Average Rate Change Afa3D7

1. **State the problem:** We need to find the average rate of change of the function $$f(x) = \sqrt{4x + 1}$$ between $$x = 0$$ and $$x = 2$$. 2. **Recall the formula:** The average rate of change of a function $$f$$ between $$x = a$$ and $$x = b$$ is given by: $$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$$ 3. **Evaluate the function at the given points:** $$f(0) = \sqrt{4(0) + 1} = \sqrt{1} = 1$$ $$f(2) = \sqrt{4(2) + 1} = \sqrt{8 + 1} = \sqrt{9} = 3$$ 4. **Apply the formula:** $$\frac{f(2) - f(0)}{2 - 0} = \frac{3 - 1}{2} = \frac{2}{2}$$ 5. **Simplify the fraction:** $$\frac{\cancel{2}}{\cancel{2}} = 1$$ 6. **Conclusion:** The average rate of change of $$f(x)$$ between $$x = 0$$ and $$x = 2$$ is $$1$$.