Posted by: Dave Richeson | October 22, 2008

E-Z Pass, speeding tickets, and the mean value theorem

On Monday I gave a lecture on the mean value theorem in my Calculus I class.  The mean value theorem says that if $f$ is a differentiable function and $a, then there exists a value $c\in (a,b)$ such that $\displaystyle f'(c)=\frac{f(b)-f(a)}{b-a}$.

That is, the average rate of change of the function $f$ over $~[a,b]$ must be achieved (as an instantaneous rate of change) at some point $c$ between $a$ and $b$.

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 $p(t)$ be the position function for the car (measured by the milage markers) and that the car passed through the Carlisle toll booth at $t=0$ hrs.  Then $p(0)=226$ and $p(1.25)=326$. Assuming $p$ is differentiable, the mean value theorem says that there is a $c\in(0,1.25)$ such that

$\displaystyle v(c)=p'(c)=\frac{p(1.25)-p(0)}{1.25-0}=\frac{326-226}{1.25}=80$ mph.

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).

Responses

1. In New York State at least, one may exceed the speed limit for 1/4 mile while passing. Let’s assume the law is the same in PA. I can imagine a scenario in which cars are spaced along the PA Turnpike in such a way and at such speeds that you could pass each one speeding for 1/4 mile, then slow down, then speed again for 1/4 mile, and so on. The average speed would be speeding, but you would at no time be breaking the law.

At least not the speeding law. If there were yellow speed limit signs, say at sections under construction, exceeding those speed limits but staying under the stated limits on white signs might earn you a ticket for reckless driving while exeeding a posted “recommended” speed limit. One might even argue that behaving as I described in paragraph 1 is de facto reckless.

So an automatically issued summons might have to say, “for either speeding or reckless driving, but the MVT won’t tell us which.”

I use the same example in my Calculus I class, and point out that the MVT only guarantees a point of speeding, not an interval of speeding.

One of my students was surprised to learn that there are functions which are 5 times differentiable but not 6 times differentiable, when I was helping her at my desk. Calculus I students see very few functions that are not elementary functions, hence that are not C^\inf.

BTW, as a word lover, I notice that w does strange things to words. Reckless driving is WITHOUT Wrecks. Whole surfaces are without Holes. Who would have guessed that W was a negative prefix?

2. $C^\inf$ — still trying to learn your blog’s LaTeX.

3. OK, \$ isn’t it and angle-bracket math close-angle-bracket isn’t it. Are your commenters allowed to post LaTeX? How?

4. I didn’t know that about legal “short interval speeding”. Is that also true on four-lane highways or just on two-lane roads in which you have to pass by driving in an oncoming lane?

5. Gene, here’s a link to the WordPress Latex guide.

6. One part of your explanation leaves me stumped; how did you calculate p(0)and what does 1.25 represent and how did you calculate p(1.25)?????