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To calculate the average rate of change of the function \( y = \cos(2x) \) on the interval \([0, \frac{\pi}{2}]\), we use the formula for the average rate of change:

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

Where \( f(x) = \cos(2x) \), and \( a = 0 \) and \( b = \frac{\pi}{2} \).

1. **Evaluate \( f(a) \):**
   \[
   f(0) = \cos(2 \times 0) = \cos(0) = 1
   \]

2. **Evaluate \( f(b) \):**
   \[
   f\left(\frac{\pi}{2}\right) = \cos\left(2 \times \frac{\pi}{2}\right) = \cos(\pi) = -1
   \]

3. **Apply the formula:**
   \[
   \text{Average Rate of Change} = \frac{f\left(\frac{\pi}{2}\right) - f(0)}{\frac{\pi}{2} - 0} = \frac{-1 - 1}{\frac{\pi}{2}} = \frac{-2}{\frac{\pi}{2}} = -\frac{4}{\pi}
   \]

The average rate of change is \( -\frac{4}{\pi} \).

Thus, the correct answer is **A**: \( -\frac{4}{\pi} \).

Would you like more details on any part of this? Here are some related questions for further practice:

1. What is the derivative of \( y = \cos(2x) \) and how does it relate to the rate of change?
2. How does the average rate of change differ from the instantaneous rate of change?
3. How do you compute the average rate of change for \( y = \sin(2x) \) on the same interval?
4. What would the average rate of change be on a different interval, like \( [0, \pi] \)?
5. How does the graph of \( y = \cos(2x) \) help in visualizing the average rate of change?

**Tip**: The average rate of change over an interval is the slope of the secant line connecting the two endpoints of the curve on that interval.

What is the average rate of change of y = cos(2x) on the interval [0, π/2]?

Learn how to calculate the average rate of change of the function y = cos(2x) on the interval [0, π/2]. This problem involves key trigonometric concepts and the formula for the average rate of change, suitable for students in Grades 10-12.

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