Assume that the amount of air in a balloon after t hours is specified by

                                            V (t ) = t ³ - 6t ² + 35

Calculate the instantaneous rate of change of the volume after 5 hours.

Solution

The first thing that we need to do is get a formula for the average rate of change of the volume.  In this case it is,

 A.R.C. = V (t ) - V (5)/(t-5)= t ³ - 6t ² + 35 -10 /t - 5  =t ³ - 6t ² + 25 /t - 5

To calculate the instantaneous rate of change of the volume at t = 5 we only have to pick values of t that are getting closer & closer to t = 5 .  Following is a table of values of t and the average rate of change for those values.

Thus, from this table it looks like the average rate of change is approaching 15 and thus we can calculate that the instantaneous rate of change is 15 at this point.