Question

How do you find the average rate of change of $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2+\mathrm{x};\mathrm{x}\in 3\mathrm{to}6$

Answer 1

Step-1: Find the values of $\mathrm{f}\left(a\right)$ and $\mathrm{f}\left(b\right)$.

The formula for the average rate change of $\mathrm{f}\left(\mathrm{x}\right)$, from $\mathrm{x}=\mathrm{a}\mathrm{to}\mathrm{x}=\mathrm{b}$ is $\mathrm{Average}\mathrm{rate}=\frac{f\left(b\right)-f\left(a\right)}{b-a}$ and here, $\mathrm{b}=6,\mathrm{a}=3$.

Now,

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2+\mathrm{x} \\ \Rightarrow \mathrm{f}\left(3\right)=3^2+3 \\ \therefore \mathrm{f}\left(3\right)=12 \end{gathered}$$

And,

$$\begin{gathered} \mathrm{f}\left(6\right)=6^2+6 \\ \Rightarrow \mathrm{f}\left(6\right)=36+6 \\ \therefore \mathrm{f}\left(6\right)=42 \end{gathered}$$

Step-2: Find the average rate change of $\mathrm{f}\left(\mathrm{x}\right)$.

Substitute the given values in the above formula to find the average value:

$\begin{array}{rcl}\mathrm{Average}\mathrm{value}& =& \frac{f\left(b\right)-f\left(a\right)}{b-a}\\ & =& \frac{f\left(6\right)-f\left(3\right)}{6-3}\\ & =& \frac{42-12}{3}\\ & =& \frac{30}{3}\\ & =& 10\end{array}$

Hence, the average rate change of $\mathrm{f}\left(\mathrm{x}\right)\mathit{}$is $10$.