The Calculus Diaries - Jennifer Ouellette [18]
The officers don’t have a radar gun, which measures velocity directly, but unfortunately for Sean, they are well versed in math. They do have a time-stamped photograph of the Prius at a similar intersection one minute before. So it’s a simple matter for the officers to show where we were at the traffic light—the two-minute mark—and subtract our position at the previous intersection (the one-minute mark) to determine how far the Prius traveled in that time: in this case, one mile. Then they can divide that by the time it took to travel that one mile, and this gives them the car’s average speed: one mile per minute, or 60 mph.
Ah, but Sean doesn’t give up so easily; he has one more argument to make. The officers are assuming the Prius was moving at a constant speed. Yet every experienced driver knows that one’s speed is rarely constant. Just because our average speed was 1 mile per minute doesn’t necessarily mean that was our instantaneous speed at the moment we crossed the intersection.
The officers remain undaunted. They don’t have access to the information recorded by our trusty speedometer and odometer, so they have supplemented this imaginary stretch of road with some pretty cutting-edge technology, dividing it into intervals at every possible distance and placing tiny nanosize traffic cameras at each and every interval. Call it willing suspension of disbelief, although at the rate nanotechnology research is currently progressing, a scenario quite close to this may one day become a reality.
Thanks to our imaginary nanocameras, the officers have an infinite number of time-stamped still shots of our humble Prius, taken at infinitesimally small intervals along this extremely high-tech futuristic road. This is incontrovertible ocular proof 12 of the car’s position at every given point in time since we left home: In calculus terminology, this is our position function. We know the position of the Prius as a function of time. The cameras reveal that there was less time between equal intervals as the Prius approached the light—which means we were actually accelerating.
The basic concept is the same whether we’re talking about driving down the imaginary highway at a constant rate or about a more complicated real-world scenario in which our speed is constantly changing. Even though the Prius is accelerating, it still has one specific speed at each instant, and I can use the same highly repetitive process of accumulating evidence to prove it, showing where the car was at all times. I run the same calculation outlined above over and over, for ever smaller intervals, to show how fast the car was going at any given moment in time.
This time, there is a crucial difference: Instead of getting the same answer each time—as in the constant-speed scenario—I get slightly different answers each time. But as the intervals get shorter and shorter, those answers get closer to a point of convergence: 2 miles per minute. The answer is never exactly 2. But the answers are clearly converging toward a single answer, to a very close approximation. The limit rears its ugly head. I can safely conclude that the car’s instantaneous speed at the moment in question must be 2 miles per minute.
Ingenious, isn’t it? Hats off to Newton, Leibniz, and untold mathematicians before and after them who repeated the same exact process of calculation, over and over again, until they’d compiled sufficient proof that the derivative formula