The average rate of change of a function $g(x)$ over an interval $[a, b]$ is given by the formula:

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

Given function:

$g(x) = \sqrt{x - 5} - 2$

Given interval:

[ [a, b] = [7, 14] ]

Step 1: Compute $g(7)$

$g(7) = \sqrt{7 - 5} - 2 = \sqrt{2} - 2$

Step 2: Compute $g(14)$

$g(14) = \sqrt{14 - 5} - 2 = \sqrt{9} - 2 = 3 - 2 = 1$

Step 3: Apply the average rate of change formula

$\frac{g(14) - g(7)}{14 - 7} = \frac{1 - (\sqrt{2} - 2)}{7}$

$= \frac{1 - \sqrt{2} + 2}{7} = \frac{3 - \sqrt{2}}{7}$

Thus, the average rate of change of $g(x)$ over $[7,14]$ is:

$\frac{3 - \sqrt{2}}{7}$

Would you like a numerical approximation or more explanation?

Related Questions:

1. What does the average rate of change represent graphically?
2. How does the average rate of change compare to the derivative at a point?
3. Can we find the instantaneous rate of change of $g(x)$ at $x = 10$?
4. How does shifting the function $g(x)$ affect its average rate of change?
5. What happens to the average rate of change if we extend the interval to $[7, 20]$?

Tip:

The average rate of change is essentially the slope of the secant line between two points on a function's graph!