## 3.4 Average Rate of Change

(This topic is also in Section 3.4 in Applied Calculus or Section 10.4 in Finite Mathematics and Applied Calculus)

Average Rate of Change of f over the interval [a, b]: Difference Quotient

The average rate of change of the function f over the interval [a, b] is

\begin{align*} \text{Average rate of change of } f &= \frac{\Delta f}{\Delta x} = \frac{f(b)-f(a)}{b-a}\\ &= \text{ Slope of line through points } P \text{ and } Q \text{ in the figure} \end{align*} Average rate of change = slope of PQ

We also call this average rate of change the difference quotient of f over the interval [a, b].

Units: The units of the average rate of change are units of f per unit of x.

Quick Examples

If f (3) = -1 zonars and f (5) = 0.5 zonars, and if x is measured in years, then the average rate of change of f over the interval [3, 5] is given by

\begin{align*} \text{Average rate of change of } f \text{ over } [3, 5] &= \frac{f(5)-f(2)}{5-3}\\ &= \frac{ 0.5 - (-1) }{2} = 0.75 \text{ zonars per year} \end{align*}

Here is one for you. Let f be specified by the following table:

 x () 0 1 2 3 4 f(x) ()
 Average rate of change of f on = Enter answer as a fraction or as a decimal accurate to 3 decimal places. Units of measurement of the rate of change are per Each time you press this "Help" button your score will decrease.

The following graph shows data on West Coast exports to East Asia: Data are approximate and randomized. Source: The New York Times April 21, 1999, p. C1.

Complete the following sentences (Note: Answers you enter must either be fractions or decimals accurate to at least 3 digits.):

• Over the period , west coast exports to east Asia were at an average rate of

• Over the period , west coast exports to east Asia were at an average rate of

Computing the Average Rates of Change over Smaller and Smaller Intervals

In preparation for the next section, we are going to look at the average rate of change of a function over smaller and smaller intervals and look for some kind of pattern or trend in the answers.

Let . We are going to compute the average rates of change of f over the following smaller and smaller intervals: , where h = 1,\ 0.1,\ 0.01,\ 0.001. This means that we are going to compute the rate of change of f over each of the following intervals:

 h = 1, so [ + h] = [ + 1] h = 0.1, so [ + h] = [ + 0.1] h = 0.01, so [2, 2+h] = [ + 0.01] h = 0.001, so [+h] = [ + 0.001]

Use technology to assist you with the calculations. You could use either the Function Evaluator & Grapher or a graphing calculator. Be sure to enter the exact values -- do not round.
 f = f = f = f = f =
 Ave. Rate of Change of f over = Ave. Rate of Change of f over = Ave. Rate of Change of f over = Ave. Rate of Change of f over =

Do you see a trend? First, we notice an interesting pattern in the decimal places as h gets smaller and smaller. Also, we see that the average rates of change are getting closer and closer to the value

Q Think about how you might interpret this "limiting value."

Now go over the examples and try some of the exercises in Section 3.4 in Applied Calculus or Section 10.4 in Finite Mathematics and Applied Calculus.

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Last Updated: June, 2009