I don’t understand, how can adoption rate change overnight if its derivative is negative? Trying to draw a parallel to get intuition, if adoption is distance, adoption rate speed, and the derivative of adoption rate is acceleration, then if I was pedal to the floor but then release the pedal and start braking, I’ll not lose the distance gained (adoption) but my acceleration will flatten then get negative and my speed (adoption rate) will ultimately get to 0 right? Seems pretty significant for an industry built on 2030 projections.
One announcement from a company or government can suddenly change the derivative discontinuously.
Derivatives irl do not follow the rules of calculus that you learn in class because they don't have to be continuous. (you could quibble that if you zoom in enough it can be regarded as continuous.. But you don't gain anything from doing that, it really does behave discontinuous)
Derivatives in actual calculus don’t have to be continuous either. Consider the function defined by f(x) = x^2 sin(1/x) for x != 0; f(0) = 0.
The derivative at 0 exists and is 0, because lim h-> 0 (h^2 sin(1/h))/h = lim h-> 0 (h sin(1/h)), which equals 0 because the sin function is bounded.
When x !=0, the derivative is given by the product and chain rules as 2x sin(1/x) - cos(1/x), which obviously approaches no limit as x-> 0, and so the derivative exists but is discontinuous.
Not sure what kinda calculus you took at least here in the states it's very standard to learn about such functions in class, and yes there is a difference between discontinuous and the slope being really large (though finite) for a brief period of time
You rarely study delta and step functions in an introductory calculus class. In this case the first derivative would be a step function, in the sense that over any finite interval it appears to be discontinuous. Since you can only sample a function in reality there's no distinguishing the discontinuous version from its smooth approximation.
(I suppose a rudimentary version of this is taught in intro calc. It's been a long time so I don't really remember.)
I'm sure it depends on who's teaching the class and what curriculum they follow, but we were doing piecewise linear functions well before differentiation so I think I do actually disagree as per your caveat. It's also possible that the courses triaged different material. As a calc for engineers not calc for math majors taker, my experience may have been heavier on deltas and steps.
I don't understand your point. It seemed like the person I was replying to didn't understand how both claims could be simultaneously true so I was elaborating.
