To solve the problem, we need to calculate the estimated average rates of change for both the square root function (let’s denote it as \( f(x) = \sqrt{x} \)) and the cube root function (denote it as \( g(x) = \sqrt[3]{x + 2} \)) over the interval \([2, 7]\).
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Calculate the average rate of change for the square root function \( f(x) = \sqrt{x} \):
The average rate of change of \( f(x) \) from \( x = 2 \) to \( x = 7 \) is given by: \[ \text{Average Rate of Change} = \frac{f(7) - f(2)}{7 - 2} \]
Now, calculate \( f(2) \) and \( f(7) \): \[ f(2) = \sqrt{2} \] \[ f(7) = \sqrt{7} \]
Thus, the average rate of change becomes: \[ \text{Average Rate of Change} = \frac{\sqrt{7} - \sqrt{2}}{5} \]
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Calculate the average rate of change for the cube root function \( g(x) = \sqrt[3]{x + 2} \):
The average rate of change of \( g(x) \) from \( x = 2 \) to \( x = 7 \) is given by: \[ \text{Average Rate of Change} = \frac{g(7) - g(2)}{7 - 2} \]
Now, calculate \( g(2) \) and \( g(7) \): \[ g(2) = \sqrt[3]{2 + 2} = \sqrt[3]{4} \] \[ g(7) = \sqrt[3]{7 + 2} = \sqrt[3]{9} \]
Thus, the average rate of change becomes: \[ \text{Average Rate of Change} = \frac{\sqrt[3]{9} - \sqrt[3]{4}}{5} \]
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Evaluate the average rates and compute the difference:
To get approximate values, we use numerical estimations: \[ \sqrt{2} \approx 1.4, \quad \sqrt{7} \approx 2.645 (use a calculator for accuracy) \] Average rate for \( f(x) \): \[ \frac{2.645 - 1.414}{5} \approx \frac{1.231}{5} \approx 0.2462 \]
For \( g(x) \): \[ \sqrt[3]{4} \approx 1.587, \quad \sqrt[3]{9} \approx 2.080 (use a calculator for accuracy) \] Average rate for \( g(x) \): \[ \frac{2.080 - 1.587}{5} \approx \frac{0.493}{5} \approx 0.0986 \]
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Compute the difference: \[ \text{Difference} = 0.2462 - 0.0986 \approx 0.1476 \]
Rounding to the nearest tenth: \[ \text{Difference} \approx 0.1 \]
Thus, the difference between the estimated average rates of change is \( \boxed{0.1} \).
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