Part 1: Expression for the average rate of change

The average rate of change of a function over an interval $[a, b]$ is given by:

$\frac{f(b) - f(a)}{b - a}$

In this case, the interval is $[2, 8]$ and the function is $f(x) = \sqrt{2x}$. The correct expression to find the average rate of change is:

$\frac{f(8) - f(2)}{8 - 2}$

This corresponds to Option D.

Part 2: Calculating the average rate of change

Now, let's calculate the average rate of change for $f(x) = \sqrt{2x}$ between $x = 2$ and $x = 8$.

1. First, find $f(8)$: $f(8) = \sqrt{2 \times 8} = \sqrt{16} = 4$

2. Next, find $f(2)$: $f(2) = \sqrt{2 \times 2} = \sqrt{4} = 2$

3. Now, compute the average rate of change: $\text{Average rate of change} = \frac{f(8) - f(2)}{8 - 2} = \frac{4 - 2}{8 - 2} = \frac{2}{6} = \frac{1}{3}$

So, the average rate of change is $\frac{1}{3}$.

Summary

The expression to find the average rate of change is: $\frac{f(8) - f(2)}{8 - 2}$ And the average rate of change of $y = \sqrt{2x}$ between $x = 2$ and $x = 8$ is: $\frac{1}{3}$

Would you like more details or have any questions?

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Here are some related questions:

1. How does the average rate of change differ from the instantaneous rate of change?
2. How would the average rate of change change if the interval were from $x = 1$ to $x = 9$?
3. What is the geometric interpretation of the average rate of change for this function?
4. How would you compute the instantaneous rate of change of $f(x) = \sqrt{2x}$ at $x = 2$?
5. How does the form of the function $f(x) = \sqrt{2x}$ affect the average rate of change as $x$ increases?

Tip: The average rate of change is essentially the slope of the secant line connecting two points on a curve.