Question
What is the average rate of change over the interval [2, 7] given the function f(x) = √x=2
.(1 point)
Responses
5
5
1
1
15
1 fifth
.(1 point)
Responses
5
5
1
1
15
1 fifth
Answers
GPT-4o mini
To find the average rate of change of the function \( f(x) = \sqrt{x} - 2 \) over the interval \([2, 7]\), we 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 calculate \( f(2) \) and \( f(7) \):
\[
f(2) = \sqrt{2} - 2
\]
\[
f(7) = \sqrt{7} - 2
\]
Now we substitute these values into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{(\sqrt{7} - 2) - (\sqrt{2} - 2)}{7 - 2}
\]
Simplifying this gives:
\[
= \frac{\sqrt{7} - \sqrt{2}}{5}
\]
Now we need to compute this value numerically.
Calculating \( \sqrt{7} \) and \( \sqrt{2} \):
- \( \sqrt{7} \approx 2.64575 \)
- \( \sqrt{2} \approx 1.41421 \)
So,
\[
\sqrt{7} - \sqrt{2} \approx 2.64575 - 1.41421 \approx 1.23154
\]
Now substituting this back:
\[
\text{Average Rate of Change} \approx \frac{1.23154}{5} \approx 0.246308
\]
Thus, the average rate of change over the interval \([2, 7]\) for the function \( f(x) = \sqrt{x} - 2 \) is approximately 0.246.
None of the given options (5, 1, 15, 1/5) match this result, which indicates there might be some miscommunication or error in options provided.
If we consider whether there might be a choice that represents the average rate of change as an exact fraction or simplified form:
To convert \( 0.246308 \) to a fraction, we can work with \( \frac{\sqrt{7}-\sqrt{2}}{5} \).
If you have options formatted with numerical values, then it appears that there's no exact match in the multiple choice options. Please double-check the function specification or the available response options.
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 2 \) and \( b = 7 \).
First, we calculate \( f(2) \) and \( f(7) \):
\[
f(2) = \sqrt{2} - 2
\]
\[
f(7) = \sqrt{7} - 2
\]
Now we substitute these values into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{(\sqrt{7} - 2) - (\sqrt{2} - 2)}{7 - 2}
\]
Simplifying this gives:
\[
= \frac{\sqrt{7} - \sqrt{2}}{5}
\]
Now we need to compute this value numerically.
Calculating \( \sqrt{7} \) and \( \sqrt{2} \):
- \( \sqrt{7} \approx 2.64575 \)
- \( \sqrt{2} \approx 1.41421 \)
So,
\[
\sqrt{7} - \sqrt{2} \approx 2.64575 - 1.41421 \approx 1.23154
\]
Now substituting this back:
\[
\text{Average Rate of Change} \approx \frac{1.23154}{5} \approx 0.246308
\]
Thus, the average rate of change over the interval \([2, 7]\) for the function \( f(x) = \sqrt{x} - 2 \) is approximately 0.246.
None of the given options (5, 1, 15, 1/5) match this result, which indicates there might be some miscommunication or error in options provided.
If we consider whether there might be a choice that represents the average rate of change as an exact fraction or simplified form:
To convert \( 0.246308 \) to a fraction, we can work with \( \frac{\sqrt{7}-\sqrt{2}}{5} \).
If you have options formatted with numerical values, then it appears that there's no exact match in the multiple choice options. Please double-check the function specification or the available response options.