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To find the average rate of change of the function \( f(x) = \cos\left(\frac{x}{2}\right) \) from \( x = \pi \) to \( x = \frac{11\pi}{3} \), we can use the formula for the average rate of change:

\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]

where:
- \( a = \pi \)
- \( b = \frac{11\pi}{3} \)

Let's compute \( f(a) \) and \( f(b) \):

### Step 1: Evaluate \( f(a) = f(\pi) \)
\[
f(\pi) = \cos\left(\frac{\pi}{2}\right)
\]
Since \( \cos\left(\frac{\pi}{2}\right) = 0 \), we have:
\[
f(\pi) = 0
\]

### Step 2: Evaluate \( f(b) = f\left(\frac{11\pi}{3}\right) \)
\[
f\left(\frac{11\pi}{3}\right) = \cos\left(\frac{\frac{11\pi}{3}}{2}\right) = \cos\left(\frac{11\pi}{6}\right)
\]
The value of \( \cos\left(\frac{11\pi}{6}\right) \) is \( \frac{\sqrt{3}}{2} \), so:
\[
f\left(\frac{11\pi}{3}\right) = \frac{\sqrt{3}}{2}
\]

### Step 3: Substitute into the Average Rate of Change Formula
\[
\text{Average rate of change} = \frac{f\left(\frac{11\pi}{3}\right) - f(\pi)}{\frac{11\pi}{3} - \pi}
\]
Substitute the values:
\[
= \frac{\frac{\sqrt{3}}{2} - 0}{\frac{11\pi}{3} - \pi}
\]
Simplify the denominator:
\[
= \frac{\frac{\sqrt{3}}{2}}{\frac{11\pi - 3\pi}{3}} = \frac{\frac{\sqrt{3}}{2}}{\frac{8\pi}{3}}
\]
To simplify, multiply by the reciprocal of the denominator:
\[
= \frac{\sqrt{3}}{2} \cdot \frac{3}{8\pi} = \frac{3\sqrt{3}}{16\pi}
\]

### Final Answer
\[
\text{The average rate of change is } \frac{3\sqrt{3}}{16\pi}
\]

Would you like further details or have any questions?

Here are 5 additional questions to explore:
1. How is the average rate of change related to the derivative for continuous functions?
2. What is the significance of the cosine function's behavior over intervals in finding rates of change?
3. Can we interpret the average rate of change geometrically?
4. How would the solution change if we considered a sine function instead?
5. What are the implications if \( f(x) \) were periodic over the interval \( [\pi, \frac{11\pi}{3}] \)?

**Tip:** For trigonometric functions, remember to simplify angles to common values when possible, as this often reveals known cosine or sine values.

Find the average rate of change of f from π to 11π/3, where f(x) = cos(x/2).

Learn how to find the average rate of change of the function f(x) = cos(x/2) from x = π to x = 11π/3. This solution involves trigonometric functions, cosine values at specific angles, and the formula for the average rate of change. Ideal for students in Grades 10-12.

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