Question

Calculate the average rate of change over the interval [2,7] given the function f(x)=x+2−−−−−√ . (3 points) (2, ) (7, ) The average rate of change is . Express your answer as a fraction.

Answer 1

To calculate the average rate of change of the function \( f(x) = \sqrt{x + 2} \) over the interval \([2, 7]\), we will use the formula for the average rate of change:

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

where \( a = 2 \) and \( b = 7 \).

First, we need to evaluate \( f(2) \) and \( f(7) \):

1. **Calculate \( f(2) \)**:
\[
f(2) = \sqrt{2 + 2} = \sqrt{4} = 2
\]

2. **Calculate \( f(7) \)**:
\[
f(7) = \sqrt{7 + 2} = \sqrt{9} = 3
\]

Now we can substitute these values into the average rate of change formula:

\[
\text{Average Rate of Change} = \frac{f(7) - f(2)}{7 - 2} = \frac{3 - 2}{7 - 2} = \frac{1}{5}
\]

Thus, the average rate of change of the function \( f(x) = \sqrt{x + 2} \) over the interval \([2, 7]\) is

\[
\boxed{\frac{1}{5}}.
\]