Answer
The value of the average rate of change of the function $f\left( x \right)=3{{x}^{2}}-x$ from the point ${{x}_{1}}=-1$ and ${{x}_{2}}=2$ is $2$.
Work Step by Step
We need to calculate the average rate of change of the function $f\left( x \right)=-3{{x}^{2}}-x$.
Use $A=\frac{f\left( {{x}_{2}} \right)-f\left( {{x}_{1}} \right)}{{{x}_{2}}-{{x}_{1}}}$
The value of $f\left( {{x}_{2}} \right)=f\left( 2 \right)$ is:
$\begin{align}
& f\left( 2 \right)=3{{x}^{2}}-x \\
& =3{{\left( 2 \right)}^{2}}-2 \\
& =3\left( 4 \right)-2 \\
& =10
\end{align}$
And $f\left( {{x}_{1}} \right)=f\left( -1 \right)$ is:
$\begin{align}
& f\left( -1 \right)=3{{x}^{2}}-x \\
& =3{{\left( -1 \right)}^{2}}-\left( -1 \right) \\
& =3\left( 1 \right)+1 \\
& =4
\end{align}$
So, the average rate of change is:
$\begin{align}
& A=\frac{f\left( {{x}_{2}} \right)-f\left( {{x}_{1}} \right)}{{{x}_{2}}-{{x}_{1}}} \\
& =\frac{10-4}{2+1} \\
& =\frac{6}{3} \\
& =2
\end{align}$
Therefore, the value of the average rate of change is $2$.
