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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
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
Here is one for you. Let f be specified by the following table:
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The following graph shows data on exports to East Asia.
Complete the following sentences:
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 f(x) = x3 + x. We are going to compute the average rates of change of f over the following smaller and smaller intervals: [2, 2+h], where h = 1, 0.1, 0.01, 0.001, 0.0001. This means that we are going to compute the rate of change of f over each of the following intervals:
| [2, 3] | h = 1, so [2, 2+h] = [2, 2+1] | |
| [2, 2.1] | h = 0.1, so [2, 2+h] = [2, 2+0.1] | |
| [2, 2.01] | h = 0.01, so [2, 2+h] = [2, 2+0.01] | |
| [2, 2.001] | h = 0.001, so [2, 2+h] = [2, 2+0.001] |
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.
Alternatively, press "game version" on the sidebar to go to the game version of this tutorial (it has different examples to try and is a lot of fun!) or press "next" to go on to the next topic.
