find the average rate of change of the function f(x)= - 1x² - 1x - 7, from x=-1 to x=4. note, the directions…

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, $a=-1$, $b = 4$, and $f(x)=-x^{2}-x - 7$.

Step2: Calculate $f(a)$

Substitute $x=-1$ into $f(x)$: $f(-1)=-(-1)^{2}-(-1)-7=-1 + 1-7=-7$.

Step3: Calculate $f(b)$

Substitute $x = 4$ into $f(x)$: $f(4)=-(4)^{2}-4-7=-16-4-7=-27$.

Step4: Calculate the average rate of change

$\frac{f(4)-f(-1)}{4-(-1)}=\frac{-27-(-7)}{4 + 1}=\frac{-27 + 7}{5}=\frac{-20}{5}=-4$.

Answer:

$-4$