Humans are notoriously terrible about estimating volumes when things are curved and volume functions are exponential.
A great example of this done in 8th grade science classes across the US is to put 100ml of water in a 100ml graduated cylinder, 150ml in a 1L beaker, and ask the class which has more. Humans are awful at estimating how much volume the increased radius adds, and usually will say the 100ml.
The problem only gets worse as we graduate from cylinders to spheres.
We can all visually see which sphere is bigger, but cannot come close to estimating how much bigger one is than another.
Already eight years ago, I complained that people were using "exponential" where it doesn't make any sense. (See these two data points? Clearly exponential growth happend there. They're so far apart!)
I believe the problem has increased exponentially since then. Now everyone is using exponentially in literally the same way as literally.
You might be interested to know that the first definition of "exponential" is "of or relating to an exponent". The second definition is, as you say, "involving a variable in an exponent". https://www.merriam-webster.com/dictionary/exponential
As this is an internet forum and not a rigorous mathematical setting, I assert that my use of "exponential" is correct in context and to claim otherwise is incorrect. :)
I'm not sure if you are kidding but just in case you are not this is very misleading and in fact misguided.
Refering to polynomials as exponential just results in confusion essentially removing any meaning from the word. Any function can be written as something involving exponents, so that statement becomes meaningless.
I don't think this word means what you think it does. Or I don't. Exponents are just the number the value is raised. Squaring a value just uses an exponent of 2 where cubing uses an exponent of 3. Polynomials are x^2 + x + 1 type of equations. But admittedly, it has been 30+ years since I've thought about them at that level, so maybe I'm the one with fuzzy groking
We can go a step further with the pedantry, and say that the commenter above is using an unreasonably narrow definition of the work "exponential" and that there are others which allow x^2 to be described as "exponential".
We could, but I would describe it as "mathematically accurate". Which is not incompatible with "unreasonably narrow", given that the definition of "exponential" has recently gotten polluted enough that it is now often synonymous with "fast growing". But what's the point of arguing over definitions if we're going to start with a baseline of saying that there is no basis upon which to argue definitions other than recent conventional usage?
> there are others which allow x^2 to be described as "exponential".
Those same definitions allow x*1000 to be described as "exponential". (x*1000000 would be "more exponential"!)
If you're describing something as exponential, then either you're just saying "fast growing", or you're trying to describe the type of growth. If you're describing the type of growth, then neither x*1000 nor x^2 is exponential. The fact that x^2 has an exponent in it is no more relevant than saying that x*1000=x*10^3 and x*10^3 has an exponent in it.
(Again, I sadly accept that in today's world, "exponentially" is being used to mean "fast growing", or sometimes more specifically "faster than linear". If I'm trying to understand what someone means, then it doesn't matter whether I find that usage to be a good idea or not.)
No; to characterize "exponential" as "fast-growing" is a misunderstanding of what I'm saying. "Faster than linear" would be a good descriptor.
> > there are others which allow x^2 to be described as "exponential".
> Those same definitions allow x1000 to be described as "exponential". (x1000000 would be "more exponential"!)
> If you're describing something as exponential, then either you're just saying "fast growing", or you're trying to describe the type of growth. If you're describing the type of growth, then neither x1000 nor x^2 is exponential. The fact that x^2 has an exponent in it is no more relevant than saying that x1000=x10^3 and x10^3 has an exponent in it.
I don't agree with this. These are categorically different.
In f(x)=x*1000, as x increases, the function's output increases linearly. The slope of the derivative is 0.
In f(x)=x^3, as x increases, the function's output increases more than linearly. The slope of the derivative is positive and linear.
In f(x)=3^x, as x increases, the function's output increases much more than linearly. The slope of the derivative is positive and is itself a function of x.
These are all categorically different, and refer to something different than "fast-growing". "Exponential" in the mathematical sense, means the derivative is a function of x. "Exponential" in the colloquial sense means that the derivative has a positive slope. "Fast growing" just means that the derivative is large, even if it is a constant.
Um, ok. Your position baffles me, because you clearly understand what exponential means mathematically, yet you insist that the word means something else colloquially. Specifically "faster than linear". Usually, people who (mathematically) misuse the term do so because they don't understand what it actually means, but that's not what is happening here.
If it's going to mean something precise, such as
> The slope of the derivative is positive and linear.
then why not pick the precise thing that the word already means?
Is x*log(x) also exponential to you? If so, then why not use the word that already exists: superlinear? If not... oh wait, the above definition I quoted wouldn't even cover x^2, since the slope of its derivative is constant, not linear. So I'm just completely confused; I can't figure out which (mathematically) non-exponential functions you would like to label as exponential. x*1000, no. x^3, yes. x^2, I don't know. x*log(x), I don't know. x^2*log(x), I don't know.
> "Exponential" in the colloquial sense means that the derivative has a positive slope.
"Exponential" in the colloquial sense means that the speaker isn't using a mathematical sense, and so isn't considering first or second derivatives. I don't buy the argument that the colloquial sense accepts x^3 and rejects x^2, and in fact I bet I could find someone using it for a linear relation ("My workload has gone up exponentially since you laid off half the team!")
> "Exponential" in the mathematical sense, means the derivative is a function of x.
No it doesn't. x^2 is not mathematically exponential, yet its derivative is a function of x. Exponential means the derivative is exponential. But that's just a detail that doesn't really change the core of your message.
The main purpose of the mathematical definition is to exclude polynomials. The main purpose of the colloquial definition seems to be something like an impressive or important increase.
A great example of this done in 8th grade science classes across the US is to put 100ml of water in a 100ml graduated cylinder, 150ml in a 1L beaker, and ask the class which has more. Humans are awful at estimating how much volume the increased radius adds, and usually will say the 100ml.
The problem only gets worse as we graduate from cylinders to spheres.
We can all visually see which sphere is bigger, but cannot come close to estimating how much bigger one is than another.