Average Rate of Change

The average rate of change of a population is the total change divided by the time taken for that change to occur. The average rate of change can be calculated with only the times and populations at the beginning and end of the period. Calculating the average rate of change is similar to calculating the average velocity of an object, but is different from calculating the instantaneous rate of change.

I want to talk about average rate of change. We've talked about average velocity. Average rate of change is a similar concept only we'll be applying it to functions that don't measure position. They'll measure something else.

Let's take a look at an example. The population F of T of gnomes in Thuringia is described by the table below. T is the year. And so we have the years from 1850 to 1900, skipping by tens, and the population starting at 3200 and ending at 22,800.

This is how we define average rate of change of F of T over an interval. It's F of B minus F of A over by minus A. Looks like average velocity. It's really the same kind of thing. It's the average rate of change.

Let's calculate the rate of change of the gnome population on this interval between 1850 and 1880. So 1850 is going to be our A value. 1880 will be our B value. And so we'll have to compute F of 1880 minus F of 1850. Over 1880 minus 1850. Okay.

So let's look at our table. In 1880, the population of gnomes was 12 -- rather 10,400. 1850 it was 3200. So 10,400. And 3200. That's divided by-- this is a 30-year difference. 30. 10,400 minus 3,200 is 7,200, and that's in gnomes. Gotta remember units. Because actually when you're dealing with average rate of change, the units will tell you, they'll tell you what the average rate of change actually means. So this is 30 years.

And so let's simplify this answer. 72 divided by 3 is 24. So this would be 240 gnomes per year. What this means is -- now you notice that the population of gnomes has increased by 7200 in this 30-year period. 240 gnomes per year is how fast the gnome population would have to grow if it were growing at a constant rate in order to make up that increase. So 240 gnomes per year.

Now let's look at the rate of change of the population from 1880 to 1900. That's going to be F of 1900 minus F of 1880 over 1900 minus 1880. So always remember, it's final population minus initial population. Or final quantity minus initial quantity, whatever the quantity is that F measures. So, again, we look at our table. At 1900, we had 22,800 gnomes. 800 gnomes. 10,400 in 1880. So 22.8. 22,800, 10,400 and this is going to be a 20-year period. So 20. So we have 22 minus 10, 12. 8 minus 4, 4. 12,400 over 20. So let me cancel the 0 here. This is going to be 620 gnomes, right, that's the units at the top, gnomes per year. Okay. So the average rate of growth of the population from 1880 to 1900 is 620 gnomes per year.

Average rate of change is exactly like average velocity. It's the change in the quantity F divided by the change in time.