So I have been thinking about resource consumption a lot after watching the most important video you will ever see. It is pretty long so I will summarize it as follows, the professor makes a strong case for sustainability on the basis that if our fossil fuels are used at an exponential rate we will find ourselves without energy very quickly.
While the argument was a strong one, there was an assumption/narrative being made/forced in the video: that the consumption of our resources is currently exponential.
I did a couple of searches through the internet, but the evidence does not support the claim that our energy use is exponential -that is in developed nations, and on the global scale. In fact the charts seen in all my searches seemed to show a linear growth, as in this chart provided by wolfram of petroleum consumption. If I am wrong feel free to send me evidence showing otherwise, as this may just be a very local view of an exponential curve.
This led my mind to wonder about classifying increasing sequences based on their percent growth rates.
Let $f:\mathbb Z \rightarrow \mathbb R$ be an increasing sequence mapping a set of integers $[\alpha,\infty)$ to $[1,\infty)$ in the reals such that $f_0 = f(\alpha)=1, f_1=f(\alpha+1), \ldots, f_n=f(\alpha +n)$
We can then define sequential percent growth between any two consecutive elements in the sequence to be: ${f_n-f_{n-1} \over f_{n-1}} \times100 \%$
As to be expected with an increasing geometric sequence: $ar^0, ar^1, \ldots, ar^n$ where the ratio between any two consecutive terms is r. The percent growth between any two consecutive terms then turns out to be: ${ ar^n -ar^{n-1} \over ar^{n-1}} \times 100 \% = (r-1) \times 100 \%$.
The interesting stuff starts happening when we try to apply this growth definition to other increasing sequences.
So for instance, we find that for $f(x) = x$, the percent growth from one element $f_{n-1}$ to the next $f_n$ is: ${1 \over n} \times 100\%$
With a little more work we can generalize for $f(x)= mx+b$ to find that the percent growth is: ${m \over m(\alpha + n-1) + b} \times 100 \%$
I know enough about math to know that for polynomial the percent growth rates will asymptotically go to zero as n gets large.
I also know about the better curve fitting methods, and know for continuous models we can use calculus, and I have heard of and know very little of big O and little O notation...
What I am wondering is the following:
1) What is this concept I have stumbled upon called? I have done searches related to it, and found nothing. BUT it is too simple to have never been explored before. The search I have done on growth for example have pretty much classified growth into the categories of: linear growth, polynomial growth, logistic growth, exponential growth, and hyperbolic growth.
2) If it is not new, in which course is this taught? I am amazed that such a concept isn't explored more commonly as I think this being used to determine given the % growth of data what sequences may fit the data, and possibly have ways of making interesting theorems about the growths of certain sequences.
3) Am I making a mountain out of a mole hill?
So far I have tried finding the percent growth of sequences of the following other functions as well: Root functions, Logistic functions, hyperbolic functions, factorials, super-exponential functions, and even the Fibonacci sequence.
I apologize if the video is seen with any degree of politicization, that is not my intent. My intent is to be able to determine what can be said about % growth data that is presented to us whether it maybe in population increase, inflation, to examine Neo-Malthusian ideas with a fair and critical lens or whatever.