Amount of Change Formula

One application for derivatives is to estimate an unknown value of a function at a point by using a known value of a function at some given point together with its rate of change at the given point. If \(f \left(\right. x \left.\right)\) is a function defined on an interval \(\left[\right. a , a + h \left]\right. ,\) then the amount of change of \(f \left(\right. x \left.\right)\) over the interval is the change in the \(y\) values of the function over that interval and is given by

The average rate of change of the function \(f\) over that same interval is the ratio of the amount of change over that interval to the corresponding change in the \(x\) values. It is given by

As we already know, the instantaneous rate of change of \(f \left(\right. x \left.\right)\) at \(a\) is its derivative

For small enough values of \(h , f^{'} \left(\right. a \left.\right) \approx \frac{f \left(\right. a + h \left.\right) - f \left(\right. a \left.\right)}{h} .\) We can then solve for \(f \left(\right. a + h \left.\right)\) to get the amount of change formula:

We can use this formula if we know only \(f \left(\right. a \left.\right)\) and \(f^{'} \left(\right. a \left.\right)\) and wish to estimate the value of \(f \left(\right. a + h \left.\right) .\) For example, we may use the current population of a city and the rate at which it is growing to estimate its population in the near future. As we can see in Figure 3.22, we are approximating \(f \left(\right. a + h \left.\right)\) by the \(y\) coordinate at \(a + h\) on the line tangent to \(f \left(\right. x \left.\right)\) at \(x = a .\) Observe that the accuracy of this estimate depends on the value of \(h\) as well as the value of \(f^{'} \left(\right. a \left.\right) .\)

Figure 3.22 The new value of a changed quantity equals the original value plus the rate of change times the interval of change: \(f \left(\right. a + h \left.\right) \approx f \left(\right. a \left.\right) + f^{'} \left(\right. a \left.\right) h.\)