Salinan 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. [br][br]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. [br][br]Even though no law enforcement officer ever saw John traveling, does the state police department have the right to issue him a ticket?[br][br]
Explain why you responded the way you did to the first question. Spare no detail! [br][br]
Describe the [b]Mean Value Theorem [/b]in your own words. As you do, consider the following:[br][br]What criterion/criteria does it require? [br]If all sufficient criteria do hold true, what does it allow us to conclude?
Calculus Students:
In each applet below, the [b][color=#9900ff]line[/color] [color=#9900ff]passing through C is tangent[/color][/b] to the graph of this function. [br]Also note the secant segment displayed. [br]
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.
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