Chapter 6.5, Problem 9E

(a)

To determine

To find: The average value of the function $f\left(x\right)=2\sin x-\sin 2x$

Answer to Problem 9E

The average value of the function is $\frac{4}{\pi }$.

Explanation of Solution

Given information:

The function is $f\left(x\right)=2\sin x-\sin 2x$,$\left[0,\pi \right]$.

Calculation:

Show the function:

$f\left(x\right)=2\sin x-\sin 2x$ (1)

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

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

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

Substitute 0 for a, $\pi$  for b and $2\sin x-\sin 2x$ for $f\left(x\right)$ in Equation (2).

$\begin{array}{c}{f}_{\text{ave}}=\frac{1}{\pi -0}{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(2\sin x-\sin 2x\right)}dx\\ =\frac{1}{\pi }{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(2\sin x-\sin 2x\right)}dx\end{array}$ (3)

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

$\begin{array}{c}{f}_{\text{ave}}=\frac{1}{\pi }{\left[-2\cos x+\frac{1}{2}\cos 2x\right]}_0^{\pi }\\ =\frac{1}{\pi }\left[2+\frac{1}{2}-\left(-2+\frac{1}{2}\right)\right]\\ =\frac{4}{\pi }\end{array}$

Thus, the average value of the function is $\frac{4}{\pi }$.

(b)

To determine

To find: The value of count c.

Answer to Problem 9E

The value of c is 1.238 or 2.808.

Explanation of Solution

Given information:

The function is $f\left(x\right)=2\sin x-\sin 2x$,$\left[0,\pi \right]$.

Calculation:

Find the value of c using the formula:

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

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

$f\left(c\right)=\frac{4}{\pi }$

Substitute c for x in Equation (1)

$f\left(c\right)=2\sin c-\sin 2c$ (5)

Substitute $\frac{4}{\pi }$ for $f\left(c\right)$ in Equation (5).

$2\sin c-\sin 2c=\frac{4}{\pi }$ (6)

Solve Equation (6) using excel solver.

$\begin{array}{c}2\sin c-\sin 2c=\frac{4}{\pi }\\ c=c_1\approx 1.238\\ c=c_2\approx 2.808\end{array}$

Thus, the value of c is 1.238 or 2.808.

(c)

To determine

To draw: The graph of the function f.

Explanation of Solution

Substitute 1.238 for x in Equation (1)

$\begin{array}{c}y=2\sin 1.238-\sin 1.238\\ =1.890-0.945\\ =0.944\end{array}$

Substitute 2.808 for x in Equation (1)

$\begin{array}{c}y=2\sin 2.808-\sin 2.808\\ =0.654-0.327\\ =0.327\end{array}$

Draw the graph for the curve function $y=2\sin x-\sin 2x$ and mark the coordinates (1.238, 0.944) (2.808, 0.327) as shown in Figure 1.