To compare the estimated average rates of the square root function and the cube root function over the interval \([2, 7]\), we need to calculate the average rate of change for each function over this interval.
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Calculating Average Rate of Change for the Square Root Function:
The square root function generally takes the form \(y = \sqrt{x}\).
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Find the values at the endpoints: \[ f(2) = \sqrt{2} \approx 1.414 \] \[ f(7) = \sqrt{7} \approx 2.646 \]
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The average rate of change is given by: \[ \text{Average Rate of Change} = \frac{f(7) - f(2)}{7 - 2} = \frac{\sqrt{7} - \sqrt{2}}{5} \] \[ \text{Average Rate of Change} \approx \frac{2.646 - 1.414}{5} \approx \frac{1.232}{5} \approx 0.2464 \]
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Calculating Average Rate of Change for the Cube Root Function:
The cube root function is represented as \(y = 3^{\sqrt{x} + 2}\).
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Find the values at the endpoints: \[ g(2) = 3^{\sqrt{2} + 2} \approx 3^{1.414 + 2} = 3^{3.414} \approx 40.29 \] \[ g(7) = 3^{\sqrt{7} + 2} \approx 3^{2.646 + 2} = 3^{4.646} \approx 138.34 \]
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The average rate of change is given by: \[ \text{Average Rate of Change} = \frac{g(7) - g(2)}{7 - 2} = \frac{3^{\sqrt{7} + 2} - 3^{\sqrt{2} + 2}}{5} \] \[ \text{Average Rate of Change} \approx \frac{138.34 - 40.29}{5} \approx \frac{98.05}{5} \approx 19.61 \]
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Finding the Difference:
Now, we find the difference between the average rates of change: \[ \text{Difference} = 19.61 - 0.2464 \approx 19.3636 \]
Rounding to the nearest tenth, the difference is: \[ \approx 19.4 \]
Final Answer: The difference between the estimated average rates of the square root function and the cube root function over the interval \([2, 7]\) is approximately 19.4.