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Find the average rate of change of the function [tex]\( f(x) = \sqrt{x} - 1 \)[/tex] on the interval [tex]\( 4 \leq x \leq 9 \)[/tex].

1. What are the coordinates for the end of the interval?
A. [tex]\( (9, 2) \)[/tex]
B. [tex]\( (9, 3) \)[/tex]
C. [tex]\( (9, 8) \)[/tex]

2. What is the average rate of change for this function on the given interval?
A. [tex]\(-\frac{1}{5}\)[/tex]
B. [tex]\(\frac{1}{5}\)[/tex]
C. [tex]\(-5\)[/tex]
D. 5


Sagot :

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]
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