Chapter 6.5, Problem 8E

(a)

To determine

To find: The average value of the function $f\left(x\right)=\frac{1}{x}$.

Answer to Problem 8E

The average value of the function is 0.549.

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{1}{x}$.

Calculation:

Show the function as follows:

$f\left(x\right)=\frac{1}{x}$ (1)

The expression to find the average value of the function $\left(f_{\text{avg}}\right)$ is shown below:

$f_{\text{avg}}=\frac{1}{b-a}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$ (2)

Here, the upper limit is b, the lower limit is a, and the function is $f\left(x\right)$.

Substitute 1 for a, 3 for b, and $\frac{1}{x}$ for $f\left(x\right)$ in Equation (2).

$f_{\text{ave}}=\frac{1}{3-1}{\displaystyle \underset{1}{\overset{3}{\int }}\frac{1}{x}}dx$ (3)

Integrate Equation (3) with respect to x and apply the limits.

$\begin{array}{c}{f}_{\text{ave}}=\frac{1}{2}{\left[\ln x\right]}_1^3\\ =\frac{1}{2}\left[\ln 3-\ln 1\right]\\ =\frac{1}{2}\left[1.0986\right]\\ =0.549\end{array}$

Thus, the average value of the function is 0.549.

(b)

To determine

To find: The value of count c.

Answer to Problem 8E

The value of c is 1.82.

Explanation of Solution

Given information:

The function is $f\left(x\right)=\frac{1}{x}$.

Calculation:

Find the value of c using the formula:

$f\left(c\right)=f_{\text{avg}}$ (4)

Substitute 0.549 for $f_{\text{ave}}$ in Equation (4).

$f\left(c\right)=0.549$ (5)

Substitute c for x in Equation (1)

$f\left(c\right)=\frac{1}{c}$ (6)

Substitute 0.549 for $f\left(c\right)$ in Equation (5)

$\begin{array}{l}0.549=\frac{1}{c}\\ c=1.82\end{array}$

Thus, the value of c is 1.82.

(c)

To determine

To draw: The graph of the function f.

Explanation of Solution

Substitute 1.82 for x in Equation (1)

$\begin{array}{c}y=\frac{1}{x}\\ =\frac{1}{1.82}\\ =0.549\end{array}$

Draw the graph for the curve function $y=\frac{1}{x}$ and mark the coordinates $\left(1.82,0.549\right)$ as shown in Figure 1.