Answer
The average rate of change of $f\left( x \right)={{x}^{2}}+2x$ from ${{x}_{1}}=3\text{ to }{{x}_{2}}=\text{5}$ is $10$.
Work Step by Step
Consider the provided function $f\left( x \right)={{x}^{2}}+2x$.
The value of the function $f\left( x \right)={{x}^{2}}+2x$ at ${{x}_{1}}=3$ is
$\begin{align}
& f\left( 3 \right)={{\left( 3 \right)}^{2}}+2\left( 3 \right) \\
& =9+6 \\
& =15
\end{align}$
The value of function $f\left( x \right)={{x}^{2}}+2x$ at ${{x}_{2}}=5$ is
$\begin{align}
& f\left( 5 \right)={{\left( 5 \right)}^{2}}+2\left( 5 \right) \\
& =25+10 \\
& =35
\end{align}$
The average rate of change of $f$ from ${{x}_{1}}=3\text{ to }{{x}_{2}}=\text{5}$ is
$\begin{align}
& \frac{\Delta y}{\Delta x}=\frac{f\left( 5 \right)-f\left( 3 \right)}{5-3} \\
& =\frac{35-15}{5-3} \\
& =\frac{20}{2} \\
& =10
\end{align}$
Thus, the average rate of change of $f\left( x \right)={{x}^{2}}+2x$ from ${{x}_{1}}=3\text{ to }{{x}_{2}}=\text{5}$ is $10$.
