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To find the average rate of change of the function \( f(x) \) from \( x = 1 \) to \( x = 5 \), we will use the formula for the average rate of change, which is essentially the slope of the secant line connecting the points at \( x = 1 \) and \( x = 5 \):

\[
\text{Average rate of change} = \frac{f(5) - f(1)}{5 - 1}
\]

From the table, the values of \( f(5) \) and \( f(1) \) are:

- \( f(5) = 63 \)
- \( f(1) = 39 \)

Now, let's substitute these into the formula:

\[
\text{Average rate of change} = \frac{63 - 39}{5 - 1} = \frac{24}{4} = 6
\]

The average rate of change from \( x = 1 \) to \( x = 5 \) is \( 6 \).

Would you like further explanation or details? Here are five related questions for further exploration:

1. What is the significance of the average rate of change in a real-world context?
2. How does the average rate of change relate to the concept of the derivative?
3. How would you calculate the average rate of change over a different interval, such as from \( x = 0 \) to \( x = 4 \)?
4. What does it mean if the average rate of change is negative?
5. How does the average rate of change compare to instantaneous rate of change?

**Tip:** When calculating the average rate of change, always identify the corresponding function values at the endpoints of the interval to ensure accuracy.

Given the table below of values for f(x), find the average rate of change from x = 1 to x = 5.

Learn how to find the average rate of change of a function using a table of values. This problem involves calculating the slope of a secant line between points x = 1 and x = 5, with the values f(1) = 39 and f(5) = 63, resulting in an average rate of change of 6. Suitable for high school students in Grades 9-12.

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