What the average rate of change means
The average rate of change of a function on an interval is how much the output changes per unit of input, on average, from one end to the other. On a graph it's the slope of the secant line joining the two endpoints. If a car's position is f(t), it's the average velocity; if f is a price, it's the average change per year.
Formula
With two points (x₁, y₁) and (x₂, y₂), it's the same thing: (y₂ − y₁)/(x₂ − x₁), the slope between them.
Worked examples
f(x) = x² on [1, 3]. f(1) = 1 and f(3) = 9, so the average rate of change is (9 − 1)/(3 − 1) = 8/2 = 4. The secant line through (1, 1) and (3, 9) is y = 4x − 3.
f(x) = sin x on [0, π/2]. f(0) = 0 and f(π/2) = 1, so the rate is 1/(π/2) = 2/π ≈ 0.63662. The calculator keeps exact values like π and fractions when it can.
Two points (1, 2) and (5, 10). Δy = 8 and Δx = 4, so the rate is 2: y goes up 2 for every 1 that x goes up.
Average vs instantaneous rate
The average rate covers a whole interval. The instantaneous rate at one point is the derivative f′(x), the limit of the average rate as the interval shrinks to zero width. The mean value theorem says that for a smooth function there is at least one point inside the interval where the instantaneous rate equals the average rate. Find derivatives with the derivative calculator, or watch the average rate approach the derivative in the difference quotient calculator.
Limitations
- The average hides what happens in between. f(x) = x³ − 3x on [−2, 2] has an average rate of 1, even though it rises, falls and rises again.
- f must be defined at both ends. The calculator checks the endpoints only, so a gap or jump in the middle (like 1/x on [−1, 1]) won't stop it, but the answer then says little about the function.
- a and b must differ: with a = b there's no change in x to divide by.