Change, the Change of Change, and the Change Thereof

In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.

—Hugo Rossi

Change we can believe in.

—2008 campaign slogan for Barack Obama

Change we can find in everything—especially a soda machine. Change comes in many denominations, and some people carry more of it in their pockets; others refuse it. Change is an ancient necessity, and absolutely a modern one. Change is the soul of existence.

Sometimes, even the lack of change is in itself a change. For example, every sentence so far has contained the word “change.” If the next sentence does not contain the sacred word, would it thereby be a change? Indeed.

And if this post is on change, then why have I included an epigram about the third derivative—a scary calculus term—in a post about change? Answer: Calculus is the study of change.

First Derivative: Change

If you heard from a stock analyst, “Google went up $24 yesterday,” then you have heard a statement about the first derivative. If—and it is a very likely possibility—you have not, then don’t worry; consider instead the subtly edited statement: “Google went up $27 yesterday.” You clearly see a change between the two sentences. This is also a first derivative.

But in any case—and this is an even more likely possibility—you should ignore the two examples above and consider the following car example, which is probably quite unbelievable, for in such examples, cars always drive on perfectly straight roads with speeds and accelerations that match formulas exactly.

Suppose a car is moving on a perfectly straight road from New York City to London, with a velocity of exactly 100 km per hour. Furthermore, suppose that someone were to ask you for the first derivative. If you answered, “100 km per hour,” congratulations. The first derivative is the change of something. In this case, the position of the car changes. By how much? The first derivative.

But this car example is a bit unbelievable, so I would like to present what I believe is a more reasonable situation, which in this case involves two groups of angry monkeys on an alien planet. The first group of angry monkeys we’ll call the Lazies, and the second group the Angries. Furthermore, suppose that the two groups are incompatible, and that for some reason, both groups are evolving more limbs over time.

At year 0, both the Lazies and the Angries have two arms per member. Due to evolution, the Lazies gain an extra arm every 100 years, and the Angries gain two extra arms every 100 years. When the difference in arm count reaches three, the group whose monkeys have more arms can defeat the other group. Suppose no human politician intervenes. Who will win, and when?

The Angries will win. They’re gaining limbs twice as fast as the Lazies: every 100 years, the Angries gain two limbs while the Lazies only gain one. After the first 100 years, the Lazies have 3 arms while the Angries have 4. After 100 more years, the score will be 4 to 6. And after another 100 years, it will be 5 to 8, and the Angries win. This occurs at year 300.

Second Derivative: The Change of Change

The second derivative is important when not only the original thing is changing, but the change itself is changing. For example, if Google’s stock rose $24 the first day, $27 the second day, $30 the next day, the second derivative is $3 per day per day. That’s not a typo: the first derivative is “per day,” the second is “per day per day.” The amount the stock changes per day is changing—per day.

For the car, acceleration is the second derivative. Let’s say the car is accelerating at 20 km per hour per hour. That is, after each hour, the car is moving 20 km per hour faster than it was before. In the first hour the car is moving at 100 km per hour, in the second it is at 120 km per hour, in the third, 140 km per hour, etc. Actually these are only approximations: in the first hour, the car is actually moving somewhere between 100 and 120, because it starts at 100 and accelerates to 120 in the span of an hour.

What if every time a monkey group gained an arm, it would gain arms faster? The actual term is rather awkward to say: “1 arm per century per century.” For example, the Lazies start out with 2, but then increase by 1 for a total of 3. The next time, they increase by 2 for a total of 5, then increase by 3 for a total of 8, and so on. The Angries also start with 2, and at first increase by 2 for a total of 4. They then increase by 3 for a total of 7, then increase by 4 for a total of 11. At this point, the Lazies only have 8, so the Angries win again at year 300.

Now what if the Angries evolved as above but the Lazies reverted back to the rule in the first section? That is, the Lazies will gain 1 limb in the first century, 1 limb in the second, and 1 limb in the third. The Angries will gain 2 limbs in the first century, 3 limbs in the second, and 4 limbs in the third. By the second century, the Angries have already won: the Angries have 2+2+3 = 7 limbs while the Lazies only have 2+1+1 = 4. The Angries win at year 200.

The Red Queen’s quote from Lewis Carroll’s Through the Looking Glass is a classic demonstration:

“Now, HERE, you see, it takes all the running YOU can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!”

If two things are moving at the same rate, it is as if neither one is moving. To go twice as fast, one must accelerate, i.e., use the second derivative. Of course I’m taking the quotation out of context.

Third Derivative: The Change of Change of Change

The third derivative is one step up from the second. It is the change of the second derivative, which is in turn the change of the first.

This level is out of the human comfort zone. It is difficult to explain the third derivative with the stock market or evolutionary examples, so we’ll go to the car once again. We examine the pressing of the gas pedal.

Suppose a fully pressed gas pedal causes the car to accelerate at 20 km per hour per hour. This is sluggishly unrealistic, but it’s for example. To go from zero acceleration to the max, the driver can step on the pedal gradually or suddenly. Now the acceleration itself is the second derivative, so the change of acceleration—the pressing of the gas pedal—is the third derivative. Pressed slowly, it exemplifies a low third derivative, and quickly, a high third derivative. The latter case doesn’t feel good.

Numerically, if the pedal is completely pushed down in 1 second, we have the third derivative as 20 km per hour per hour per second. If it’s done in 5 seconds, the third derivative is just 4 km per hour per hour per second. If “the third derivative” sounds tedious, there is a scientific name for this: jerk. I’m not kidding. Jerk is the change in acceleration, which is in turn the change in velocity, in turn the change in position.

Let us now return to Rossi’s quote: “In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.”

Inflation is the first derivative, i.e. the change in prices. The “rate of increase of inflation” is the second derivative. Then to say that the “rate of increase of inflation was decreasing” is to say that the third derivative was negative, as it caused the second derivative to decrease.

Concluding Remarks

Next time somebody says, “Spare some change,” be more willing to share some first derivatives. After all, to restock on the first derivative, try a soda machine. You never know what you’ll find in them.

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