Average Rate of Change of f over [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.
If f(3) = -1 tons. and f(5) = 0.5 tons, 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.
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.
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