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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.

What is the average value of cos(x) on the interval [-2, 7]?

Learn how to calculate the average value of cos(x) over the interval [-2, 7] using calculus. This detailed solution involves evaluating a definite integral and applying the average value formula for continuous functions.

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