To solve the problem:

We need to find the average rate of change of the function $h(x) = -x^2 + 7x + 19$ over the interval $1 \leq x \leq 10$. The formula for the average rate of change is:

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

where $a = 1$ and $b = 10$.

Steps:

1. Evaluate $h(1)$: $h(1) = -(1)^2 + 7(1) + 19 = -1 + 7 + 19 = 25.$

2. Evaluate $h(10)$: $h(10) = -(10)^2 + 7(10) + 19 = -100 + 70 + 19 = -11.$

3. Plug into the formula: $\text{Average Rate of Change} = \frac{h(10) - h(1)}{10 - 1} = \frac{-11 - 25}{10 - 1} = \frac{-36}{9} = -4.$

Final Answer:

The average rate of change is -4.

Would you like more details about this calculation or clarification on any step?

Related Questions:

1. How do you interpret the meaning of the average rate of change in real-life scenarios?
2. How does the average rate of change compare to the instantaneous rate of change?
3. Can we use the same process to compute average rate of change for other types of functions, such as exponentials?
4. What are the implications of a negative average rate of change in the given interval?
5. How can this concept be used in physics, particularly for velocity or acceleration?

Tip:

The average rate of change essentially represents the slope of the straight line connecting the two points on the graph of the function within the interval.