Derivatives as Rates of Change: Learn It 1

  • Calculate how quantities change on average over time
  • Use rates of change to figure out how an object’s position, speed, and acceleration are changing over time
  • Estimate future population sizes using current data and how fast the population is growing
  • Use derivatives to determine the cost and revenue of producing one more unit in a business

Amount of Change Formula

One application of derivatives is to estimate an unknown value of a function at a point by using a known value of the function at some given point together with its rate of change at that given point.

If f(x) is a function defined on an interval [a,a+h], then the amount of change of f(x) over the interval is the change in the y values of the function over that interval and is given by:

f(a+h)f(a)

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:

f(a+h)f(a)h

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

f(a)=limh0f(a+h)f(a)h

For small enough values of h,f(a)f(a+h)f(a)h. We can then solve for f(a+h) to get the amount of change formula:

f(a+h)f(a)+f(a)h

We can use this formula if we know only f(a) and f(a) and wish to estimate the value of f(a+h). 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 1, we are approximating f(a+h) by the y coordinate at a+h on the line tangent to f(x) at x=a. Observe that the accuracy of this estimate depends on the value of h as well as the value of f(a).

On the Cartesian coordinate plane with a and a + h marked on the x axis, the function f is graphed. It passes through (a, f(a)) and (a + h, f(a + h)). A straight line is drawn through (a, f(a)) with its slope being the derivative at that point. This straight line passes through (a + h, f(a) + f’(a)h). There is a line segment connecting (a + h, f(a + h)) and (a + h, f(a) + f’(a)h), and it is marked that this is the error in using f(a) + f’(a)h to estimate f(a + h).
Figure 1. The new value of a changed quantity equals the original value plus the rate of change times the interval of change: f(a+h)f(a)+f(a)h.

average rate of change

The average rate of change of a function f over the interval [a,a+h] is the ratio of the amount of change in f over that interval to the corresponding change in x values.

The average rate of change is given by:

f(a+h)f(a)h

Here is an interesting demonstration of rate of change.

If f(3)=2 and f(3)=5, estimate f(3.2).