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I am writing up a small text about (increasing) exponential functions versus linear functions. I want to make the point that it is not true that if an exponential is bigger than a linear function that it will not always stay that way.

To make my text more readable, I want to label the images with some relatable things that grow exponentially versus linearly. Now, I have two problems: First, I have a really, really hard time to think about things not covid-related. Secondly, I want to have an example where we want the exponential growth to exceed the linear growth.

I have thought about repopulation of depleted environments, but the examples feel forced. Interest on a bank account is certainly "good" exponential growth, but I do not see a natural linear growth for comparison.

So, do you have very common recognizable examples of exponential versus linear growth where we are happy that the exponential growth exceeds the linear one?

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    $\begingroup$ The money under your bed grows linearly with respect to time if you put a fixed amount there every month. $\endgroup$ – Steven Gubkin May 6 at 10:41
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    $\begingroup$ it is not true that if an exponential is bigger than a linear function that it will not always stay that way --- This is rather muddled and I don't know what you intended. I think you want to say that although we all know that exponential growth is eventually above linear growth, it is still possible for exponential to be above linear at one point and then later have linear above exponential (but of course not stay above exponential). This is easy to see since exponential can start out nearly horizontal and then get steep, so use something like $y = e^x$ and $y = 100x$ (for $x \geq 0).$ $\endgroup$ – Dave L Renfro May 6 at 11:58
  • $\begingroup$ Maybe this old handout of mine on exponential growth that I managed to find on the internet will be of use, especially Section VI Some Consequences of Continued Exponential Growth (3rd page of the .pdf document). In particular, the money example involving Karen and Megan is a fairly extreme example of how strongly exponential growth dominates linear growth in the long run. $\endgroup$ – Dave L Renfro May 6 at 12:04
  • $\begingroup$ @StevenGubkin Money under your bed is not a good example for the interaction of exponential and linear because the money in the bank is always more than the money under your bed as long as interest stays fixed. $\endgroup$ – user11235 May 6 at 15:59
  • $\begingroup$ What is the intended audience for this write up? $\endgroup$ – Andrew Chin May 6 at 16:25
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One example for "good" exponential growth is cryptographics. Linear key length growth vs. exponential growth for the effort to break the key. In this example we are kept safe by the exponential growth.

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Chain reactions in explosives. I mean, when you want one of those. After all, people develop and use them. Without the chain reaction and resultant exponential growth, you don't get the desired effect. (Linear would be too slow and/or require too high of an initial input, because of less amplification.)

https://en.wikipedia.org/wiki/Chain_reaction

Also avalanche breakdown in Geiger counters:

https://en.wikipedia.org/wiki/Geiger_counter#Principle_of_operation

and

https://en.wikipedia.org/wiki/Townsend_discharge

The extreme amplification is needed to detect and signal individual particles.

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