Rates of Change or instantaneous rate of change ; Now we need to look at is the rate of change problem.  It will turn out to be one of the most significant concepts .

We will consider a function, f(x), which represents some quantity which varies as x varies.  For example, maybe f(x) represents the amount of water into a holding tank after x minutes. Or possibly f(x) is the distance traveled through a car after x hours.  In both of these example we utilized x to represent time.  Certainly x doesn't need to represent time, however it makes for examples that are easy to visualize.

What we desire to do here is determine just how fast f(x) is verifying at some point, say x = a.  It is called the instantaneous rate of change or sometimes just rate of change of f(x) at x = a .

As along the tangent line problem all that we're going to be capable to do at this point is to find out the rate of change.  Hence let's continue with the instance above and think of f(x) as something i.e. changing in time and x being the time measurement. Again x doesn't need to represent time but it will make the clarification a little easier. Whereas we can't calculate the instantaneous rate of change at this point we can find the average rate of change.

To calculate the average rate of change of f(x) at x = a all we have to do is to select another point, say x, and then the average rate of change will be,

                A.R.C. = change in f ( x ) /change in x

                              = f ( x ) - f ( a ) /x - a

Then to estimate the instantaneous rate of change at x = a all we have to do is to decide values of x getting closer & closer to x = a (don't forget to decide them on both sides of x = a ) and calculate values of A.R.C. Then we can estimate the instantaneous rate of change from that.