To find the average rate of change of the function $f(x) = -5x^3 - 4x^2 - 4x - 2$ from $x = -1$ to $x = 1$, we use the formula:

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

where $a = -1$ and $b = 1$.

Let's calculate:

1. Find $f(1)$: $f(1) = -5(1)^3 - 4(1)^2 - 4(1) - 2 = -5 - 4 - 4 - 2 = -15$

2. Find $f(-1)$: $f(-1) = -5(-1)^3 - 4(-1)^2 - 4(-1) - 2 = 5 - 4 + 4 - 2 = 3$

3. Now, calculate the average rate of change: $\text{Average Rate of Change} = \frac{f(1) - f(-1)}{1 - (-1)} = \frac{-15 - 3}{1 + 1} = \frac{-18}{2} = -9$

The average rate of change of the function from $x = -1$ to $x = 1$ is $-9$.

Would you like more details or have any questions?

Here are 5 related questions for practice:

1. How would the average rate of change change if the interval was $x = 0$ to $x = 2$?
2. What is the instantaneous rate of change of the function at $x = 1$?
3. Can you calculate the slope of the tangent line to the curve at $x = 0$?
4. How does the degree of the polynomial influence the overall behavior of the function?
5. What is the value of $f(x)$ at $x = 0.5$?

Tip: The average rate of change gives a sense of the overall trend of the function over an interval, similar to the concept of slope for a linear function.