Definition: Let f(x) be a differentiable function on [a,b]. Then there is at least one point c between a and b such that
Again, it is very important that a function is differentiable; if the function is not differentiable, then the mean value theorem does not make sense.
Example: A certain stretch of highway is 60 miles long, and the speed limit is 60 mph. There are camera at the beginning and at the end of this stretch of highway, and they takes pictures of cars as they pass each camera, and record the time it takes each car to traverse the entire stretch of highway. Any car that completes the stretch of highway in under 48 minutes is automatically ordered a speeding ticket. Bob recieves such a ticket and complains that the ticket is unfair, since they cannot prove that he was ever in excess of 60 mph. Is Bob’s ticket fair, or is Bob right?
We can consider Bob’s location as a function of time. The rate of change of Bob’s location, his speed, is the derivative of this function. If Bob completes the stretch of highway in 48 minutes, .8 hours then the slope of his speed is 60 miles/0.80 hours = 75 mph (miles per hour). According to the Mean Value Theorem, Bob must have been going at least 75 mph at least once over the course of his journey, so Bob’s ticket is certainly fair!
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