Is there a name for the point of a exponential curve where the y axis significantly increases?

The English idiom is "the knee in the curve."

This doesn't have much, if anything, to do with mathematics, however.

For a good explanation of why it's a subjective issue and not a mathematical one, you can look at this article: http://www.growth-dynamics.com/articles/Kurzweil.htm (archived) About 1/3 down the page there are a few graphs one over the other with the title "where is the knee?" They have different $y$ axes, but show the same function. You can see that your $y$ axis determines where you think the "knee" should fall.


This is an old question, but a very good rule of thumb for such a point would be the minimum of the radius of curvature function.

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The exponential function and its radius of curvature:

It takes a minimum value at $x=-\frac{\log (2)}{2}=-0.346574$

Another example, arctan function:

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The radius is smallest at $x=\pm 0.831576$

Tags:

Algebra Precalculus

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