To find the average rate of change of the function $g(x) = 8\sqrt{x + 14}$ over the interval $[-1, 6]$, we use the formula:

$\text{Average rate of change} = \frac{g(b) - g(a)}{b - a}$

where $a = -1$ and $b = 6$.

1. First, calculate $g(-1)$:

$g(-1) = 8\sqrt{-1 + 14} = 8\sqrt{13}$

2. Next, calculate $g(6)$:

$g(6) = 8\sqrt{6 + 14} = 8\sqrt{20}$

3. Now, substitute these into the formula for the average rate of change:

$\text{Average rate of change} = \frac{g(6) - g(-1)}{6 - (-1)} = \frac{8\sqrt{20} - 8\sqrt{13}}{7}$

4. Simplify the expression:

$\text{Average rate of change} = \frac{8(\sqrt{20} - \sqrt{13})}{7}$

So, the average rate of change of the function $g(x) = 8\sqrt{x + 14}$ over the interval $[-1, 6]$ is:

$\boxed{\frac{8(\sqrt{20} - \sqrt{13})}{7}}$

This expression gives the exact average rate of change over the interval.

Would you like more details or have any questions?

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Here are some related questions:

1. What is the significance of the average rate of change in a real-world context?
2. How would the rate of change differ if the interval was [0, 6] instead of [-1, 6]?
3. What happens to the average rate of change if the interval includes a point where the function is undefined?
4. How can you approximate the average rate of change using a graph?
5. What is the relationship between the average rate of change and the derivative of the function?

Tip: The average rate of change is a useful concept for understanding how a function behaves over a certain interval, and it can give you insight into the overall trend of the function.