[ I cannot and do not try to explain it, but yesterday was a busier-than-average day around these parts, with a surprising number of references coming from an Entertainment weekly article about the House series finale for some reason. In this context a “surprising” number is “any number other than zero” since I don’t know why anyone would go from there to here. I watched House, sometimes, sure, and liked it, but kind of drifted away when there was other stuff to do, you know? ]
That’s enough time spent establishing the heck out of the idea of a polynomial. Let’s actually put one in place. My goal back when was estimating what the population of Charlotte, North Carolina, was around 1975. I had some old Census data from 1970 and 1980 giving its population on the first of April, the earlier year, as 840,347; and the first of April, 1980, as 971,391.
I’m going to let the variable y be the number of people in Charlotte at a particular time. I’m going to let the variable x the number of years since the first of April, 1950, because I think I might be able to find census data going back that far easily enough. What I want is to create a polynomial of the form 
Except we supposed this equation was going to describe the population of Charlotte, and we want it to exactly fit the population of Charlotte at two times. In particular when x is 20 (the year 1970), the population y should be 840,347. When x is 30 (the year 1980), the population y should be 971,391. We can plug in those particular real numbers for x and y in this equation, and we get a pair of equations, both of which we want to be true:
There are two numbers in this pair of equations we don’t know: a0 and a1. But we have two different equations describing them, and we want those equations to both be true for the same pair of a0 and a1. This means we’re able to solve for some unique pair of those numbers. For a change I’ll spare you the process of finding those numbers; trust me please that a0 is 578259, while a1 is 13104.4. This makes my interpolation of Charlotte’s population
And so for 1975 — specifically, for the first of April, 1975, when x is 25 — then 
According to Wikipedia, Charlotte covers an area of about 297.7 square miles — let’s call that 300 square miles, since there’s so much approximation being done here anyway. I don’t know offhand whether Charlotte’s borders have grown or shrunk since 1975, so I’ll deal with that problem by ignoring it altogether. If we suppose Charlotte covered the same 300 square miles in 1975, then, when I lived there, it had a population density of about 3,020 people per square mile. On my list of population densities for places I’ve lived this puts it just at the median, actually, with as many places less dense as more. (It comes in just barely on the upper half when we include places I’ve merely worked.) Curiously, it comes in higher than most of the places in New Jersey where I’ve worked, as well.
| Municipality | Population per Square Mile |
|---|---|
| Charlotte, NC (1975) | 3,020 |
| South Amboy, NJ | 5,100 |
| Old Bridge, NJ | 1,700 |
| Marlboro, NJ | 1,300 |
| New Brunswick, NJ | 9,500 |
| Piscataway, NJ | 2,700 |
| Troy, NY | 4,500 |
| Jackson, NJ | 550 |
| Singapore | 18,900 |
| Municipality | Population per Square Mile |
| Manalapan, NJ | 1,300 |
| Sayreville, NJ | 2,300 |
| Albany, NY | 5,500 |
| Trenton, NJ | 11,100 |
| Toms River, NJ | 1,400 |
nebusresearch 