Remember that the average rate of change of a function over an interval is the slope of the straight line connecting the end points of the interval. To find those slopes, we are going to use the slope formula: [tex]m= \frac{y_{2}-y_{1}}{x_2-x_1} [/tex]
Rate of change of [tex]a[/tex]:
From the graph we can infer that the end points are (0,1) and (2,4). So lets use our slope formula to find the rate of change of [tex]a[/tex]:
[tex]m= \frac{y_{2}-y_{1}}{x_2-x_1} [/tex]
[tex]m= \frac{4-1}{2-0} [/tex]
[tex]m= \frac{3}{2} [/tex]
[tex]m=1.5[/tex]
The average rate of change of the function [tex]a[/tex] over the interval [0,2] is 1.5
Rate of change of [tex]b[/tex]:
Here the end points are (0,0) and (2,2)
[tex]m= \frac{2-0}{2-0}[/tex]
[tex]m= \frac{2}{2} [/tex]
[tex]m=1[/tex]
The average rate of change of the function [tex]b[/tex] over the interval [0,2] is 1
Rate of change of [tex]c[/tex]:
Here the end points are (0,-1) and (2,0)
[tex]m= \frac{0-(-1)}{2-0}[/tex]
[tex]m= \frac{1}{2} [/tex]
[tex]m=0.5[/tex]
The average rate of change of the function [tex]c[/tex] over the interval [0,2] is 0.5
Rate of change of [tex]d[/tex]:
Here the end points are (0,0.5) and (2,2.5)
[tex]m= \frac{2.5-0.5}{2-0} [/tex]
[tex]m= \frac{2}{2} [/tex]
[tex]m=1[/tex]
The average rate of change of the function [tex]d[/tex] over the interval [0,2] is 1
We can conclude that the function that has the greatest rate of change over the interval [0, 2] is the function a.
