Certainly! To find the average rate of change of the function [tex]\( f(x) = \sqrt{x} - 1 \)[/tex] on the interval [tex]\( [4, 9] \)[/tex], we need to follow these steps:
1. Identify the function values at the endpoints of the interval:
We are given the interval [tex]\([4, 9]\)[/tex]:
- At [tex]\(x = 4\)[/tex], the coordinate is already given as [tex]\((4, 3)\)[/tex].
- We need to find the coordinate at [tex]\(x = 9\)[/tex].
2. Calculate the function value at [tex]\( x = 9 \)[/tex]:
[tex]\[
f(9) = \sqrt{9} - 1
\][/tex]
[tex]\[
= 3 - 1
\][/tex]
[tex]\[
= 2
\][/tex]
So, the coordinate at [tex]\( x = 9 \)[/tex] is [tex]\( (9, 2) \)[/tex].
3. Determine the coordinates for the end of the interval:
Given the value we just calculated, the coordinate for the end of the interval is [tex]\( (9, 2) \)[/tex].
4. Calculate the average rate of change:
The formula for the average rate of change of a function [tex]\( f(x) \)[/tex] over the interval [tex]\([a, b]\)[/tex] is:
[tex]\[
\frac{f(b) - f(a)}{b - a}
\][/tex]
Plugging in the values from our interval [tex]\([4, 9]\)[/tex]:
[tex]\[
\frac{f(9) - f(4)}{9 - 4}
\][/tex]
We know:
[tex]\[
f(9) = 2 \quad \text{and} \quad f(4) = 3
\][/tex]
So:
[tex]\[
\frac{2 - 3}{9 - 4} = \frac{-1}{5} = -0.2
\][/tex]
Thus, the average rate of change of the function [tex]\( f(x) = \sqrt{x} - 1 \)[/tex] on the interval [tex]\( [4, 9] \)[/tex] is:
[tex]\[
\frac{-1}{5} \text{ or } -0.2
\][/tex]
### Summary:
- The coordinates for the end of the interval are:
[tex]\[
(9, 2)
\][/tex]
- The average rate of change of the function [tex]\( f(x) \)[/tex] on the interval [tex]\([4, 9]\)[/tex] is:
[tex]\[
-0.2
\][/tex]
Therefore, the correct answers are:
- Coordinates for the end of the interval: [tex]\((9, 2.0)\)[/tex]
- Average rate of change: [tex]\(-\frac{1}{5} \text{ or } -0.2\)[/tex]
