As well as the average speed, we can also define the rate of change of a function y=f(x) during an interval [x₁,x₂] . This is expressed as follows:

The graph of the function which was used in the previous exercise to illustrate a car's journey will also be used now to analyse the rate of change of a function y=f(x) during different intervals. The rate of change of the function during the time interval [10,17] is the gradient of the line which cuts through points A and B on the graph. 

1.-  You can see that in both cases the rate of change is continuously changing.

At what points during the interval is the gradient of the line AB at its smallest and greatest?

Can you see a connection between the gradient of the line and the average speed we looked at in the previous exercise?

2.- As the intervals gets shorter and shorter the rate of change usually reaches a fixed value. Try and find this value when x = 15.

The Descartes window makes it easier to work out the rate of change for polynomial functions for zero, first, second and third grade equations. By changing the values of the coefficients a, b, c and d we can obtain the function of a cubic, quadratic or simple equation.

3.- Find the rate of change  of the following functions during the intervals given:

• y=x+2 during the intervals: [0,1] and [-3,-1].

• y=x²-2x-3 during the intervals: [1,3] and [-1,0].

• y=x³-2x²-x+1 during the intervals: [0,2], [-2,3] and [-1,1].