This is one of those questions which require an answer that does not take practical limitations into account. It is a theoretical physics question, perhaps. If there are any loopholes used, please explicitly state them.
If the position is known as $x(t)$ from t=0 to t=1 second, how do I get the velocity at the initial and end points, since velocity at the end point will require $x(1-(\Delta t)/2)$ and $x(1+ (\Delta t)/2)$, which are added and divided by $\Delta t$ ?
It gets worse if I want to know the acceleration at the end point, which requires the $v(1+(\Delta t)/2)$ which in turn requires $x(1+(\Delta t))$, which is simply not available.
Is this an order thing or is it just neglected in calculus?