Velocity : Recall that it can be thought of as special case of the rate of change interpretation. If the situation of an object is specified by f(t ) after t units of time the velocity of the object at t = a is given by f ′ ( a ) .

Example   Assume that the position of an object after t hours is specified by,

                                          g (t ) =t/t+1

Answer following

(a) Is the object moving towards the right or the left at t = 10 hours? 

(b) Does the object ever stop moving? 

 Solution; The derivative is,               g ′ (t ) = 1 /(t + 1)²

 (a) Is the object moving towards the right or the left at t = 10 hours?

To find out if the object is moving to the right (velocity is positive) or left (velocity is negative) we require the derivative at t = 10 .

                                                             g′ (10) =1 /121

Thus the velocity at t = 10 is positive and hence the object is moving to the right at t = 10 .

(b) Does the object ever stop moving?

If the velocity is ever zero then the object will stop moving.  Though, note that the only way a rational expression will ever be zero is if the numerator is zero. As the numerator of the derivative (and therefore the speed) is constant it can't be zero.

Then, the object will not at all stop moving.

Actually, we can say little more here. The object will be moving always to the right as the velocity is always +ve.

Here we've seen three major interpretations of the derivative.  You must remember these, specially the rate of change, as they will continually show up throughout this course.