Calculate the average rate of change of f(x) = 1 x - x2 - 2 for 3 ≤ x ≤ 6. A) -163 18 B) -18 163 C) 163 18 D) 18 163

To solve this we are going to use the average rate of change formula: [tex]A(x)= \frac{f(b)-f(a)}{b-a} [/tex]
where
[tex]A(x)[/tex] is the average rate of change of the function
[tex]f(a)[/tex] is the position function evaluated at [tex]a[/tex]
[tex]f(b)[/tex] is the position function evaluated at [tex]b[/tex]
[tex]a[/tex] is the first point in the interval
[tex]b[/tex] is the second point in the interval

We can infer for our problem that the first point is 3 and the second point is 6, so [tex]a=3[/tex] and [tex]b=6[/tex]. Lets replace those values in our formula:
[tex]A(x)= \frac{f(b)-f(a)}{b-a} [/tex]
[tex]A(x)= \frac{f(6)-f(3)}{6-3} [/tex]
[tex]A(x)= \frac{6-6^2-2-(3-3^2-2)}{3}[/tex]
[tex]A(x)= \frac{-32-(-8)}{3}[/tex]
[tex]A(x)= \frac{-32+8}{3}[/tex]
[tex]A(x)= \frac{-24}{3} [/tex]
[tex]A(x)=-8[/tex]

We can conclude that the average rate of change of the function f(x) = 1 x - x2 - 2 for 3 ≤ x ≤ 6 is -8