In 3blue1brown: derivative paradox.
supposed car moving with:
$S(t) = t^3$
And velocity is:
$V(t) = 3t^2$
He asked when t = 0 velocity is 0 m/s , does that car move at that time ? And here his answer:
Take a moment to unpack what it actually means for the derivative of the distance function to be 0 at this point. It means the best constant approximation for the car’s velocity around that point is 0 meters per second. For example, between t=0 and t=0.1 seconds, the car does move, it moves 0.001 meters. That’s very small, and importantly it’s very small compared to the change in time, an average speed of only 0.01 meters per second.
For smaller and smaller nudges in time, this ratio of the change in distance over change in time approaches 0, though in this case it never actually hits it. So would you say this qualifies as moving at the time t=0? I would argue the question makes no sense, movement is something which happens between two points in time, so has no meaning in a given instant.
It’s tempting to say the derivative gives this notion meaning, and many people would happily say the car is not moving at t=0, but it is moving for all times t>0. For my part, I’d recommend not taking the phrase “instantaneous rate of change” too seriously, instead thinking of it as a conceptual shorthand for “the best constant approximation for the rate of change”
What does this part mean? Does he said that zero instantaneous velocity is not static car?
