To find the estimated average rates of change of the functions \( f(x) = \sqrt{3x - 4} \) and \( g(x) = \sqrt[3]{2x - 43} \) over the interval \([2, 3]\), we first need to evaluate each function at the endpoints of the interval.
Step 1: Calculate \( f(2) \) and \( f(3) \)
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Calculate \( f(2) \): \[ f(2) = \sqrt{3(2) - 4} = \sqrt{6 - 4} = \sqrt{2} \]
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Calculate \( f(3) \): \[ f(3) = \sqrt{3(3) - 4} = \sqrt{9 - 4} = \sqrt{5} \]
Step 2: Calculate the average rate of change of \( f(x) \)
The average rate of change of \( f(x) \) over \([2, 3]\) is given by: \[ \text{Average rate of change of } f = \frac{f(3) - f(2)}{3 - 2} = f(3) - f(2) = \sqrt{5} - \sqrt{2} \]
Step 3: Calculate \( g(2) \) and \( g(3) \)
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Calculate \( g(2) \): \[ g(2) = \sqrt[3]{2(2) - 43} = \sqrt[3]{4 - 43} = \sqrt[3]{-39} \]
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Calculate \( g(3) \): \[ g(3) = \sqrt[3]{2(3) - 43} = \sqrt[3]{6 - 43} = \sqrt[3]{-37} \]
Step 4: Calculate the average rate of change of \( g(x) \)
The average rate of change of \( g(x) \) over \([2, 3]\) is: \[ \text{Average rate of change of } g = \frac{g(3) - g(2)}{3 - 2} = g(3) - g(2) = \sqrt[3]{-37} - \sqrt[3]{-39} \]
Step 5: Estimate and Compare the Rates
To compute the actual numerical values of the averages:
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Estimate \( f(3) - f(2) \):
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \) \[ \sqrt{5} - \sqrt{2} \approx 2.236 - 1.414 \approx 0.822 \]
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Estimate \( g(3) - g(2) \):
- \( \sqrt[3]{-37} \approx -3.332 \)
- \( \sqrt[3]{-39} \approx -3.389 \) \[ \sqrt[3]{-37} - \sqrt[3]{-39} \approx -3.332 + 3.389 \approx 0.057 \]
Step 6: Find the Difference in Average Rates
Finally, we calculate the difference: \[ \text{Difference} = (\sqrt{5} - \sqrt{2}) - (\sqrt[3]{-37} - \sqrt[3]{-39}) \approx 0.822 - 0.057 \approx 0.765 \]
Step 7: Round to the nearest tenth
The difference between the estimated average rates of change is approximately: \[ \boxed{0.8} \]
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