Question: Find the average rate of change for the given function.
f(x) = x^2 + 9x between x = 0 and x = 9
The average rate of change is □. (Simplify your answer.)
1. **State the problem:** We need to find the average rate of change of the function $$f(x) = x^2 + 9x$$ between $$x=0$$ and $$x=9$$.
2. **Formula for average rate of change:**
$$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$$
where $$a=0$$ and $$b=9$$.
3. **Calculate $$f(a)$$ and $$f(b)$$:**
$$f(0) = 0^2 + 9 \times 0 = 0$$
$$f(9) = 9^2 + 9 \times 9 = 81 + 81 = 162$$
4. **Apply the formula:**
$$\frac{f(9) - f(0)}{9 - 0} = \frac{162 - 0}{9} = \frac{162}{9}$$
5. **Simplify the fraction:**
$$\frac{\cancel{162}}{\cancel{9}} = 18$$
6. **Interpretation:** The average rate of change of the function $$f(x) = x^2 + 9x$$ from $$x=0$$ to $$x=9$$ is $$18$$.
**Final answer:** $$18$$