The averate rate of change of a function from x = 2 to x = 5 is the ratio of f(5) - f(2) to 5-2 =3, where f(x) is the considered function.
Let the thing that is changing be y and the thing with which the rate is being compared is x, then we have the average rate of change of y as x changes as:
[tex]\text{Average rate} = \dfrac{y_2 - y_1}{x_2 - x_1}[/tex]
where when
[tex]x = x_1, y = y_1\\and \\x = x_2, y = y_2[/tex]
This represents that the independent variable y changes averagely [tex]y_2 -y_1[/tex] units per [tex]x_2 - x_1[/tex] change in the independent variable x. Or, that the independent variable y changes averagely [tex](y_2 -y_1)/(x_2-x_1)[/tex] units per unit (1) change in variable x.
For this case, the dependent variable is y = f(x) where f(x) represents the considered function.
As we have:
[tex]x = 2, y = f(2)\\and \\x = 5, y = f(5)[/tex]
Thus, we get:
[tex]\text{Average rate} = \dfrac{f(5) - f(2)}{5-2} = \dfrac{f(5) - f(2)}{3}[/tex]
Thus, the averate rate of change of a function from x = 2 to x = 5 is the ratio of f(5) - f(2) to 5-2 =3, or say,
[tex]\text{Average rate} = \dfrac{f(5) - f(2)}{5-2} = \dfrac{f(5) - f(2)}{3}[/tex]
where f(x) is the considered function.
This represents that the function f(x) changes averagely [tex]f(5) - f(2)[/tex] units per 3 units change in the independent variable x. Or, that the function f(x) changes averagely [tex](f(5) - f(2) )/3[/tex] units per unit (1) change in variable x.
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