Answer
The average rate of change of $f\left( x \right)=3{{x}^{2}}-5$ from ${{x}_{1}}=6$ to ${{x}_{2}}=10$ is $\underline{48}$.
Work Step by Step
Calculate the values of $f\left( x \right)$ at both points as shown below:
$\begin{align}
& f\left( {{x}_{2}} \right)=f\left( 10 \right) \\
& f\left( {{x}_{1}} \right)=f\left( 6 \right)
\end{align}$
$\begin{align}
& f\left( 10 \right)=3{{\left( 10 \right)}^{2}}-5 \\
& =300-5 \\
& =295.
\end{align}$
$\begin{align}
& f\left( 6 \right)=3{{\left( 6 \right)}^{2}}-5 \\
& =108-5 \\
& =103.
\end{align}$
Now, to obtain the average rate of change, put the values in the formula.
$\begin{align}
& \text{Average rate of change of }f\left( x \right)=\frac{f\left( 10 \right)-f\left( 6 \right)}{10-6} \\
& =\frac{295-103}{10-6} \\
& =\frac{192}{4} \\
& =48
\end{align}$
Therefore, the average rate of change of $f$ from ${{x}_{1}}=6$ to ${{x}_{2}}=10$ is $48$.
