Rate of change of a function

Average speed between two points

When a car travels 120 km in 2 hours, its average speed is 120/2 = 60 km/h. In mathematics, we generalise this idea to any function f using the rate of change.

For a function f and two distinct real numbers a and b (where b is different from a), the rate of change between a and b is:

T = (f(b) - f(a)) / (b - a)

This is the change in f divided by the change in the variable: it measures the average change in f over the interval [a; b].

Graphical interpretation

On the graph of f, this rate corresponds exactly to the slope of the line (secant) connecting the points A(a; f(a)) and B(b; f(b)). The larger this rate, the steeper the line (AB).

Practical example

Let f(x) = x². Let us calculate the rate of change between a = 1 and b = 3:

T = (f(3) - f(1)) / (3 - 1) = (9 - 1) / 2 = 8/2 = 4

a	b	f(a)	f(b)	Rate of change
1	3	1	9	4
2	4	4	16	6
0	2	0	4	2

Common pitfall

Do not confuse f(b) – f(a) (change in the function) with b – a (change in the variable): the order must be the same in both differences, otherwise the sign of the result will be incorrect.