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.