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

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

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

### Step 1: Calculate \( f(a) \) and \( f(b) \)
- \( f(a) = f(0) = \cos(2 \times 0) = \cos(0) = 1 \)
- \( f(b) = f\left(\frac{\pi}{2}\right) = \cos\left(2 \times \frac{\pi}{2}\right) = \cos(\pi) = -1 \)

### Step 2: Apply the formula
Now we substitute these values into the formula for average rate of change:

\[
\text{Average Rate of Change} = \frac{-1 - 1}{\frac{\pi}{2} - 0} = \frac{-2}{\frac{\pi}{2}} = \frac{-2 \times 2}{\pi} = \frac{-4}{\pi}
\]

Thus, the average rate of change is:

\[
\boxed{-\frac{4}{\pi}}
\]

This matches option (A).

Would you like more details, or do you have any questions? 

Here are five related questions:
1. What is the geometric meaning of the average rate of change?
2. How would the average rate of change differ for \( y = \sin(2x) \) on the same interval?
3. Can the average rate of change ever be zero for a trigonometric function?
4. What is the derivative of \( y = \cos(2x) \) and how does it relate to the average rate of change?
5. How do you interpret the negative sign in the average rate of change?

**Tip**: The average rate of change gives the slope of the secant line between two points on the graph of the function.

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 y = cos(2x) on the interval [0, π/2]. This step-by-step solution covers key trigonometric concepts, including the cosine function and how to apply the formula for the average rate of change.

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