To find the average value of $f(x) = -\cos(x)$ over the interval $\left[ -\frac{\pi}{4}, \frac{\pi}{4} \right]$, we use the formula for the average value of a function:

$\text{Average Value} = \frac{1}{b-a} \int_a^b f(x) \, dx$

Step 1: Set up the integral

Here, $a = -\frac{\pi}{4}$, $b = \frac{\pi}{4}$, and $f(x) = -\cos(x)$. The formula becomes:

$\text{Average Value} = \frac{1}{\frac{\pi}{4} - \left(-\frac{\pi}{4}\right)} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} -\cos(x) \, dx$

Simplify $b - a$:

$b - a = \frac{\pi}{4} - \left(-\frac{\pi}{4}\right) = \frac{\pi}{2}$

Thus, the formula becomes:

$\text{Average Value} = \frac{1}{\frac{\pi}{2}} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} -\cos(x) \, dx$

Step 2: Solve the integral

The integral of $-\cos(x)$ is:

$\int -\cos(x) \, dx = -\sin(x)$

Evaluate the definite integral:

$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} -\cos(x) \, dx = -\sin\left(\frac{\pi}{4}\right) - \left(-\sin\left(-\frac{\pi}{4}\right)\right)$

Using $\sin(-x) = -\sin(x)$:

$-\sin\left(\frac{\pi}{4}\right) - \left(-\sin\left(-\frac{\pi}{4}\right)\right) = -\sin\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{4}\right)$

Thus:

$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} -\cos(x) \, dx = 0$

Step 3: Calculate the average value

Substitute into the formula:

$\text{Average Value} = \frac{1}{\frac{\pi}{2}} \cdot 0 = 0$

Final Answer:

$\text{The average value is } 0.$

Graph:

I will now generate a graph of $f(x) = -\cos(x)$ over the interval $\left[ -\frac{\pi}{4}, \frac{\pi}{4} \right]$ and indicate the average value.import numpy as np import matplotlib.pyplot as plt

Define the function

x = np.linspace(-np.pi/4, np.pi/4, 500) y = -np.cos(x)

Plot the function

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r"$f(x) = -\cos(x)$", linewidth=2) plt.axhline(0, color="red", linestyle="--", label="Average Value = 0") plt.axvline(-np.pi/4, color="gray", linestyle=":", label=r"$x = -\frac{\pi}{4}$") plt.axvline(np.pi/4, color="gray", linestyle=":", label=r"$x = \frac{\pi}{4}$")

Add labels and legend

plt.title("Graph of $f(x) = -\cos(x)$ with Average Value Indicated") plt.xlabel("$x$") plt.ylabel("$f(x)$") plt.legend() plt.grid() plt.show() <<ImageDisplayed>>successThe graph of $f(x) = -\cos(x)$ over the interval $\left[ -\frac{\pi}{4}, \frac{\pi}{4} \right]$ is shown above. The red dashed line at $y = 0$ represents the average value of the function.

Do you want further details or have any questions?

Related Questions:

1. How does the average value change if the interval is expanded?
2. What is the geometric interpretation of the average value in terms of the graph?
3. Can the average value of a function be negative? Why or why not?
4. How would the calculation differ if the function were $\cos(x)$ instead of $-\cos(x)$?
5. How do you compute the average value if the function has discontinuities?

Tip:

Always double-check the symmetry of the function and interval—it can simplify the computation significantly.