Respuesta :
Answer:
The average rate of change of [tex]f\left ( x \right )[/tex] is 34
Step-by-step explanation:
The formula for average rate of change is given as,
[tex] Average\:rate\:of\:change=\dfrac{f\left(b\right)-f\left(a\right)}{b-a}[/tex]
Given that [tex]f\left ( x \right )=x^{3}-9x[/tex]
The [tex]f\left ( x \right )[/tex] lies between interval 1 and 6. So a = 1 and b = 6.
Substituting the values in the formula,
[tex] Average\:rate\:of\:change=\dfrac{f\left(6\right)-f\left(1\right)}{6-1}[/tex]
Now to calculate [tex]f\left ( a \right )[/tex] and [tex]f\left ( b \right )[/tex].
To calculate [tex]f\left ( a \right )[/tex]
[tex]f\left ( a \right )=f\left ( 1 \right )=1^{3}-9\left ( 1 \right )[/tex]
[tex]\therefore f\left ( 1 \right )=1-9[/tex]
[tex]\therefore f\left ( 1 \right )=-8[/tex]
To calculate [tex]f\left ( b \right )[/tex]
[tex]f\left ( b \right )=f\left ( 6 \right )=6^{3}-9\left ( 6 \right )[/tex]
[tex]\therefore f\left ( 6 \right )=216-54[/tex]
[tex]\therefore f\left ( 6 \right )=162[/tex]
Substituting the values in the formula,
[tex]Average\:rate\:of\:change=\dfrac{162-\left(-8\right)}{6-1}[/tex]
[tex] Average\:rate\:of\:change=\dfrac{162+8}{6-1}[/tex]
[tex] Average\:rate\:of\:change=\dfrac{170}{5}[/tex]
[tex] Average\:rate\:of\:change=34[/tex]
