find the average rate of change for the function over the given interval. y=x³ + x² - 8x - 7 between x = 0…

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, $a = 0$, $b = 2$, and $f(x)=x^{3}+x^{2}-8x - 7$.

Step2: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$: [ \begin{align*} f(2)&=(2)^{3}+(2)^{2}-8\times2 - 7\ &=8 + 4-16 - 7\ &=12-16 - 7\ &=-4 - 7\ &=-11 \end{align*} ]

Step3: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$: [ \begin{align*} f(0)&=(0)^{3}+(0)^{2}-8\times0 - 7\ &=-7 \end{align*} ]

Step4: Calculate the average rate of change

[ \begin{align*} \frac{f(2)-f(0)}{2 - 0}&=\frac{-11-(-7)}{2}\ &=\frac{-11 + 7}{2}\ &=\frac{-4}{2}\ &=-2 \end{align*} ]

Answer:

B. -2