To find the average rate of change of the function $f(x)$ over the interval $5 \leq x \leq 6$, we use the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

where $a = 5$ and $b = 6$ in this case.

From the table:

• $f(5) = 33$
• $f(6) = 33$

Substitute these values into the formula:

$\text{Average Rate of Change} = \frac{f(6) - f(5)}{6 - 5} = \frac{33 - 33}{6 - 5} = \frac{0}{1} = 0$

Answer:

The average rate of change of the function $f(x)$ over the interval $5 \leq x \leq 6$ is $0$.

Would you like further details or have any questions?

Here are some related questions to explore:

1. What does an average rate of change of zero indicate about the function over the interval?
2. How do we interpret the average rate of change when it is positive or negative?
3. Can the average rate of change be applied to non-linear functions? How does it differ?
4. How would we find the instantaneous rate of change at a point instead?
5. How does the average rate of change help in understanding the behavior of a function over a specific interval?

Tip: The average rate of change over an interval can reveal whether a function is increasing, decreasing, or constant within that range.