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If you were given a set of data, such as population vs time, for example:

(years) 0, 10, 20, 30, 40, 50, 60

(population)5, 10, 20, 40, 80, 160, 320

Would you get the overall average rate of change by calculating each individual average rate of change and then averaging that? Or just by dividing the last population variable, 320, by the last time variable, 60?

(this was part of a functions chapter in my textbook)

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3 Answers 3

I think you would look at $$\frac{\Delta P}{\Delta t} = \frac{320-5}{60-0}$$

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This depends on the rate of change between which two times; Damien's answer generalizes. Given population value $P(t_i)$ and $P(t_j)$ , for times $t_i,t_j$ , i.e., P(0)=5, p(10)=10 , etc.(i.e., $t_i,t_j$ are values in {$0,10,20,30,40,50,60$}, i.e., $t_o$=0, $t_1=10$ , etc., the change between time $t_i$ and time $t_j$ (assume $t_i>t_j$ is:

$\frac {P(t_i)-P(t_j)}{t_i-t_j}$

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If the rate of change is one value minus the previous, then the average rate of change is the average of those, I get the average rate of change is 63=(320-5)/5. You should be able to convince yourself this is the same averaging pop(this year)-pop(last year) over the data.

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