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To determine on which interval the average rate of change of the function is the greatest, we need to calculate the average rate of change for each interval provided in the table. The average rate of change of a function over an interval \([x_1, x_2]\) is given by the formula:

\[
\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's compute this for each interval in the table.

1. **Interval \([0, 3]\)**:
   \[
   \frac{15 - 4}{3 - 0} = \frac{11}{3} \approx 3.67
   \]

2. **Interval \([3, 12]\)**:
   \[
   \frac{20 - 15}{12 - 3} = \frac{5}{9} \approx 0.56
   \]

3. **Interval \([12, 20]\)**:
   \[
   \frac{27 - 20}{20 - 12} = \frac{7}{8} = 0.875
   \]

4. **Interval \([20, 24]\)**:
   \[
   \frac{40 - 27}{24 - 20} = \frac{13}{4} = 3.25
   \]

Comparing the calculated rates of change:
- Interval \([0, 3]\): 3.67
- Interval \([3, 12]\): 0.56
- Interval \([12, 20]\): 0.875
- Interval \([20, 24]\): 3.25

The greatest average rate of change occurs on the interval \([0, 3]\) with a rate of approximately 3.67.

Would you like more details or have any questions about this? Here are some related questions:

1. How do you calculate the average rate of change for a nonlinear function?
2. What does the average rate of change tell us about a function's behavior?
3. How is the average rate of change related to the slope of a linear function?
4. How can we estimate the instantaneous rate of change from a table of values?
5. Can the average rate of change be negative, and what does it signify?

**Tip:** For linear functions, the average rate of change is constant over any interval, but for nonlinear functions, it can vary widely depending on the chosen interval.

A function is shown in the table below. On which interval of x is the average rate of change of the function the greatest?

In this problem, you will learn how to calculate the average rate of change for a function based on a given table of values. The solution includes using the formula (y2 - y1) / (x2 - x1) and comparing rates across different intervals. The greatest average rate of change occurs between x = 0 and x = 3.

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