Consider the cubic function d(x)=2x−1−−−−−√3 and the square root function g(x)=2x+1−−−−−√

Question

. Select the interval where the square root function has the lower estimated average rate of change.(1 point)
Responses

[2,6]
left bracket 2 comma 6 right bracket

[1,5]
left bracket 1 comma 5 right bracket

[0,4]
left bracket 0 comma 4 right bracket

[3,7]
left bracket 3 comma 7 right bracket

Answer 1

To determine where the square root function \( g(x) = \sqrt{2x + 1} \) has the lower estimated average rate of change over the given intervals, we need to calculate the average rate of change for \( g(x) \) on each interval.

The average rate of change of a function \( g(x) \) on the interval \([a, b]\) is given by:

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

Now, let's calculate the average rate of change for each specified interval.

1. For the interval \([2, 6]\):

   • \( g(2) = \sqrt{2(2) + 1} = \sqrt{5} \)
   • \( g(6) = \sqrt{2(6) + 1} = \sqrt{13} \)
   • Average Rate of Change = \(\frac{\sqrt{13} - \sqrt{5}}{6 - 2} = \frac{\sqrt{13} - \sqrt{5}}{4}\)

2. For the interval \([1, 5]\):

   • \( g(1) = \sqrt{2(1) + 1} = \sqrt{3} \)
   • \( g(5) = \sqrt{2(5) + 1} = \sqrt{11} \)
   • Average Rate of Change = \(\frac{\sqrt{11} - \sqrt{3}}{5 - 1} = \frac{\sqrt{11} - \sqrt{3}}{4}\)

3. For the interval \([0, 4]\):

   • \( g(0) = \sqrt{2(0) + 1} = 1 \)
   • \( g(4) = \sqrt{2(4) + 1} = \sqrt{9} = 3 \)
   • Average Rate of Change = \(\frac{3 - 1}{4 - 0} = \frac{2}{4} = \frac{1}{2}\)

4. For the interval \([3, 7]\):

   • \( g(3) = \sqrt{2(3) + 1} = \sqrt{7} \)
   • \( g(7) = \sqrt{2(7) + 1} = \sqrt{15} \)
   • Average Rate of Change = \(\frac{\sqrt{15} - \sqrt{7}}{7 - 3} = \frac{\sqrt{15} - \sqrt{7}}{4}\)

After calculating the average rates of change for all intervals, let's summarize:

• \([2, 6]\): \(\frac{\sqrt{13} - \sqrt{5}}{4}\)
• \([1, 5]\): \(\frac{\sqrt{11} - \sqrt{3}}{4}\)
• \([0, 4]\): \(\frac{1}{2}\)
• \([3, 7]\): \(\frac{\sqrt{15} - \sqrt{7}}{4}\)

To find the one with the lowest average rate of change, we need to evaluate which one gives the smallest result. Without exact numerical evaluations, we would need to compare approximate numeric values for the square roots:

1. \(\sqrt{13} - \sqrt{5} \approx 3.605 - 2.236 = 1.369\)
2. \(\sqrt{11} - \sqrt{3} \approx 3.317 - 1.732 = 1.585\)
3. \(\frac{1}{2} = 0.5\)
4. \(\sqrt{15} - \sqrt{7} \approx 3.873 - 2.646 = 1.227\)

Comparing all values:

• \([2, 6]\): \(\frac{1.369}{4} \approx 0.342\)
• \([1, 5]\): \(\frac{1.585}{4} \approx 0.396\)
• \([0, 4]\): \(0.5\)
• \([3, 7]\): \(\frac{1.227}{4} \approx 0.307\)

The lowest average rate of change appears to be for the interval \([3, 7]\), as it results in the least value.

Thus, the answer is:

Interval [3, 7].