4
Find the relative extrema of the function Justify.

11. Example 5
Find the x-coordinates of the relative extrema of the function Justify.