|Rachel Dolezal, now-and-then|
|Figure 1. Function over time|
|Figure 2. Constant function|
If the line is flat (horizontal), the quantity that the line is describing (the function) is not changing over time. As you move right or left on the line (forwards or backwards in time), the line does not go up or down. The quantity is neither increasing or decreasing in time. In math we refer to this as a “constant” function. This is shown in Figure 2.
If the quantity increases as we go forward in time, the line goes up. A line describing PC-ness would necessarily go up (to the right) because it is always increasing (over time).
|Figure 3. Linear function|
|Figure 4. Exponential function|
|Figure 5. Logistic function|
When CDs were first produced the technology was new. The cost to switch over to the new music medium was expensive. Both the CDs and the machines you needed to play them on cost more than the current music media, records and cassettes. As more and more people gradually discarded their old relics of the previous technology, dubbing-cassette boomboxes and record players and the like, the sales of CD players and CDs went up. As sales went up, the costs went down. As the cost went down, more people switched to the new media. Rinse and repeat. What you have here my friend is an exponential-like growth pattern.
As this goes to press, the President has used the latest Islamist beheadings as an opportunity to once again lecture the nation on the dangers of “Islamophobia;” the White House has been bathed in rainbow colors celebrating the Supreme Court’s decision legalizing homosexual marriage; and the latest list of college “microaggressions” is making the rounds on Facebook. I know it’s a tired cliché, but you can’t make this stuff up.
|Figure 6. Comparison of Exponential and Logistic function|
Perhaps two years from now our hypothetical measuring system will show the increase in PC units from 2016 to 2017 is less than it was from 2015 to 2016. If no such evidence is forthcoming, then unfortunately we are still in an uninhibited growth model (Figure 4) with no end in sight—or we are still on that bottom half of the “S” curve. A side-by-side comparison of exponential and logistic growth is shown in Figure 6.
Driving: Distance, Speed and Acceleration
Another example of functions and derivatives
An everyday experience that is often used to explain the concept of the derivative, a fundamental concept in calculus, is driving.
You go for a drive. You drive a certain distance. The (main) function corresponds to your position. How many miles had you driven at some particular point in time on the trip? It will be a line/curve going up. As the line moves to the right we are going forward in time, and we are moving, so our overall distance traveled is always increasing.
The 1st derivative of this position function gives the rate of change of the distance traveled over time, the rate of our motion: our speed. How many miles per hour were you driving at that particular time?
The 2nd derivative of the position function is the derivative of the 1st derivative; the rate of change of the 1st derivative. The rate of change of your speed. That’s acceleration. Acceleration is the 2nd derivative of the position function. When you find the 2nd derivative you can plug in a number for the time variable and it will tell you what your acceleration was at that instant in time, how many miles per hour per hour, or miles per hour2. This will be a positive number if you are speeding up or a negative number if you are braking or slowing down.
# # #
And now, finally, the magic is revealed. How do we find out where the inflection, or tipping point is, i.e., when it occurs? How do we find the date when the rate of increase begins to decrease?
Just before the inflection point the rate of change of PC-ness has its maximum value! Just beyond the inflection point the change in PC-ness has a slightly lower value. What has happened in between is that the rate of change of the increase in PC-ness has gone over from a tiny positive number to a tiny negative number. It has crossed the threshold of zero change in the rate of increase in PC-ness.
Henceforth: the Dolezal Point.
|Figure 7. Cubic Polynomial function|