# Rate of change of a function

The rate of change of a function indicates the rate at which the values of the function change

Example 1

Below is a function graph that describes Liisa-Petter's mood as a function of time. Determine the average rate of change between points A and B. Point A is at time 3h and the mood value is 11. Point B is at time 10h and the mood value is 39. Draw a line through the points. The average rate of change is described by the slope of this line. On average, Liisa-Petter's mood rises by 4 units per hour between 3h and 10h.

If we want to know the instantaneous rate of change at both point A and point B, we set tangents to the points, that is, lines that intersect the function only at that point. Select two points on the lines and find the slope. At point A, the instantaneous rate of change is 0.83 units/h and at point B, the instantaneous rate of change is 11 units/h. The steeper the graph of a function, the faster its values change.

Example 2

Liisa-Petter's brain activity is described by the function f. The function variable x is the time in hours of awakening. Find the average rate of change in brain activity between the waking hours of 1 and 4. Brain activity increases by an average of 8 units per hour between waking hours 1 and 3.

Determine the average rate of change between the waking hours of 10 and 11. The rate of change is negative, meaning that brain activity decreases by an average of 9 units per hour.

The average rate of change of a function in interval [a, b] 