On Monday I gave a lecture on the mean value theorem in my Calculus I class. The mean value theorem says that if is a differentiable function and , then there exists a value such that .
That is, the average rate of change of the function over must be achieved (as an instantaneous rate of change) at some point between and .
As an example, I gave them a hypothetical means of using E-Z Passes to issue speeding tickets. (An E-Z Pass is an electronic device in your car that reads when you pass through highway toll booths. It is tied to your credit card and is used to automatically collect tolls.)
Suppose a car that is equipped with an E-Z Pass drives from the toll plaza in Carlisle, PA (milage marker 226 on the PA Turnpike) to the one in Valley Forge (marker 326) in 1 hour and 15 minutes. A few days later the driver receives a speeding ticket in the mail. How did the PA state troopers know that the driver was speeding?
If we let be the position function for the car (measured by the milage markers) and that the car passed through the Carlisle toll booth at hrs. Then and . Assuming is differentiable, the mean value theorem says that there is a such that
In other words, at some time during the drive, the car was traveling 80 mph (the speed limit of the PA Turnpike is 65 mph).
I ended the discussion by commenting that it would be refreshing to hear a state trooper cite the mean value theorem in a court room if the ticket was challenged!
After class, a student told me that a friend of his got just such a speeding ticket. I was skeptical, so I searched on the internet to see what I could find on the topic. Sure enough, the urban legends website Snopes.com had a posting on this very topic.
We have yet to find any verified accounts of municipalities (in any state) automatically issuing traffic citations based on transit times recorded by electronic toll collection systems. Although many people maintain they have received such citations (or know someone who has), those claims have so far always proved to be misunderstandings: Motorists who travel too fast as they pass by E-Z Pass toll collection points may receive letters warning them to slow down while they use E-Z Pass lanes or else risk cancellation of their E-Z Pass accounts, but those letters are not law enforcement citations, nor are they based on speeds calculated by recording times of passage between two checkpoints.
However, law enforcement officials have used the mean value theorem to nab speeders. In 2005 Scotland began a system that used cameras to compute the average velocity of cars at locations along a 28-mile stretch of A77. They used this information to issue speeding tickets. The cars were identified by their license plates (the cameras were equipped with optical character recognition software).