# Mean Value Theorem: Quick Intuitive Tests

John went on the entrance ramp of a highway. The overhead highway radar detected he got on the highway at 12:38PM. He drove for a bit, and the overhead radar on the exit ramp detected he got off the highway at 12:55PM. In this time, he drove a total distance of 30 miles. During his travel time, he never drove under an overhead speed detector. Yet 2 weeks later, John got a speeding ticket in the mail. The ticket claimed he was speeding in great excess over the allotted speed limit of 70 mi/hr. Even though no law enforcement officer ever saw John traveling, does the state police department have the right to issue him a ticket?

Check all that apply

Explain why you responded the way you did to the first question. Spare no detail!

Describe the Mean Value Theorem in your own words. As you do, consider the following: What criterion/criteria does it require? If all sufficient criteria do hold true, what does it allow us to conclude?

## Calculus Students:

In each applet below, the line passing through C is tangent to the graph of this function. Also note the secant segment displayed.

## Create your own: Move A, B, and the 2 unlabeled points around and challenge your classmates!

How does the Mean Value Theorem relate to John's traveling story above? Does it somehow suggest that John should be issued a ticket? If so, explain. If not, explain how it doesn't relate.