[j7ake]

Yeah the constant is a poor approximation of the exponential. Maybe the analogy works better if I start at the second term .

Zeroth order approximations are rarely useful because the dont capture the local gradient.

But in the life analogy, there are people who assume where they are in life will be same in future (eg the “new” normal during pandemic). So maybe it’s not so terrible after all.

[froj]

Well, 0th order approximations are fine if the gradient is indeed about zero for all intents and purposes. It's just very much not the case for exponential functions.

Maybe your analogy could work if you want to say that exponential growth might feel like linear growth locally?