4.3a Increasing and Decreasing Functions And the First Derivative Test State Standards – 9.0b Students can identify maxima, minima of graphs of functions. – 9.0d Students find the intervals of which the graphs of functions are increasing or decreasing Objective – To be able to find max, min, and where the graph is increasing or decreasing.

In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator or computer to draw complicated graphs, it is still important to understand the relationships between derivatives and graphs.

First Derivative Test Increasing/Decreasing Test If f’(x)>0 on an interval, then f is increasing on that interval. If f’(x)<0 on an interval, then f is decreasing on that interval.

Just to reiterate: A function is increasing over an interval if the derivative is always positive. A function is decreasing over an interval if the derivative is always negative.

3 Example 1: 4 Neg Pos Dec Min Inc Intervals or CN Test pt k Identify the intervals that are increasing and Decreasing for: First derivative test: Set Intervals or CN 3 Test pt k Conclusion 4 Neg Pos Dec Min Inc

4 Example 2: Pos 5 Neg Inc Max Dec Intervals or CN Test pt k Identify the intervals that are increasing and Decreasing for: First derivative test: Intervals or CN 4 Test pt k Conclusion Set Pos 5 Neg Inc Max Dec

4 Example 3: -1 1 5 Pos Neg Pos Inc Max Dec Min Inc Intervals or CN Identify the intervals that are increasing and Decreasing for: First derivative test: Set Intervals or CN 4 Test pt k Conclusion -1 1 5 Pos Neg Pos Inc Max Dec Min Inc

4.3a Homework Identify the open intervals on which the function is increasing or decreasing. 1) 7) 2) 8) 3) 9) 4) 10) 5) 11) 6) 12)