Definition: The average rate of change of a function over an interval is the change in output divided by the change in input, equal to the slope of the secant line.
What does it measure?
It tells how fast the output changes, on average, as x moves from one value to another. For a linear function it is constant. For a curve, it depends on the interval you pick.
What is the formula?
Over the interval from a to b, the average rate is (f(b) – f(a)) / (b – a). Units are output units per input unit, such as dollars per month.
Example. Let f(x) = x2 + 3x. On the interval from 1 to 5: f(1) = 4 and f(5) = 40. The average rate is (40 – 4) / (5 – 1) = 36/4 = 9. On the interval from 5 to 7: f(7) = 70, so the rate is (70 – 40) / 2 = 15. The function speeds up.
What errors are common?
Dividing by the sum of the endpoints instead of their difference, or subtracting the outputs in one order and the inputs in the other. A negative result means the function is decreasing on average. The rate over an interval says nothing about what happens inside it.