$$ 1

Question

$$ 1< $$ Find the average value of $$ f(x)=-\cos x $$ over the interval $$ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $$. Draw a graph of the function and indicate the average value.

UpStudy AI · Accepted Answer

To find the average value of $$ f(x)=-\cos x $$ on the interval $$ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $$, we use the formula for the average value of a function:

$$
f_{\text{avg}}=\frac{1}{b-a}\int_{a}^{b} f(x)\,dx
$$

Here, $$ a=-\frac{\pi}{2} $$ and $$ b=\frac{\pi}{2} $$, so the length of the interval is

$$
b-a=\frac{\pi}{2}-\left(-\frac{\pi}{2}\right)=\pi.
$$

Substitute $$ f(x)=-\cos x $$ and the limits into the formula:

$$
f_{\text{avg}}=\frac{1}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(-\cos x\right)\,dx.
$$

Factor out the constant $$-1$$:

$$
f_{\text{avg}}=\frac{-1}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x\,dx.
$$

Compute the integral of $$ \cos x $$:

$$
\int \cos x\,dx=\sin x.
$$

Evaluate the definite integral:

$$
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos x\,dx=\sin x \Big|_{-\frac{\pi}{2}}^{\frac{\pi}{2}}=\sin\left(\frac{\pi}{2}\right)-\sin\left(-\frac{\pi}{2}\right)=1-(-1)=2.
$$

Substitute back into the expression for the average value:

$$
f_{\text{avg}}=\frac{-1}{\pi}(2)=-\frac{2}{\pi}.
$$

Thus, the average value of $$ f(x)=-\cos x $$ over the interval $$ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $$ is

$$
-\frac{2}{\pi}.
$$

Below is a sketch of the graph of $$ f(x)=-\cos x $$:

- The graph of $$ \cos x $$ is a wave starting at $$1$$ when $$ x=0 $$ and decreasing towards $$0$$ at $$ \pm \frac{\pi}{2} $$.
- Inverting the graph with a negative sign, $$ -\cos x $$ starts at $$-1$$ when $$ x=0 $$ and increases towards $$ 0 $$ at $$ x=\pm\frac{\pi}{2} $$.

The sketch with the average value line is shown below:

```
         f(x)
          |
      0   |          --------
          |         /        \
 -0.5     |   ------+--------+------
          |         \        /
      -1  |----------\______/-----
          |
          |-------------------------- x
              -π/2   0   π/2
          
Average value: y = -2/π ≈ -0.64 (drawn as a horizontal line intersecting the graph)
```

On the graph, the horizontal line $$ y=-\frac{2}{\pi} $$ (approximately $$ y=-0.64 $$) represents the average value of the function over the interval.
The average value of $$ f(x)=-\cos x $$ over the interval $$ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $$ is $$ -\frac{2}{\pi} $$.

Find the average value of over the interval . Draw a graph of the function and indicate the average value.

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