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

Here, $a = -\frac{\pi}{2}$, $b = \frac{\pi}{2}$, and $f(x) = -\cos x$. Substituting into the formula:

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

The length of the interval is:

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

So the formula simplifies to:

$\text{Average value} = \frac{1}{\pi} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} -\cos x \, dx$

Step 2: Compute the integral

The integral of $-\cos x$ is:

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

Now evaluate the definite integral:

$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} -\cos x \, dx = \left[-\sin x\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}$

Substitute the bounds:

$-\sin\left(\frac{\pi}{2}\right) - \left(-\sin\left(-\frac{\pi}{2}\right)\right)$

Since $\sin\left(\frac{\pi}{2}\right) = 1$ and $\sin\left(-\frac{\pi}{2}\right) = -1$, we have:

$-\sin\left(\frac{\pi}{2}\right) - (-\sin\left(-\frac{\pi}{2}\right)) = -1 - (-(-1)) = -1 + 1 = 0$

Step 3: Multiply by $\frac{1}{\pi}$

The average value is:

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

Final Answer:

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

$\boxed{0}$

Graph and Explanation:

I'll now plot the graph of $f(x) = -\cos x$ over the interval $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$ and indicate the average value.import numpy as np import matplotlib.pyplot as plt

Define the function

def f(x): return -np.cos(x)

Define the interval

x = np.linspace(-np.pi/2, np.pi/2, 500) y = f(x)

Plot the function

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r"$f(x) = -\cos(x)$") plt.axhline(y=0, color='red', linestyle='--', label="Average Value (0)") plt.title("Graph of $f(x) = -\cos(x)$ with Average Value Indicated") plt.xlabel("$x$") plt.ylabel("$f(x)$") plt.axhline(0, color='black', linewidth=0.8, linestyle=':') # x-axis plt.axvline(0, color='black', linewidth=0.8, linestyle=':') # y-axis plt.legend() plt.grid(True) plt.show() <<ImageDisplayed>>successHere is the graph of $f(x) = -\cos(x)$ over the interval $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$. The red dashed line represents the average value, which is $0$.

Let me know if you'd like further clarification or additional computations.

Related Questions:

1. What is the physical interpretation of the average value of a function?
2. Can we compute the average value of other trigonometric functions over similar intervals?
3. How would the average value change if the interval is shifted or widened?
4. What happens to the average value of $f(x) = -\cos(x)$ over an interval longer than $2\pi$?
5. How does symmetry of the function influence the average value?

Tip:

The integral of an odd function over a symmetric interval $[-a, a]$ is always $0$, as their contributions on the left and right sides cancel out. This can save computation in many cases.