5.5 The Mean Value Theorem

In this lesson, you will learn about the Mean Value Theorem.

Lesson Objectives

Upon completion of the lesson 5.5, you will be able to:

• • Visually find the points that are guaranteed to exist by the Mean Value Theorem when given the graph of a function that satisfies the hypotheses of the Mean Value Theorem.

  • Determine whether the Mean Value Theorem applies to a given function on a given interval and if so, analytically find or approximate the point(s) that are guaranteed to exist by the Mean Value Theorem.

  • Use the Mean Value Theorem to show that two functions whose derivatives are equal differ by a constant.

  • Recognize that many graphing techniques are based on the Mean Value Theorem.

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

• 1. YouTube video: The Mean Value Theorem

• 1. YouTube videos: Mean Value Theorem

  2. Part 1:

• 1. Part 2:

1. Application of the Derivative: Lesson 31- Rolle's Rule & the Mean Value Theorem [Internet Archive]

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.

• 1. The Mean Value Theorem [Paul's Online Math Notes]

  2. (Rolle's theorem and Mean Value Theorem with examples and solutions)

  3. Proofs of Derivative Applications Facts/Formulas [Paul's Online Math Notes]

  4. (proofs dealing with the first derivative and intervals of increase or decrease and with the second derivative and intervals of concavity) - Read Fact, the Shape of a Graph, Part 1 - Proof of 1, 2 and 3. - Read Fact, the Shape of a Graph, Part II - Proof of 1 and 2. - Read the Second Derivative Test - Proof of 1 and 2.

  5. Chapter 4: 4.2- Mean Value Theorem

  6. Note: Read the PowerPoint until the end of example 6.

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

• 1. Practice Problems: The Mean Value Theorem [Paul's Online Math Notes]

  2. Note: Skip problem #4

  3. Drill - Mean Value Theorem [University of Tennessee]

  4. Note: Skip the 6th and 8th problems.

1. Calculus Applet - Mean Value Theorem [calculusapplets.com]

Additional Resources

Below are additional resources that help reinforce the content for this module.

• 1. The Mean Value Theorem, Fall 2011, Clark University

  2. (A technical lemma, zero derivatives at extreme values, extreme values and critical points, Rolle's theorem, the Mean Value Theorem, theorems on increasing and decreasing functions, functions with equal derivatives)

  3. Note: Read pages 1 - 3. Stop reading when you come to the First Derivative Test on page 3.

  4. The Mean Value Theorem [Harvey Mudd College]

  5. (Rolle's theorem, proof of Rolle's Theorem, Mean Value Theorem, Proof of the Mean Value Theorem, and key concepts)

1. The Mean Value Theorem [CalculusQuest]