### Does an exponential curve have a corner?

A recent Washington Post article has the title “When a danger is growing exponentially, everything looks fine until it doesn’t.” The article talks about how suddenly exponential growth seems to go from vanishingly tiny to passing a major threshold. This made me wonder if there is some natural moment in time at which we can say “Now is when the exponential growth has turned the corner and gone from barely growing to growing outrageously fast.” It turns out that the answer is both “Yes” and “No” — “Yes” in a way that is surprisingly quantifiable, and “No” in a way that I find personally ominous.

Here’s a sketch of an exponential function:

Exponential functions have the defining feature that the bigger they get, the faster they grow. On the left, the height of the function is still very small, so it’s growing super-slowly: the graph looks flat. On the right, the height is large, so it grows quickly and looks like a steep almost-straight line. In the middle is where the exponential curve looks the most bent:

Can we quantify where exactly that corner is? One mathematical tool that tells us how sharply a curve is turning is called the “radius of curvature”, which at any given point is the radius of the best-fitting circle at that point. Here’s a very curvy curve for comparison:

And here are the best-fitting circles at three of the points:

We could define a “corner” to be a place where the radius of curvature is as small as possible: if the curve were a road you were driving a car on, its “corner” would be where you have to turn your steering wheel the hardest.

On our exponential curve, that would put the corner here:

[Note: On the standard exponential function $y=e^x$, this point is located at $x= -\ln(\sqrt 2)$. A fun calculus puzzle is to calculate the location of the corner on a general exponential curve $y = A b^x$! You can play with an interactive demo I built here.]

So does that mean for any real-life phenomenon modeled by an exponential curve, there will be a moment where it objectively turns a corner and goes from Fine to Not Fine? Not so fast: where the radius of curvature is smallest depends on what scales we use for the x-axis and y-axis. Let’s say we are tracking, oh I don’t know, the number of cases of a disease outbreak, and suddenly we switch from reporting numbers on a scale of hundreds to numbers on a scale of thousands. Then all the heights on the graph would appear to scale down by a factor of 10:

Now what used to be the corner of the curve has been flattened (and not in a good way). Where is the corner now? For an exponential curve, scaling it vertically is just the same as shifting the whole curve horizontally:

That shifts the corner to the right as well:

What does that mean for us? It means that as our sense of the scale of an exponentially growing problem shifts from tens to hundreds to thousands and beyond, the point at which we feel like the problem has gone from Fine to Not Fine will shift too. As of this writing, my personal experience of the COVID-19 outbreak has looked something like this, and every step has felt like a transition from Fine to Not Fine:

• Countries adopt “shelter in place” orders.
• It appears in countries I’ve been to.
• Countries I’ve been to adopt “shelter in place” orders.
• It appears in my country.
• Places in my country adopt “shelter in place” orders.
• My location adopts a “shelter in place” order.
• I know people who know people who have it.
• I know people who know people who have died of it.
• I know people who have it.
• I know people who… not yet, at least, but I imagine it won’t be long.

I am hoping that the measures we are taking will keep there from being many more steps in this awful sequence, but we’ll find out soon enough.

## 2 thoughts on “Does an exponential curve have a corner?”

1. David Holmes says:

This post is awesome! It’s something I’d vaguely wondered about for ages, but never actually thought properly about, and you explain things super-clearly.

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