The one thing about exponentials that many of us would remember from school calculus is that they have the special property of remaining unaltered on differentiation: the derivative of e^x is also e^x. But we may not have fully absorbed the significance of this property for grasping the notion of exponential growth. I certainly hadn't, until very recently.
So let's look at it this way. Consider first a quantity n which grows linearly with time. This means that the growth function is n(t) = n0 + at, where n0 is the value of n at t=0, and 'a' is the growth per unit time or the slope of the growth line. The first time derivative of this function is n'(t) = a. And the second time derivative is n''(t) = 0. So, using the analogy with physical motion, one can think of linear growth as having a constant speed (a) and zero acceleration. Sounds pretty unremarkable, right?
Now let's think of exponential growth in the same way. Here the growth function is n(t) = n0.e^(at); 'a' in this case becomes proportional to the inverse of the doubling time (a phrase we have heard quite frequently of late). Consider the time derivatives again: from the above-mentioned special property, we can say straight away that the first time derivative will also be an exponential, and hence the second time derivative will also be an exponential, and so on ad infinitum. To write them down exactly: n'(t) = (n0.a).e^(at) and n''(t) = (n0.a^2).e^(at). So both of them are exponentials with exactly the same doubling time as for n(t); just the initial values have changed. In terms of our physical analogy, exponential growth corresponds to the speed of growth itself growing exponentially (doubling) at the same rate, which in turn implies the acceleration also doubling at the same rate, which in turn implies; put this way, it's rather remarkable (and scary), is it not?
The concept of exponential growth can be somewhat hard for us to develop an intuition for, and this might be one of the reasons why many people and governments did not initially take the pandemic very seriously. It was not so easy to fully absorb how quickly things would get much, much worse. Perhaps thinking and talking in terms of physical analogies like the above can help us address this intuitive gap to some extent.
So let's look at it this way. Consider first a quantity n which grows linearly with time. This means that the growth function is n(t) = n0 + at, where n0 is the value of n at t=0, and 'a' is the growth per unit time or the slope of the growth line. The first time derivative of this function is n'(t) = a. And the second time derivative is n''(t) = 0. So, using the analogy with physical motion, one can think of linear growth as having a constant speed (a) and zero acceleration. Sounds pretty unremarkable, right?
Now let's think of exponential growth in the same way. Here the growth function is n(t) = n0.e^(at); 'a' in this case becomes proportional to the inverse of the doubling time (a phrase we have heard quite frequently of late). Consider the time derivatives again: from the above-mentioned special property, we can say straight away that the first time derivative will also be an exponential, and hence the second time derivative will also be an exponential, and so on ad infinitum. To write them down exactly: n'(t) = (n0.a).e^(at) and n''(t) = (n0.a^2).e^(at). So both of them are exponentials with exactly the same doubling time as for n(t); just the initial values have changed. In terms of our physical analogy, exponential growth corresponds to the speed of growth itself growing exponentially (doubling) at the same rate, which in turn implies the acceleration also doubling at the same rate, which in turn implies; put this way, it's rather remarkable (and scary), is it not?
The concept of exponential growth can be somewhat hard for us to develop an intuition for, and this might be one of the reasons why many people and governments did not initially take the pandemic very seriously. It was not so easy to fully absorb how quickly things would get much, much worse. Perhaps thinking and talking in terms of physical analogies like the above can help us address this intuitive gap to some extent.
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