Calculus Group Activity: Mean Value Theorem

Goals

This lab has three main goals:

1. Aid in the understanding of the Mean Value Theorem and its consequences.
2. Assist in clarifying the logical reasoning that leads to the Mean Value Theorem.
3. Provide practice in reading mathematical arguments and synthesizing the progression of ideas behind mathematical results.

Main Topics

Sections 4.1 and 4.2 of Stewart's Calculus (Sixth Edition) identifies the following main phrases and results (in the order given):

• absolute maximum (or global maximum) and maximum value
• absolute minimum (or global minimum) and minimum value
• extreme values of a function
• local maximum (or relative maximum)
• local minimum (or relative minimum)
• Extreme Value Theorem
• Fermat's Theorem
• critical number
• Closed Interval Method
• Rolle's Theorem
• Mean Value Theorem
• Theorem 5 (If f'(x) = 0 throughout (a, b), then f(x) is constant.)
• Theorem 7 (If f'(x) = g'(x) throughout (a, b), then f(x) = g(x) + c for some constant c.)

Step 1: Examples

Write an example to illustrate each of the above phrases and results.

Order of Presentation

Principle: A logical argument begins with known definitions, properties, or results. The argument proceeds one step at a time, in which each step follows by a logical explanation based on previous steps (which already have been shown to be true).

Following this principle, when one statement is supposed to follow from previous statements, the foundational results must be stated (and proved) before the foundational results can be used to prove something else. For example, if results A and B are used in the argument of the truth of C, then the argument must establish the truth of A and B before the argument for C can start.

However, if arguments C and D both rely upon A and B, but neither the argument for C nor the argument for D references the other, then the arguments for C and D could be given in either order.

Step 2: Order of Presentation in Sections 4.1 and 4.2

Review the various phrases and results from Sections 4.1 and 4.2, as listed at the start of this lab. For each phrase or result, determine what material the phrase or result depends upon. Using this analysis as background, determine the extent to which the topics in Sections 4.1 and 4.2 could have been presented in another order.

• If each topic depends upon all previous material, then the order of presentation could not have been changed.
• If two topics do not depend upon each other, then they could have been discussed in either order.

Ideas Behind Results

In mathematics, terms are given specific meanings; a definition simply provides a meaning to a word or phrase. As a definition, the meaning of a term cannot be considered open to discussion — this is what the word or phrase means. We could discuss whether or not the definition was useful, but not whether the definition is true. A definition is an agreement that for the purposes of discussion, a word will have a specified meaning. From this perspective, a definition can be considered as an axiom; we can assume the term has the meaning specified.

For results that are based on a logical argument (i.e., for any statement other than a definition or axiom), the conclusion must depend upon a logical argument. In many (most?) cases, the argument for a mathematical result is based on a simple idea, although writing out the details of that idea might require considerable length. With this observation, a mathematical proof may extend for several paragraphs or even several pages, but typically the idea can be summarized in just a few sentences.

Step 3: Main Ideas for Sections 4.1 and 4.2

Review the logical progression of ideas in Sections 4.1 and 4.2. For each term or result given at the part of this lab, indicate the main idea behind why the statement is true.

• If the term is a definition, it is sufficient to state this is an axiom — we assume this definition provides an acceptable meaning for the term.
• If the term is a result based on a logical argument, summarize the main idea behind the argument in 3-5 sentences.

Work to Turn In

You must work in groups of 2 or 3 for this group assignment.

• Write-ups for each step should be clearly written and placed in a logical order (e.g., answers to Step 1, then Step 2, and then Step 3).

This document is available on the World Wide Web as

http://www.walker.cs.grinnell.edu/courses/131.fa09/labs/lab-mean-value.shtml

created 29 September 2012 last revised 29 September 2012
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.