2.1 Modeling Rates of Change

Summary

This introductory section is intended to introduce some of the issues involved in modeling rates of change, and to set the agenda for the rest of the chapter. Given a set of time and distance data for a falling object, average speeds are calculated. On the basis of mathematical analysis, the distance function is shown to be quadratic. The concept of an average rate of change is introduced for an arbitrary function, and is computed for the function e ^x.

By the end of your studying, you should know:

How to calculate the average speed of an object from a table of time and distance data.
With average speed called the first derived, how to calculate the second derived.
The formula for the distance reached by an object falling near the earth.
The definition of the average rate of change of a function over an interval.
How to calculate the average rate of change (also called the first derived), and the second derived for a given function.
The difference between average rate of change and instantaneous rate of change.

On-screen applet instructions: This applet shows the average rate of change at x = a, where a can be chosen from the pull down list. A single click in the graph gives an enlarged picture around the point x = a. Another click restores the original size. The value of h can be set on the slider. Click here for further instructions.

Examples

While skiing, Josh rides a chair lift moving at 3 miles per hour up a hill. After he has skied down the hill, his average velocity for the round trip is 5 miles per hour. How fast did he ski?

Fill in the missing numbers in a derived table.

A automobile testing organization runs cars around a speed track to test performance. One test involves running a car around a 1200-meter track at high speed, and recording the individual lap times and the average speed for the entire test. On one occasion, the results are recorded, but are missing the final lap time. The first 4 laps show times of 37 seconds, 25 seconds, 29 seconds, 27 seconds, and the average speed for the test is recorded as 112 miles per hour. Show that even without knowing the final lap time, we can conclude an error has most likely been made in recording the results.