Just a small caveat that velocity ( nor the derivative in general) is not necessarily changing. Speed may be constant. Please ignore if you're aware of this.You mean like v=32t. Here the v at 5sec is 160ft/s and it will be approximately 160 around 5sec with some error. So we will be able to calculate approximate actual changes between two nearby points. we say the velocity of body from say 5sec to 5.000001 sec is approximately 160ft/sec. In this way we can calculate the approximate change of position in between them as 0.00016 ft.
We can take any value of t and approximate the actual change between t and some near point assuming nothing unexpected happens.
because the velocity keeps on changing. Right? That’s why “approximate change” and not exact change.
Let’s say 160.016 ft/s is good enough. I got that speed by taking my other point as 5.001sec. So now this approximation works for many points near 5.001 hoping there is no any sudden major changes in speed. Points like 5.01sec,5.03sec,etc
In the same way if we take the limit we get precisely 160 at 5. This is very exact than 160.016 ft/s. Now this works for points near 5. Right!