Average Rates of Change

 

 

 
Objective: The goal of this demo is to provide students with a concrete understanding of the average rate of change for physical situations and for functions described in tabular or graphic formats.

Level: Precalculus and calculus courses in high school or college.

Prerequisites: Familiarity with the concept of slope of a line and computing the slope of a line.

Platform: No particular software package is required. Support for a viewer of gif or mov files is required. Viewers within a browser, Windows media player, QuickTime, or a commercial program can be used. It is recommended that a viewer that contains a stop/start feature be used when incorporating the animation in a lecture format or when students view the animation on an individual basis.

Instructor's Notes: In mathematics average rate of change is a stepping stone to instantaneous rates of change and the fundamental concept of a limit. Thus it is important to provide a variety of learning experiences about average rates of change so that students understand this fundamental notion of change. This demo provides visual experiences which connect to the algebraic expressions for average rates of change.

To provide a focus, we start with two common visualizations; a falling ball and a moving vehicle. Figure 1 shows an animation of a ball falling from rest under the influence of gravity, while Figure 2 displays a car traveling along a straight track with constant acceleration. (Click here to download a zipped file containing the animations in Figures 1 and 2 in both gif and QuickTime formats.)

 

Figure 1.

Figure 2.

Average rate of change is often introduced by saying it is the change in distance of the change in time:

Let s denote distance and t denote time, then we use the symbols for change in distance and for change in time. Thus we have

In situations involving the motion of an object like in Figures 1 and 2 we use the terminology average velocity in place of average rate of change. In such cases we denote average velocity by and we have

We can compute the average velocity of the falling ball in Figure 1 between two marks on the ruler displayed beside the path of the ball. For instance, we can measure the time it takes the ball to drop from the top (0 meter mark) to the 3 meter mark. In this case we have

and so the average velocity of the ball from s = 0 to s = 3 is

(The symbol means approximately equal to, since your calculator will display more than two decimal digits when you compute the ratio of 3 to 0.72. Displaying several decimal places is sufficient for our work.)

Many students don't recognize that things like falling bodies, moving vehicles, rising populations, and decaying radioactive materials do not change at constant rates. For instance, the falling ball has different average velocities as it passes various meter marks. Figure 3 shows a sample of the average velocities of the ball.

Figure 3.

A display of the average velocities as the ball passes meter marks is available as an animation. By clicking here you can download a zipped file containing both an animated gif and a QuickTime file that illustrates the different average velocities of the falling ball. We recommend that when you show it to a class that you use the QuickTime file. This will let you start and stop the animation so that you can discuss portions with your students. Figure 4 contains a segment of the animation. The full animation starts tracking the average velocities as the ball starts at the top.

Figure 4.

A similar display of the average velocities as the car passes 10 meter marks is available as an animation. By clicking here you can download a zipped file containing both an animated gif and a QuickTime file that illustrates the different average velocities of the car. A preview of this animation appears in Figure 5.

Figure 5.

 

The average rate of change of a function f(x) is used as an introductory step to viewing a derivative as an instantaneous rate of change. A common definition is stated as

the average rate of change of y = f(x) with respect to x over interval [a, b] is given by the quotient

This is then illustrated geometrically as shown in Figure 1 and we say

Figure 1.

that the quotient  Dy / Dx is the slope of the secant line from (a, f(a)) to (b, f(b)). The concept of average rate of change is then illustrated through examples and exercises that use applications like average velocity, average acceleration, average weight gain, average cost, and so on. These are followed by tying the average rate of change to the instantaneous rate of change by a limit process.

To lay a foundation for related rates which come after various differentiation techniques like power rule, chain rule, and implicit differentiation are developed, we can use the motion of an escalator.  One approach is the following.

An escalator employs a right triangle so that people can move from a lower floor to an upper floor (or vice versa) as shown in Figure 2. We have labeled the base of the triangle horizontal (denoted H), the altitude of the triangle vertical (denoted V), and the hypotenuse of the triangle people (denoted P, for the distance people travel between floors).

 

 

 

 

Credits

David R. Hill
Department of Mathematics 
Temple University

and is included in Demos with Positive Impact with his permission.