To calculate the average rate of change of the function $h(x) = -x^2 + 5x + 8$ over the interval $0 \leq x \leq 7$, we use the formula:

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

where $a = 0$ and $b = 7$. The values of $h(0)$ and $h(7)$ need to be computed first:

Step 1: Find $h(0)$

Substitute $x = 0$ into the function $h(x)$:

$h(0) = -(0)^2 + 5(0) + 8 = 8$

Step 2: Find $h(7)$

Substitute $x = 7$ into the function $h(x)$:

$h(7) = -(7)^2 + 5(7) + 8 = -49 + 35 + 8 = -6$

Step 3: Apply the Average Rate of Change Formula

Now that we have $h(0) = 8$ and $h(7) = -6$, we can find the average rate of change:

$\text{Average Rate of Change} = \frac{h(7) - h(0)}{7 - 0} = \frac{-6 - 8}{7} = \frac{-14}{7} = -2$

So, the average rate of change of the function over the interval $0 \leq x \leq 7$ is -2.

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Do you need further clarification or a deeper explanation? Here are some related questions you might find useful:

1. How do you find the instantaneous rate of change of a function?
2. What is the derivative of the function $h(x) = -x^2 + 5x + 8$?
3. How does the average rate of change differ from the instantaneous rate of change?
4. What is the significance of the average rate of change in real-world applications?
5. How would the average rate of change change if the interval was $x = 2$ to $x = 5$?

Tip: When calculating the average rate of change, always check if the interval is closed and use the correct values for $a$ and $b$.