1. **State the problem:** We are given points on a function and need to find the interval where the average rate of change is the smallest. 2. **Recall the formula for average rate of change:** $$\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$$ This measures the slope of the line connecting two points on the graph. 3. **Calculate the average rate of change for each interval:** - From $x=3$ to $x=7$: $$\frac{213 - 75}{7 - 3} = \frac{138}{4} = 34.5$$ - From $x=7$ to $x=19$: $$\frac{656 - 213}{19 - 7} = \frac{443}{12} \approx 36.92$$ - From $x=19$ to $x=22$: $$\frac{1081 - 656}{22 - 19} = \frac{425}{3} \approx 141.67$$ - From $x=22$ to $x=38$: $$\frac{1218 - 1081}{38 - 22} = \frac{137}{16} \approx 8.56$$ 4. **Compare the values:** The average rates of change are approximately 34.5, 36.92, 141.67, and 8.56. 5. **Conclusion:** The smallest average rate of change is approximately $8.56$ on the interval from $x=22$ to $x=38$. **Final answer:** The average rate of change is smallest on the interval $x=22$ to $x=38$.