Answer
$$
y=f(x)=6x^{3}+2 \quad \text { from} \quad x=1\quad \text { to} \quad x=4
$$
The average rate of change of $ f(x) $ with respect to $x$ as $x$ changes from $a=1 $ to $ b=4$ is
$$
\begin{split}
\text {Average rate of change}
&=\frac{f(b)-f(a)}{b-a}\\
& =\frac{f(4)-f(1)}{4-1}\\
& =126\\
\end{split}
$$
Instantaneous rate of change at $x= 1$:
$$
y^{\prime}(1)=18(1)^{2}=18
$$
Work Step by Step
$$
y=f(x)=6x^{3}+2 \quad \text { from} \quad x=1\quad \text { to} \quad x=4
$$
The average rate of change of $ f(x) $ with respect to $x$ as $x$ changes from $a=1 $ to $ b=4$ is
$$
\begin{split}
\text {Average rate of change}
&=\frac{f(b)-f(a)}{b-a}\\
& =\frac{f(4)-f(1)}{4-1}\\
& =\frac{(6(4)^{3}+2)-(6(1)^{3}+2)}{4-1}\\
& =\frac{(384+2)-(6+2)}{3}\\
& =\frac{(386)-(8)}{3}\\
& =\frac{(378)}{3}\\
& =126\\
\end{split}
$$
and we can find that :
$$
y^{\prime}(x)=18x^{2}
$$
Instantaneous rate of change at $x= 1$:
$$
y^{\prime}(1)=18(1)^{2}=18
$$
