Find the average rate of change of the function from \( x_{1} \) to \( x_{2} \). \( f(x)=\sqrt{2 x} \) from \( x_{1}=2 \) to \( x_{2}=8 \)

To find the average rate of change of the function \( f(x) = \sqrt{2x} \) from \( x_1 = 2 \) to \( x_2 = 8 \), we first evaluate the function at these points: 1. \( f(2) = \sqrt{2 \cdot 2} = \sqrt{4} = 2 \) 2. \( f(8) = \sqrt{2 \cdot 8} = \sqrt{16} = 4 \) Next, we use the formula for the average rate of change: \[ \text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{f(8) - f(2)}{8 - 2} = \frac{4 - 2}{8 - 2} = \frac{2}{6} = \frac{1}{3}. \] Thus, the average rate of change of the function from \( x = 2 \) to \( x = 8 \) is \( \frac{1}{3} \). To reinforce your understanding, remember that the average rate of change essentially gives you the slope of the secant line between two points on the graph of the function. You can think of it as the "flatness" of the curve over that interval — a handy concept for visualizing how functions behave! As for real-world applications, understanding the average rate of change can help in various fields such as physics, where you might calculate acceleration, or economics where you could analyze trends in stock prices. It's the tool that helps you decipher how things shift over time!