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

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

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

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

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

[2,6]

1 answer

To find the interval where the square root function \( g(x) = \sqrt{2x + 1} \) has the lower estimated average rate of change, we first need to calculate the average rate of change for \( g(x) \) over each of the provided intervals. The average rate of change of a function \( g \) over the interval \([a, b]\) is given by:

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

Now, we will calculate the average rate of change for each interval.

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

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

  2. 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 = \(\frac{\sqrt{15} - \sqrt{7}}{7 - 3} = \frac{\sqrt{15} - \sqrt{7}}{4}\)

  3. 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 = \(\frac{\sqrt{11} - \sqrt{3}}{5 - 1} = \frac{\sqrt{11} - \sqrt{3}}{4}\)

  4. 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 = \(\frac{\sqrt{13} - \sqrt{5}}{6 - 2} = \frac{\sqrt{13} - \sqrt{5}}{4}\)

Next, we need to compare the calculated average rates of change. To do this, we will first compute approximate values:

  • \( \frac{1}{2} = 0.5 \)

  • For \([3, 7]\):

    • \( \sqrt{15} \approx 3.87 \)
    • \( \sqrt{7} \approx 2.65 \)
    • \( \frac{\sqrt{15} - \sqrt{7}}{4} \approx \frac{3.87 - 2.65}{4} \approx \frac{1.22}{4} \approx 0.305 \)

  • For \([1, 5]\):

    • \( \sqrt{11} \approx 3.32 \)
    • \( \sqrt{3} \approx 1.73 \)
    • \( \frac{\sqrt{11} - \sqrt{3}}{4} \approx \frac{3.32 - 1.73}{4} \approx \frac{1.59}{4} \approx 0.3975 \)

  • For \([2, 6]\):

    • \( \sqrt{13} \approx 3.61 \)
    • \( \sqrt{5} \approx 2.24 \)
    • \( \frac{\sqrt{13} - \sqrt{5}}{4} \approx \frac{3.61 - 2.24}{4} \approx \frac{1.37}{4} \approx 0.3425 \)

Now comparing the calculated average rates:

  • \( [0, 4] : 0.5 \)
  • \( [3, 7] : 0.305 \)
  • \( [1, 5] : 0.3975 \)
  • \( [2, 6] : 0.3425 \)

The lowest average rate of change is in the interval \( [3, 7] \).

Thus, the interval where the square root function has the lower estimated average rate of change is:

[3, 7]