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To determine the average rate of change of the function \( f(x) = x^3 - 2x - 2 \) over the interval \([-1, 5]\), we use the formula:

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

where \( a = -1 \) and \( b = 5 \).

1. **Evaluate \( f(-1) \):**
   \[
   f(-1) = (-1)^3 - 2(-1) - 2 = -1 + 2 - 2 = -1
   \]

2. **Evaluate \( f(5) \):**
   \[
   f(5) = (5)^3 - 2(5) - 2 = 125 - 10 - 2 = 113
   \]

3. **Calculate the average rate of change:**
   \[
   \text{Average Rate of Change} = \frac{f(5) - f(-1)}{5 - (-1)} = \frac{113 - (-1)}{6} = \frac{113 + 1}{6} = \frac{114}{6} = 19
   \]

So, the average rate of change of the function over the interval \([-1, 5]\) is **19**.

Would you like more details or have any questions?

Here are five related questions you might find interesting:
1. What is the instantaneous rate of change at \( x = 0 \) for the function \( f(x) = x^3 - 2x - 2 \)?
2. How would the average rate of change differ if the interval were \([0, 5]\)?
3. Can the average rate of change be negative for some intervals? How does that relate to the behavior of the function?
4. What is the significance of the point where the derivative equals zero for this function?
5. How does the average rate of change compare to the slope of the secant line on the graph of the function?

**Tip:** The average rate of change of a function over an interval can be interpreted as the slope of the secant line connecting the points corresponding to the interval's endpoints on the graph of the function.

Learn how to calculate the average rate of change of the function f(x) = x^3 - 2x - 2 over the interval [-1, 5]. This problem involves evaluating the function at endpoints, applying the average rate of change formula, and interpreting its significance. Suitable for advanced high school students in Grades 11-12.

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