I am currently relearning calculus. (Since I did not understand the fundamentals very well). We've just talked about how limits were relevant in the real world.
The teacher showed us a case where he calculated the instantaneous velocity of a rocket from its position function using limits. (Using the following formula: V = $\lim_{t\to t_0} \frac{f(t) - f(t_0)}{t-t_0}$)
I think I understand the logic behind it. What I have a hard time wrapping my mind around is:
How do you actually get this position function in real life ?
Let's say I build a robot that moves only in one direction, and that this robot gives me its position once every 0.5 seconds.
Well then, I think I would basically have to "guess" what the curve of the position function is doing between each data point, which means that the function would not be totally accurate.
So, wouldn't that prevent me from finding the actual instantaneous velocity at 0.75s, 1s, etc.. ?
Or maybe it would "work" but only for the precise numbers that the robot gave me ? (It would only works for multiples of 0.5 ?)
If you can only measure these things by polling at discrete intervals, then it is true that you can only ever get a discrete approximation of your measurement. This is true of a lot of things in the real world, though - our measurements are always limited by some kind of precision, whether that's the tick marks on a measuring tape or the second hand on a clock or the frequency of a particular kind of radiation.
However, there are more precise measurements and there are less precise measurements, and calculus lets us imagine about the scenario where we're able to get more and more precise measurements about something. So you can imagine polling your robot about its position once per 10 seconds, then compare that to polling it once per second, then once per millisecond, and so on until you magically have exact knowledge of its position at any arbitrary point in time.
There are also various methods that let you estimate what those smooth measurements might look like, based on the discrete data point you can actually observe, under the assumption that you're taking measurements at a reasonable frequency (so in our example, polling the robot once per minute might not be good enough if it can do an entire lap of the track in that time and report its position as not having changed, but once per second might be fine).