3.5 Issues in Curve Sketching

Summary

The first and second derivatives are used to sketch a curve by hand. Hence, f' and f'' are being used to get graphical information about f. For example, where is f increasing or decreasing, and where does it have local maxima or minima?

By the end of your studying, you should know:

The definitions of concepts connected with curve sketching. For example, local max/min, absolute max/min, critical point, singular point, endpoint, concave up/down, inflection point.
The possibiltiies for z if f is differentiable and has a local max (or min) at z.
The First Derivative Test for local max/min and how to apply it in tabular form.
The Second Derivative Test for concavity and how to apply it in tabular form.
The Second Derivative Test for local max/min.
How to sketch a curve by putting together the foregoing information.

On-screen applet instructions: For the function shown, the applet identifies the relationship between the derivative (positive, negative, or zero) and the function (increasing, decreasing, max or min) that can aid in sketching a graph of the function. Use the slider to move the point P along the curve.