f(x)=x^{3}-9x\nwhat is the average rate of change of f over the interval 1,6?

Answer

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 1$, $b = 6$, and $f(x)=x^{3}-9x$.

Step2: Calculate $f(6)$

Substitute $x = 6$ into $f(x)$: $f(6)=6^{3}-9\times6=216 - 54=162$.

Step3: Calculate $f(1)$

Substitute $x = 1$ into $f(x)$: $f(1)=1^{3}-9\times1=1 - 9=-8$.

Step4: Calculate the average rate of change

$\frac{f(6)-f(1)}{6 - 1}=\frac{162-(-8)}{5}=\frac{162 + 8}{5}=\frac{170}{5}=34$.