MATH 121 – APPLIED CALCULUS – DR. NARDO
AVERAGE RATE OF CHANGE
In class, we will use various formulas for an average rate of change; however, they are all really the same. These formulas all boil down to:
.
You use this one simple formula whether you are calculating values out of a table, off a graph, or from a formula.
Example 1: Consider the data for the two variables D and M below.
D |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
M |
50 |
50 |
45 |
43 |
40 |
30 |
0 |
Suppose that D represents the day of the week and that M represents the amount of money you possess at the beginning of the day.
Calculate the average rate of change for M with respect to D from D = 1 to D = 7.
We see, via the definition of a function, that M is clearly a function of D. So, instead of writing a separate output variable M, we could write the function . Then the formula will look slightly different – but give the same value.
Questions:
(The units, $ per day, given above should help you with this last one.)
Consult your class notes (or textbook) for help on these questions! Also, now that we have covered calculating an average rate of change for a function given numerically, you should consult your class notes (or textbook) for help in calculating an average rate of change for a function given graphically.
Now, let’s turn our attention to functions given by formula.
Example 2: Let be a function.
We will calculate several average rates of change.
From x = 0 to x = 10
From x = 5 to x = 8
Suppose that we have several of these rates of change to calculate. Is there a way to keep from doing the same basic calculation over and over again? Yes! We’ll use generic starting and stopping points. We’ll start at x and move over h units to get to the stopping point of (x + h). Let’s see how that works.
From x to x + h
If this is a valid approach which is executed correctly, then the last formula above should match the results of the first two rates we calculated!
Instead of using and , we can translate the phrase “from 0 to 10.” It means that we have a starting point of x = 0 and move a total of h = 10 units. Thus, the “generic” average rate of change formula of “” from the last box above gives: . This is the same as our first concrete calculation!
Similarly, the phrase “from 5 to 8” translates to a starting point of x = 5 and moving a total of
h = 3 units. As before, the “generic” formula gives: . This is the same as our second concrete calculation!