1. Chapter 4 Curve Sketching Using Derivatives instead of Plotting Points

2. Lets relate a graph to its derivative Find the first derivative. Put zeroes on a number line Find the second derivative. Put zeroes on a number line Lets look at the graph here:here:

3. Slope y′ = slope, correct? Logic says (as did the graph): y′ > 0 m+ Graph rises y′ < 0 m– Graph falls y ′ = 0 m = 0 Graph hits peak or valley

4. Curvature y′′ = curvature We saw on the graph y ′′ > 0 concave up y ′′ < 0 concave down y ′′ = 0 inflection point

5. Some vocabulary Critical Point where y′ = f′ = 0 or DNE Absolute Maximum f(c) ≥ f(x) for all x in domain (highest point overall) Absolute Minimum f(c) ≤ f(x) for all x in domain (lowest point overall) Relative Maximum/Minimum f(c) ≥ f(x)/f(c) ≤ f(x) for x near c (high or low somewhere in a small area)

6. How to graph this way? Lets see if we can figure it out, with an example: Please may I have another? Of course you may.