The Calculus Diaries - Jennifer Ouellette [17]
2
Drive Me Crazy
We apprehend time only when we have marked motion . . . not only do we measure movement by time, but also time by movement because they define each other.
—ARISTOTLE
Abrooding shot of a long, straight desert highway, shimmering slightly in the heat and stretching far into the horizon, opens Ridley Scott’s classic 1991 film Thelma and Louise. It’s become an iconic image, foreshadowing the women’s glorious demise as they drive off a cliff in the Grand Canyon in their 1966 Thunderbird convertible, immortalized forever in celluloid history.
The portion of I-15 that runs between Los Angeles and Las Vegas doesn’t stretch quite so dramatically into infinity, but after three long hours of driving under a relentless midsummer sun, it’s starting to feel like it could go on forever, particularly since traffic has slowed to a crawl. We have road construction to thank for the delay: The state of California is adding a southbound lane just for trucks, and for some reason, this is also slowing down traffic on the northbound side. Thelma and Louise would have just floored it and blasted their way out, but we are wimpy, law-abiding citizens, and meekly accept our fate.
We’re on I-15 in our shiny red Prius because we’re hard-core Vegas fans: Taking a weekend jaunt now and then to play some poker, do some shopping, and perhaps indulge in a spot of fine dining or a spa treatment proves quite refreshing. But the road trip also provides an excellent example of calculus in action. Calculus deals with rates of change. Motion is, in essence, change in position with respect to time—however slowly that position is currently changing thanks to the impeded traffic. In fact, at this point, we’re barely moving at all, inching along at a scant 10 mph while our stomachs rumble in anticipation of savoring the world’s best gyros11 and falafel at the Mad Greek Cafe in the tiny town of Baker, California (population 600).
There’s precious little to do on a road trip, creeping along a desert highway while breathing in exhaust fumes, with nothing but dusty hills, tumbleweeds, and a long line of rear bumpers as scenery. So I figure it’s as good a time as any to muddle through the basics of derivatives and integrals; I’m already bored, hungry, and cranky. Also, it occurs to me that our predicament is reminiscent of Zeno’s paradox, outfitted in contemporary garb—exchanging Zeno’s trademark toga and sandals for acid-washed Levi’s and snazzy ostrich-skin boots, if you will.
Think about it: If our motion is divided into infinitely smaller increments of time and distance—as it would be in a calculus class—in what sense can I claim we are “moving” at all? I can solve this modern paradox by using the tools of calculus to determine our instantaneous speed—how fast we are going at any brief, fixed moment in time—even though our position in time and space is constantly changing. Assuming I know our instantaneous speed (velocity) at every possible moment, can I then use that information to determine how far we’ve traveled—our position—without cheating and looking at our trusty odometer? Calculus says I can.
ROAD TO NOWHERE
Let’s start with a bit of precalculus to demonstrate the concept of instantaneous speed, using the simplest possible example with highly idealized conditions. In my mind’s eye, I-15 magically morphs into that endless, perfectly straight road in Thelma and Louise, except rather than stretching into eternity, it runs between our home in Los Angeles and the Luxor Hotel in Las Vegas, with an infinite number of points in between.
Imagine that a squad car pulls up as we drive into the Luxor entrance. The officers claim Sean ran a red light a few miles away. Sean denies it. As proof, they show us