Whether digital or analog, kilometers or miles per hour, a speedometer tells you how fast you are going at any point in time.
Suppose you are going to visit a friend in a different city. The trip of 192 miles will take 3 hours of driving.
As you leave your house, and as you approach your friend’s house, you are driving through the neighborhoods fairly slowly.
On the highway access roads, as you enter and as you leave the highway, you are driving somewhat faster.
On the highway you are driving as fast as you can, up to what the speed limit allows.
What is your average speed for the trip? It isn’t just a matter of adding up five speeds and dividing by five,
because you were likely driving on the highway for a much longer time than you were driving in the neighborhoods or on the access roads.
But if you know the distance to your friend’s house and kept track of the time you were driving, your average speed can be found by dividing the total distance by the elapsed time.
In fact, such an average can be found for any segment of your trip by dividing the change in distance by the change in time.
This concept can be extended to other situations, not just the situation of covering ground. For a continuous function f, the changes are found by subtraction.
This tool lets you experiment with the average rate of change for several different functions.
Select the desired function with the buttons.
Set the x-values a and b by using the sliders, by typing values directly into the boxes, or by dragging the points on the graph to the left or right.
If you want the x-values a and b to remain the same distance apart, check the “Lock” box.
When the box is checked, you can change either of the x-values and the other will be automatically updated so that the distance between them is always the same.
The average rate of change of the selected function over the closed interval with endpoints a and b will be continuously updated.
Look for trends in the value of the average rate of change as you experiment. Try to make and verify predictions about the value based on the shape of the function you select. Have fun and learn! 1