Chapter 3, Problem 1CRE

To determine

Find the average rate of change of $f\left(x\right)$ over $\left[0,2\right]$. Also, write the graphical interpretation of this average rate.

Answer to Problem 1CRE

Solution:

The average rate of change is 3.

Explanation of Solution

Given:

The graph of the function is

Formula used:

The average rate of change of $f\left(x\right)$ over $\left[a,b\right]$ is,

$\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Calculation:

The average rate of change of $f\left(x\right)$ over $\left[0,2\right]$.

The formula is $\frac{f\left(b\right)-f\left(a\right)}{b-a}$.

Where $a=0\text{and}b=2$.

Find the average rate of change.

$\frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{f\left(2\right)-f\left(0\right)}{2-0}$

By the graph, the function is $f\left(2\right)=7\text{and}f\left(0\right)=1$.

$\begin{array}{c}\frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{7-1}{2-0}\\ =\frac{6}{2}\\ =3\end{array}$

By the graphical representation, the average rate of change is the slope of the secant line via the points $\left(2,7\right)\text{and}\left(0,1\right)$.

Hence, the average rate of change is 3.

Conclusion:

With the help of the formula of the average rate of change, we can figure out the average rate of change.