Lessons in Freeway Calculus

By Grace Peng

As an immigrant, I hate to see my chosen team (America) dissed by know-nothings.

The views of the governor of Pennsylvania, after an Eagles-Vikings playoff game in Philadelphia was postponed because of snow:

"We've become a nation of wusses. The Chinese are kicking our butt in everything," he added. "If this was in China do you think the Chinese would have called off the game? People would have been marching down to the stadium, they would have walked and they would have been doing calculus on the way down."

Gov. Ed Rendell clearly doesn't understand calculus. If he did, then he'd know that calculus was basically invented to describe motion. (He should have read Steven Strogatz's Change We Can Believe In.) Therefore, anyone in motion -- whether walking to the stadium or driving in a SUV with heated leather seats -- is doing calculus.

(Thanks to my favorite math website, Wolfram MathWorld, for the figures and theorem info.)

On a more serious note, I want the general public to understand that math is a huge subject encompassing much more than arithmetic. People shouldn't give up their math education too soon just because they don't like arithmetic. Many mathematicians are only so-so at arithmetic.

Truth be told, after I finished the introductory lower-division undergraduate math courses at UC Berkeley, I hardly ever worked with actual numbers. My math homework consisted mainly of Greek symbols and other abstract notation along with copious arguments in English describing the steps of the proofs. It usually ended with relief and a big Q.E.D. as I finally signed off and could go to sleep.

Math is also so broad that mathematicians do not understand all areas of math. It would be like expecting a historian to know about the history of everything. It can't be done, though it might be fun to try.

It would be more productive to think of math as both a liberal art and a science. Not only is math the language of science, giving us a framework for describing our physical world, but it is also a construct of the human mind. As such, lessons learned from math can help us understand the human condition, moving it into the realm of humanities.

A friend who had been an English literature major and I discussed how we thought a book was mainly about one thing, and then reread it years later and thought it was mainly about something else. Was our past judgment so wrong? Or had experience and circumstance changed our perceptions?

Much as I enjoyed Steven Strogatz on the Elements of Math, I hope that readers will go beyond that. I will use the meaning of calculus as an example

When I first encountered calculus in high school, I mechanically went through the motions of "turning the crank" to learn the rules of differential and integral calculus. I thought that was what it was all about. Sure, there were pages and pages of material about limits in the textbooks, but they were just a prelude to the real stuff of finding the derivative (slope) or the integral (the area under a plot) of a function. So, if you had asked me back then, I would have agreed with Strogatz.

But then a boy that was a year ahead of me in math at Cal told me that he didn't understand calculus until he took math analysis and "proved" calculus. That freaked me out; was I so clueless and shallow that I misunderstood the whole point of calculus?

The panic intensified when I took the class he referred to: Math 104, aka "Real Analysis and Introductory Topology." Our class spent the entire semester on proofs, including six weeks to prove the compactness theorem. What did that have to do with calculus?

Looking back, I can laugh about it. I see now that calculus is such a broad topic, and touches so many aspects of our lives, that it can mean different things in different contexts.

All those weeks spent proving the limit of an infinite series exists and is unique? That tells us that there is a solution of the integral described by the series.

The other weeks spent proving that some limits are reached more quickly than others? The mystery was revealed when I took numerical analysis (solving math problems with computer algorithms). Those fast-approaching limits will converge before lunch. Slower ones may take overnight. The really slowly-converging limits might converge if you made a lucky guess at the initial condition, but will more likely go shooting off into infinity (actually the dreaded floating point overflow error). It's really embarrassing when that happens.

My last big calculus insight occurred while driving across the San Mateo-Hayward Bridge near San Francisco with my daughter. A friend had warned me not to speed on the bridge. She said that cameras photograph the license plates of cars as they enter and exit the bridge; you need not be seen by a cop to receive a speeding ticket by mail.

How can they prove someone was speeding from two photographs?

With calculus, using the basic form of the mean-value theorem!

Let f(x) be differentiable on the open interval (a,b) and continuous on the closed interval [a,b]. Then there is at least one point c in (a,b) such that


This is the formula for a derivative on an interval. If a and b are the times your car was photographed (at the entrance and exit of the bridge), and f(a) and f(b) are the locations where your car was photographed, then your average speed is f'(c). The bridge, or rather the distance between the two camera locations) is a fixed length. If not enough time elapses between the two timestamped photos of your car, then your average speed exceeded the speed limit.

You can compute the average speed with simple algebra; you don't actually need calculus for that. A motorist can try to argue that s/he was not actually observed driving above the speed limit. But, assuming the time clocks on both cameras were well synced, the mean value theorem says that, even without an observation at the critical time, the motorist must have
exceeded the speed limit somewhere along the bridge. The basis of the state's case rests upon a foundation of calculus.

As my daughter and I discussed how this system of cameras might work (and why she should have told me she needed to make a pit stop before we got on the bridge), I had an epiphany.

There is a whole family of mean-value theorems. One in particular leapt to mind, the intermediate value theorem.

If f is continuous on a closed interval [a,b], and c is any number between f(a) and f(b) inclusive, then there is at least one number x in the closed interval such that f(x)=c.

This has major implications in integral calculus, but that is not why I am mentioning it.

I explained here how I get performance reviews for both my market work (from my boss) and my family work (from my daughter, who claims to be my boss). When I am an ideal worker, putting in long hours at my market (paid) work, I am not at home doing my family work. The opposite is also true.

The term "work-life balance" implies that it is possible to maintain a steady-state ideal balance. In real life, one always gets more than the other, though which one gets more may vary over time. As I vacillate between which demands more of my attention right this instant, I used to feel like I was always failing. But, now, I take solace in calculus.

The intermediate value theorem tells me that -- somewhere between those two positions -- I pass through the state of my ideal self.

Grace Peng has a day job at the intersection of science, technology and governance/policy. She also does a split shift at home as a mother and wife to a field scientist. She blogs about science, the culture of making, and work/life issues at Bad Mom, Good Mom.