## Precalculus (6th Edition) Blitzer

The average rate of change of $f\left( x \right)={{x}^{2}}+2x$ from ${{x}_{1}}=3\text{ to }{{x}_{2}}=\text{5}$ is $10$.
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 $10$.