Calculus with Applications (10th Edition)

$$y=f(x)=-2x^{3}-3x^{2}+8 \quad \text { from} \quad x=-2\quad \text { to} \quad x=6$$ The average rate of change of $f(x)$ with respect to $x$ as $x$ changes from $a=-2$ to $b=6$ is $$\begin{split} \text {Average rate of change} &=\frac{f(b)-f(a)}{b-a}\\ & =\frac{f(6)-f(-2)}{6-(-2)}\\ & =-68\\ \end{split}$$ Instantaneous rate of change at $x= -2$: $$y^{\prime}(x)=-6(-2)^{2}-6(-2)=6(4)+12=-12$$
$$y=f(x)=-2x^{3}-3x^{2}+8 \quad \text { from} \quad x=-2\quad \text { to} \quad x=6$$ The average rate of change of $f(x)$ with respect to $x$ as $x$ changes from $a=-2$ to $b=6$ is $$\begin{split} \text {Average rate of change} &=\frac{f(b)-f(a)}{b-a}\\ & =\frac{f(6)-f(-2)}{6-(-2)}\\ & =\frac{(-2(6)^{3}-3(6)^{2}+8 )-(-2(-2)^{3}-3(-2)^{2}+8 )}{8}\\ & =\frac{(-532-12)}{8}\\ & =\frac{(-544)}{8}\\ & =-68\\ \end{split}$$ and we can find that : $$y^{\prime}(x)=-6x^{2}-6x$$ Instantaneous rate of change at $x= -2$: $$y^{\prime}(x)=-6(-2)^{2}-6(-2)=6(4)+12=-12$$