Overview
Many statements about the shape of a graph and its local and global extrema depend on several important theoretical results, which also enable us to predict the number of solutions to an equation, as well as proving certain identities. These results are known as: The Intermediate Value Theorem, Rolle’s Theorem, and The Mean Value Theorem.
These theorems are considered part of the subject of Mathematical Analysis, a modern area of research that evolved from Calculus and now forms the rigorous theoretical foundation of Calculus.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
 State the Intermediate Value Theorem.
 Use the Intermediate Value Theorem to verify that a continuous function has a root within an interval.
 State Rolle’s Theorem.
 Use Rolle’s Theorem to verify that a differentiable function has an extremum on an interval.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
 Use the Intermediate Value Theorem to show that an equation has at least a solution.
 Use Rolle’s Theorem to find the maximum number of solutions that an equation can have.
 State the Mean Value Theorem.
 Use the Mean Value Theorem to show that if \(f'(x) = 0\) for all \(x \in (a,b)\) then \(f\) is a constant function on \((a,b)\).
 Show that if \(f'(x) = g'(x)\) for all \(x \in (a,b)\), then \(f\) and \(g\) only differ by a constant on that interval, that is \(f(x) = g(x) + k\).
To prepare for class

Read these notes (in PDFformat) about the Intermediate Value Theorem.

Watch the following video, making yourself a list of at least 3 things you learned from the videos or your reading  and include this as part of your feedback:

Read these notes (in PDFformat) about Rolle’s Theorem and the Mean Value Theorem.

Watch the following video about Rolle’s Theorem and the Mean Value Theorem (video created at CCSL):

Do the Preview Activity on WeBWorK (if required by your teacher).
After class

Do some experimentation with the following interactive applets to get a better intuitive understanding of these theorems:

Intermediate Value Theorem: https://www.geogebra.org/m/j2qPFmmx (by Philip Walker)

Mean Value Theorem:
 https://www.geogebra.org/m/EE6nAgGC (loghind)
 https://www.geogebra.org/m/aajVcC9u (MrQuinn)
 https://www.geogebra.org/m/QHJJQX99 (MrQuinn)
