# 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?

Explain why (in a way that someone who has not taken calculus would understand).

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?

**line passing through C is tangent**to the graph of this function. Also note the secant segment displayed.

## Can you determine a location for point C on the graph of this continuous function that is guaranteed by the Mean Value Theorem?

## How about here?

## 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.