Knowing that a function increases is only half the story. How fast does it increase? The average rate of change of a function over an interval of its domain is the ratio of the change in the output values to the change in the input values over that interval.
It is also the constant rate of change that yields the same change in the output as the function actually produced on that interval. A cyclist who speeds up and slows down can be replaced, for bookkeeping, by one who rides at a single steady speed and covers the same distance in the same time. That steady speed is the average rate of change, and it is the slope of the line through the two points (the secant line).
Pick a function and drag the two inputs. The dashed line through the two points is the constant-rate stand-in: a function that changes at one steady rate and still goes from the first output to the second. Its slope is the average rate of change.
A table gives you the outputs at certain inputs, so choose two columns and apply the same formula. Only the two endpoints of your interval matter. Columns in between don't change the average rate over the whole interval.
A cyclist's distance D (miles) from home is recorded at uneven times t (hours). Choose two columns. The rate divides by the change in t, not by how many columns apart they are.
| t (h) | 0 | 1 | 3 | 4 | 6 |
| D (mi) | 0 | 12 | 30 | 38 | 50 |
The rate of change at a point quantifies how fast the output would change were the input to change at that exact point. You can't use the formula on an interval of length zero, but you can approximate the rate by using the average rate of change over small intervals containing the point, if such values exist.
We can't divide by an interval of length 0, so we estimate the rate of change at the point with the average rate over a small interval that contains it. Start with [a, a + h] and shrink h to watch the averages settle. (For x², the two-sided interval happens to give exactly the same value for every h, so try both one-sided modes too.)
| h | interval | average rate of change |
|---|---|---|
| 1 | [1.00, 2.00] | 3.000 |
| 0.5 | [1.00, 1.50] | 2.500 |
| 0.25 | [1.00, 1.25] | 2.250 |
| 0.1 | [1.00, 1.10] | 2.100 |
| 0.05 | [1.00, 1.05] | 2.050 |
| 0.01 | [1.00, 1.01] | 2.010 |
The rates of change at two points can be compared by estimating each one with an average rate over a sufficiently small interval containing that point, then comparing the two numbers. Use intervals of the same small size around each point so the comparison is fair.
f(x) = x³ − 3x. Each point's rate is estimated with an interval of length 0.2 centered on it. Move the points and compare. The dashed lines show the slope each estimate found.
A positive rate of change indicates that as one quantity increases or decreases, the other quantity does the same. They move together. A negative rate of change indicates that as one quantity increases, the other decreases. They move in opposite directions.
The graph shows the temperature T (°F) of a room t hours after the heater is switched on. Pick any interval and read the sign of the average rate of change.
Rates of change quantify how two quantities vary together. In a real situation the rate always carries units: output units per input unit. A complete description names the interval, the quantity, the value of the rate, and the units.
Choose a situation and an interval. The sentence below is the pattern to copy on the exam: name the interval, the quantity, the rate, and the units.