1. **State the problem:** Calculate the average rate of change of the function $f(x) = \sqrt{x} + 2$ over the interval $[2,7]$. 2. **Formula:** The average rate of change of a function $f(x)$ over the interval $[a,b]$ is given by: $$\frac{f(b) - f(a)}{b - a}$$ 3. **Evaluate the function at the endpoints:** - $f(2) = \sqrt{2} + 2$ - $f(7) = \sqrt{7} + 2$ 4. **Calculate the difference in function values:** $$f(7) - f(2) = (\sqrt{7} + 2) - (\sqrt{2} + 2) = \sqrt{7} - \sqrt{2}$$ 5. **Calculate the difference in $x$ values:** $$7 - 2 = 5$$ 6. **Compute the average rate of change:** $$\frac{\sqrt{7} - \sqrt{2}}{5}$$ 7. **Express as a fraction:** This is already a fraction with numerator $\sqrt{7} - \sqrt{2}$ and denominator $5$. **Final answer:** $$\frac{\sqrt{7} - \sqrt{2}}{5}$$