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To find the **average rate of change** of a function over a given interval, we use the formula:

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

where \( f(t) \) is the function, and \( [a, b] \) is the interval.

The function provided is \( f(t) = 11 + \cos(t) \), and we are asked to find the average rate of change over two intervals:

### Part (a): Interval \([0, \frac{\pi}{2}]\)

1. Find \( f(0) \) and \( f\left( \frac{\pi}{2} \right) \):
   \[
   f(0) = 11 + \cos(0) = 11 + 1 = 12
   \]
   \[
   f\left( \frac{\pi}{2} \right) = 11 + \cos\left( \frac{\pi}{2} \right) = 11 + 0 = 11
   \]

2. Apply the average rate of change formula:
   \[
   \frac{f\left( \frac{\pi}{2} \right) - f(0)}{\frac{\pi}{2} - 0} = \frac{11 - 12}{\frac{\pi}{2}} = \frac{-1}{\frac{\pi}{2}} = \frac{-2}{\pi}
   \]

### Part (b): Interval \([- \pi, \pi]\)

1. Find \( f(-\pi) \) and \( f(\pi) \):
   \[
   f(-\pi) = 11 + \cos(-\pi) = 11 + (-1) = 10
   \]
   \[
   f(\pi) = 11 + \cos(\pi) = 11 + (-1) = 10
   \]

2. Apply the average rate of change formula:
   \[
   \frac{f(\pi) - f(-\pi)}{\pi - (-\pi)} = \frac{10 - 10}{2\pi} = \frac{0}{2\pi} = 0
   \]

### Summary of Answers:
- Part (a): Average rate of change is \( \frac{-2}{\pi} \).
- Part (b): Average rate of change is 0.

Would you like further details or have any other questions?

Here are 5 related questions to explore further:
1. What is the physical interpretation of the average rate of change for periodic functions?
2. How does the average rate of change relate to the derivative at a point?
3. What happens to the average rate of change for cosine functions over different intervals?
4. How does symmetry in the interval affect the average rate of change?
5. What is the average rate of change of \( \sin(t) \) over similar intervals?

**Tip**: The average rate of change is essentially the slope of the secant line connecting two points on the graph of the function.

Find the average rate of change of the function f(t) = 11 + cos(t) over the given interval.

Find the average rate of change for the function f(t) = 11 + cos(t) over the intervals [0, π/2] and [-π, π]. This solution explains the step-by-step application of the average rate of change formula and utilizes key trigonometric properties of the cosine function. Suitable for high school calculus students.

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