However, what I don't get in your example is the implied connection between the timescale and the order of the Taylor polynomial. Could you elaborate?
Then as you zoom out, you have to include higher order terms to account for the exponential increase.
If you're exponentially growing, but currently it's approximately constant, then you're at t = -inf and you'll probably be dead by the time you achieve something significant.
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.
Maybe your analogy could work if you want to say that exponential growth might feel like linear growth locally?
This means that, if a process like skill development is growing exponentially, then when zoomed in at very small times (e.g. daily), growth looks flat.
But if looked at longer times, it starts to look linearly increasing.
Then finally when looked upon after many years, things look exponential.
It's also related to the quote that people overestimate progress in the short-term but underestimate it in the long-term.