Guided Practice for 3.1: Extreme values of a function

Overview

This section starts a chapter focusing on applications of the derivative. This section in particular sets the stage with a number of important terms and conceptual tools for working with derivatives in an applied setting. We introduce the notion of relative and absolute maximum and minimum values of a function as well as critical numbers of a function and the First Derivative Test and Second Derivative Test for finding relative extreme values of a function. A method involving sign charts is introduced for tracking both the increasing/decreasing behavior of a function and the concavity of a function. Finally, the important notion of an inflection point is introduced.

Learning objectives

Basic objectives: Each student is responsible for gaining proficiency with each of these tasks prior to engaging in class discussions, through the use of the learning resources (below) and through the working of exercises (also below).

• Define the concepts of local maximum, local minimum, global maximum, and global minimum.
• State the First Derivative Test and explain both its purpose and how it is used.
• State the Second Derivative Test and explain both its purpose and how it is used.
• Explain how to make a sign chart for either the first or second derivative and then use it to determine the location of extreme values of a function.
• Define the concept of a point of inflection.
• Explain how a sign chart might be used to determine the intervals on which a function is concave up or concave down and to find inflection points.

Advanced objectives: The following objectives are the subject of class discussion and further work; they should be mastered by each student during and following class discussions.

• Construct sign charts for functions to find where they are increasing and decreasing, and concave up or concave down, and consequently to find extreme values and inflection points.

Learning resources

To gain proficiency in the learning objectives, use the following resources. You may include other resources if you wish, in addition to or in replacement of the following.

Textbook: In Active Calculus, read Section 3.1. Make sure to read actively, working through examples and activities as you go.

Video: Watch the following videos at the MTH 201 YouTube playlist (http://bit.ly/GVSUCalculus).

• Quick review: Extreme values of functions (4:11)
• Finding local and global extrema (5:06)
• Identifying critical values graphically (4:21)
• Finding critical values (3:12)
• Using the First Derivative Test (11:02)
• Second Derivative Test (8:02)
• Identifying inflection points graphically (7:03)
• Finding inflection points (7:31)

Activities

The following activity is to be done during and following your reading and viewing of the resources. Go to student.desmos.com and join using your name in the format Last, First. For example, I would enter my name as Ballif, Serge. Complete each part of the activity. Some of these problems will require you to work them out on paper before entering your answer. Practice producing high quality work so that your work is readable and meaningful. You will receive a mark of Pass if each item response shows a good-faith effort to be right and is submitted prior to the deadline. Remember to use the Piazza discussion board to ask about any questions you have.