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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.

Find the average rate of change of g(x) = 4x^3 - 5x^2 between x = -0.1 and x = 0.1

Learn how to find the average rate of change of the polynomial function g(x) = 4x^3 - 5x^2 between x = -0.1 and x = 0.1. This tutorial covers the concepts of calculus, specifically focusing on the rate of change for polynomial functions, with a detailed step-by-step solution.

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