The amount of air in a balloon at any time t is given by V(t). Determine if the balloon is being filled with air at any moment of time t.

Here the amount of air in the balloon at any moment of time t is
given as V(t). Now if the balloon is being filled or not can be determined by finding the value
of the differential of the function V(t).

At any t if the balloon is
being filled, the amount of air at t+ dt is going to be greater than the amount of air at
t.

=> V(t+ dt) > V(t)

Now
if dt --> 0, we get the instantaneous change in the volume of air at any time
t.

V'(t)= lim dt--> 0 [{V(t+dt) - V(t)} / dt
]

So we need to find V'(t).

Substitute
the value of time t at which you want to know if the balloon is being filled or not. If V'(t)
> 0 , the balloon is being filled, if V'(t)< 0, the balloon is being emptied and if
V'(t) = 0 it implies there is no change in the volume of air.