Given:
Consider the function is:
[tex]f(x)=\dfrac{x^2}{3}[/tex]
To find:
The average rate of change over the interval 2 ≤ x ≤ 4.
Solution:
We have,
[tex]f(x)=\dfrac{x^2}{3}[/tex]
At [tex]x=2[/tex],
[tex]f(2)=\dfrac{2^2}{3}[/tex]
[tex]f(2)=\dfrac{4}{3}[/tex]
At [tex]x=4[/tex],
[tex]f(4)=\dfrac{4^2}{3}[/tex]
[tex]f(4)=\dfrac{16}{3}[/tex]
The average rate of change of a function f(x) over the interval [a,b] is:
[tex]m=\dfrac{f(b)-f(a)}{b-a}[/tex]
So, the average rate of change over the interval 2 ≤ x ≤ 4 is:
[tex]m=\dfrac{f(4)-f(2)}{4-2}[/tex]
[tex]m=\dfrac{\dfrac{16}{3}-\dfrac{4}{3}}{2}[/tex]
[tex]m=\dfrac{\dfrac{16-4}{3}}{2}[/tex]
On further simplification, we get
[tex]m=\dfrac{12}{3\times 2}[/tex]
[tex]m=\dfrac{12}{6}[/tex]
[tex]m=2[/tex]
Therefore, the average rate of change over the interval 2 ≤ x ≤ 4 is 2.
