To find the average rate of change for the function $f(x) = x^2 + 10x$ between $x = 0$ and $x = 9$, we can use the formula provided in option C:

$\text{Average rate of change} = \frac{f(9) - f(0)}{9 - 0}$

Step-by-Step Solution:

1. Calculate $f(9)$: $f(9) = 9^2 + 10 \cdot 9 = 81 + 90 = 171$

2. Calculate $f(0)$: $f(0) = 0^2 + 10 \cdot 0 = 0$

3. Apply the formula: $\text{Average rate of change} = \frac{f(9) - f(0)}{9 - 0} = \frac{171 - 0}{9} = \frac{171}{9} = 19$

Answer:

The average rate of change is $19$.

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Related Questions:

1. How does the average rate of change differ from the instantaneous rate of change?
2. What is the geometric interpretation of the average rate of change on a graph?
3. Can the average rate of change be used to estimate values outside the interval $[0, 9]$?
4. How would the answer change if the interval were from $x = 3$ to $x = 9$?
5. How does this calculation apply to other quadratic functions?

Tip:

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