To find the average rate of change of the function $f(x) = x^2 + 9x$ between $x = 0$ and $x = 5$, we use the formula for the average rate of change:

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

Here, $a = 0$ and $b = 5$. So, we need to find $f(5)$ and $f(0)$.

1. Calculate $f(0)$: $f(0) = 0^2 + 9(0) = 0$

2. Calculate $f(5)$: $f(5) = 5^2 + 9(5) = 25 + 45 = 70$

Now, substitute these values into the formula: $\text{Average rate of change} = \frac{f(5) - f(0)}{5 - 0} = \frac{70 - 0}{5 - 0} = \frac{70}{5} = 14$

Thus, the average rate of change is $14$.

Would you like further details or explanations?

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Here are 5 related questions to expand on this:

1. What is the geometric interpretation of the average rate of change?
2. How is the average rate of change related to the derivative?
3. How would you find the instantaneous rate of change at $x = 2$?
4. Can the average rate of change be negative, and what would it indicate?
5. What happens if you change the interval to $x = 1$ and $x = 5$?

Tip: The average rate of change of a function over an interval is the slope of the secant line between the two points on the graph.