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Calculus Cartoon

December 15, 2009 1 comment

Ted Rall’s latest cartoon (above) is the rare political cartoon with an embedded calculus lesson. We might well wonder, as Rall does, why we are so excited by the news that unemployment continues to increase, only at a slower rate.

Rates of change are exactly what calculus allows us to discuss precisely. Rather than being some forbidden subject, calculus is simply the language to quantify and discuss such matters. But even without studying (or remembering) calculus, we all understand the basic issues, at least qualitatively.

Given a quantity we wish to measure, like the number of people unemployed, or how far we have traveled from home, the derivative tells us how quickly this quantity is changing (increase or decrease in unemployment figures per month; increase or decrease in distance from home per hour, otherwise known as velocity). And the second derivative, which is what has been in the news lately with regard to unemployment figures, measures how quickly that first rate of change is changing. This is indeed a subtle notion, but one we talk about all the time. In the unemployment example, it is the rate at which the change in unemployment is going up or down. Thus, unemployment may still be increasing this month, but perhaps it is increasing more slowly than it has increased in recent months. That slower rate of increase is measured by the second derivative, and its slowing means the second derivative is negative. In the example of leaving home, we might be driving away, but the velocity at which we are driving is decreasing — perhaps we braked but haven’t yet come to a stop. This means the second derivative of the distance from home is negative. And we have a familiar name for that second derivative. Acceleration. So acceleration is negative when the velocity is going down, even though the velocity may still be large and we may still be moving rapidly away from home. Just less rapidly.

So that’s that. Simple enough ideas, but fundamental.

By the way, let’s say I desperately wanted to get home for dinner, but I had taken a wrong turn and found myself entering the freeway in the wrong direction, taking me farther away from home. I had no choice but to drive to the next exit, another five miles away. Alas, a couple of miles down the road, traffic slowed because of a car stuck on the side of the road. I would have to brake gently and begin to drive 20 mph rather than 55 mph. Thus, the rate at which my distance from home is increasing would have begun to decrease. I would continue to get farther from home, that’s for sure, but at a slower speed. Is this good news?

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Categories: Cartoons, Economy, Math