2.1.2 Average Rates of Change for More General Functions

In calculus we study rates of change in general, not just for linear functions. Our first step is to rewrite the equation

${\displaystyle \mathrm{slope}=\frac{\mathrm{rise}}{\mathrm{run}}}$

in the equivalent form

${\displaystyle {\displaystyle \mathrm{average\; rate\; of\; change}={\displaystyle \frac{\mathrm{the\; change\; in\; the\; dependent\; variable}}{\mathrm{the\; change\; in\; the\; independent\; variable}}}}}$.

A linear function has only one rate of change: the slope of its graph. In this case, we may talk about the rate of change. However, the average rate of change of a more general function varies from one interval to another. In the next activity, we ask you to compute some average rates of change in a familiar setting.

Activity 2

An object is dropped from a height. Table 1 records at each second the distance it has fallen.

Table 1   Falling body data Time (seconds) Distance (meters) 1.0 5.0 2.0 19.4 3.0 44.1 4.0 78.0 5.0 122.8 6.0 175.8 7.0 240.0 8.0 312.8 9.0 396.1 10.0 489.0

1. What is the average speed of the object for the first five seconds? For the first three seconds? For the first second?

2. Estimate the average speed for the first \(4.5\) seconds.

3. Suppose the height from which the object is dropped is \(489\) meters; what is the average speed for the last two seconds of its fall?

4. Tap the button below to graph the data in Table 2.1. Change your window to plot just the portion of the graph that represents the last two seconds. How does your answer to (c) relate to what you see in the graph?

5. Does it make sense to talk about a single rate of change for this function? Why or why not?

Comment on Activity 2

Checkpoint 2