To find the average rate of change of the function $y = \sqrt{x}$ over the interval $x = 2$ to $x = 8$, we can use the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

where $a = 2$, $b = 8$, and $f(x) = \sqrt{x}$.

Step 1: Evaluate the function at $x = 8$ and $x = 2$.

• $f(8) = \sqrt{8} = 2\sqrt{2}$
• $f(2) = \sqrt{2}$

Step 2: Apply the formula.

$\text{Average Rate of Change} = \frac{f(8) - f(2)}{8 - 2} = \frac{2\sqrt{2} - \sqrt{2}}{6} = \frac{\sqrt{2}}{6}$

Final Answer:

The average rate of change of the function $y = \sqrt{x}$ over the interval from $x = 2$ to $x = 8$ is:

$\frac{\sqrt{2}}{6}$

Would you like a more detailed explanation or further clarification?

Related questions:

1. How would the average rate of change change if the interval was from $x = 1$ to $x = 9$?
2. How can the rate of change be interpreted geometrically for this function?
3. How would the average rate of change for a quadratic function differ from that of a square root function?
4. What is the instantaneous rate of change for $y = \sqrt{x}$ at $x = 4$?
5. How would the average rate of change be calculated for the function $y = x^2$ over the same interval?

Tip:

When calculating the average rate of change, always ensure you are using the correct values for $f(b)$ and $f(a)$, and check your subtraction and simplification carefully!