Let's find the average rate of change of the function [tex]\( f(x) = x - 2 \sqrt{x} \)[/tex] on the interval [tex]\([1, 9]\)[/tex] step-by-step.
### Step 1: Evaluate [tex]\( f(x) \)[/tex] at the endpoints of the interval
First, we need to determine the values of the function at the endpoints [tex]\( x = 1 \)[/tex] and [tex]\( x = 9 \)[/tex].
At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 1 - 2 \sqrt{1} = 1 - 2 \cdot 1 = 1 - 2 = -1 \][/tex]
So, [tex]\( f(1) = -1 \)[/tex].
At [tex]\( x = 9 \)[/tex]:
[tex]\[ f(9) = 9 - 2 \sqrt{9} = 9 - 2 \cdot 3 = 9 - 6 = 3 \][/tex]
So, [tex]\( f(9) = 3 \)[/tex].
### Step 2: Use the formula for the average rate of change
The average rate of change of a function over an interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\][/tex]
In this problem, [tex]\( a = 1 \)[/tex] and [tex]\( b = 9 \)[/tex].
### Step 3: Plug in the function values and calculate
Substituting the values we found:
[tex]\[
\text{Average Rate of Change} = \frac{f(9) - f(1)}{9 - 1} = \frac{3 - (-1)}{9 - 1} = \frac{3 + 1}{9 - 1} = \frac{4}{8} = 0.5
\][/tex]
### Final Result
The average rate of change of [tex]\( f(x) \)[/tex] on the interval [tex]\([1, 9]\)[/tex] is [tex]\( 0.5 \)[/tex].
