To solve for the average value of $\cos(x)$ on the interval $[-2, 7]$, we use the formula for the average value of a continuous function $f(x)$ over an interval $[a, b]$:

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

In this case, $f(x) = \cos(x)$, $a = -2$, and $b = 7$. Plugging these into the formula, we have:

$\text{Average value of } \cos(x) = \frac{1}{7 - (-2)} \int_{-2}^7 \cos(x) \, dx$

Simplifying:

$\text{Average value of } \cos(x) = \frac{1}{9} \int_{-2}^7 \cos(x) \, dx$

Now, we calculate the integral of $\cos(x)$:

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

So, we evaluate $\sin(x)$ at the limits:

$\left[ \sin(x) \right]_{-2}^7 = \sin(7) - \sin(-2)$

Since $\sin(-x) = -\sin(x)$, this becomes:

$\sin(7) + \sin(2)$

Now the average value is:

$\frac{1}{9} \left( \sin(7) + \sin(2) \right)$

Using approximate values for $\sin(7) \approx 0.657$ and $\sin(2) \approx 0.909$, we get:

$\text{Average value} \approx \frac{1}{9} \times (0.657 + 0.909) = \frac{1}{9} \times 1.566 \approx 0.174$

So, the average value of $\cos(x)$ on the interval $[-2, 7]$ is approximately 0.174.

Would you like further details or have any questions?

Here are 5 related questions:

1. What is the geometric meaning of the average value of a function?
2. How does the average value of $\sin(x)$ differ from that of $\cos(x)$?
3. Can the average value formula be used for non-trigonometric functions? How?
4. How does changing the interval affect the average value of $\cos(x)$?
5. What is the average value of $\cos(x)$ on $[0, 2\pi]$?

Tip: Understanding how to evaluate definite integrals is crucial for finding the average value of a function.