To find the average rate of change of $g(x) = 4x^3 - 5x^2$ between $x = -0.1$ and $x = 0.1$, we can use the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}$

where $x_1 = -0.1$ and $x_2 = 0.1$.

Step 1: Calculate $g(-0.1)$

We substitute $x = -0.1$ into the function: $g(-0.1) = 4(-0.1)^3 - 5(-0.1)^2 = 4(-0.001) - 5(0.01) = -0.004 - 0.05 = -0.054$

Step 2: Calculate $g(0.1)$

Now substitute $x = 0.1$ into the function: $g(0.1) = 4(0.1)^3 - 5(0.1)^2 = 4(0.001) - 5(0.01) = 0.004 - 0.05 = -0.046$

Step 3: Apply the formula for average rate of change

$\text{Average Rate of Change} = \frac{g(0.1) - g(-0.1)}{0.1 - (-0.1)} = \frac{-0.046 - (-0.054)}{0.1 + 0.1} = \frac{-0.046 + 0.054}{0.2} = \frac{0.008}{0.2} = 0.04$

Thus, the average rate of change of $g(x)$ between $x = -0.1$ and $x = 0.1$ is 0.04.

Would you like further details or explanations?

Here are 5 related questions:

1. How do you interpret the average rate of change geometrically?
2. What does a positive or negative rate of change imply about the function?
3. How would you find the instantaneous rate of change at a specific point?
4. How would this process differ if the function included a higher degree term?
5. What are some real-life applications of average rate of change?

Tip: The average rate of change is equivalent to the slope of the secant line between two points on the graph of the function.