## 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.

### Examples

Consider a continuous function f with the following properties:
1. f(–2) = 0
2. f(0) = 1
3. f(2) = 0
4. f has a local maximum at x = 0
Choose a possible graph of f'.

Consider a function f which is defined for all real numbers except x = 1, and assume f has a given list of properties. Draw a possible graph of f.

Match graphs of three functions with their derivatives.

### Applets

Curve Sketching: Increasing/Decreasing
Curve Sketching: Concavity

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 3.5 Issues in Curve Sketching (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel