e is the base rate of growth shared by all continually growing processes.
I don't think that's a very good explanation because all the heavy lifting is being done by the phrases "base rate" and "continually growing", nether of which are well defined. "Continually growing" could reasonably be interpreted to mean "monotonically increasing" in which case f(x)=x qualifies and the whole explanation falls apart.
I think the idea of finding a function that is its own derivative, discovering that there is a polynomial series that meets that requirement, and evaluating that series at x=1 is a much more natural explanation.
You don't need or even want well defined terms when you're trying to explain why a mathematical formula is useful or naturally applicable to real world situations.
Comments
See the great article on e from Better Explained:
https://betterexplained.com/articles/an-intuitive-guide-to-e...
I don't think that's a very good explanation because all the heavy lifting is being done by the phrases "base rate" and "continually growing", nether of which are well defined. "Continually growing" could reasonably be interpreted to mean "monotonically increasing" in which case f(x)=x qualifies and the whole explanation falls apart.
I think the idea of finding a function that is its own derivative, discovering that there is a polynomial series that meets that requirement, and evaluating that series at x=1 is a much more natural explanation.
You don't need or even want well defined terms when you're trying to explain why a mathematical formula is useful or naturally applicable to real world situations.
Maybe you don't want well-defined terms in your pedagogy, but I do.