 # Rolle’s Theorem and the Mean Value Theorem

##### This Rolle’s Theorem and the Mean Value Theorem presentation also includes:

Using the derivative to apply the Mean Value Theorem and its more specific cousin, Rolle's Theorem, is valuable practice in determining differentiability and continuity on an interval. This presentation and accompanying worksheet walk the class through steps for applying both theorems and then provide examples of all types of functions for practice. Homework problems reinforce the theorems' applications, again with an emphasis on differentiable/continuous and nondiffentiable/discontinuous functions. A great stand-alone or add-in activity and presentation to an existing calculus curriculum.

##### Instructional Ideas

• Consider having learners graph the functions and slope between the interval endpoints, then estimate the interior point that satisfies the given theorem
• Link with a science unit on velocity and gravity/projectile motion. Using velocity as a derivative of distance with respect to time, Rolle's Theorem and the Mean Value Theorem apply with context
• Divide the class into pairs, and have one partner apply Rolle's Theorem and one apply the Mean Value Theorem to given functions and intervals. Each pair should compare notes and explain the difference in results
##### Classroom Considerations

• Notesheets, homework, and answer key to match main presentation is available under Included Materials
• Notesheets divide the steps for applying the two theorems into separate applications, while presentation combines the information
• PDF of presentation would make an effective interactive whiteboard presentation with parts of examples revealing as you work the problems
##### Pros

• Clear steps for applying each theorem provided in each section of notes and worksheets
• Fully worked example problems and homework answer key provided
• Worked examples of all kinds of problems where both theorems, just the Mean Value Theorem, and neither theorem apply
##### Cons

• Presentation does not differentiate between Rolle's Theorem and Mean Value Theorem clearly in examples, though the notes and worksheet do