To find the estimated average rates of change of the functions \( f(x) = \sqrt{3x - 4} \) and \( g(x) = \sqrt[3]{2x - 4} \) over the interval \([2, 3]\), we use the formula for average rate of change:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = 2 \) and \( b = 3 \).
Let's calculate the average rate of change for \( f(x) \) first.
For \( f(x) = \sqrt{3x - 4} \):
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Calculate \( f(2) \): \[ f(2) = \sqrt{3(2) - 4} = \sqrt{6 - 4} = \sqrt{2} \]
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Calculate \( f(3) \): \[ f(3) = \sqrt{3(3) - 4} = \sqrt{9 - 4} = \sqrt{5} \]
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Calculate the average rate of change: \[ \text{Average Rate of Change of } f = \frac{f(3) - f(2)}{3 - 2} = \frac{\sqrt{5} - \sqrt{2}}{1} = \sqrt{5} - \sqrt{2} \]
For \( g(x) = \sqrt[3]{2x - 4} \):
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Calculate \( g(2) \): \[ g(2) = \sqrt[3]{2(2) - 4} = \sqrt[3]{4 - 4} = \sqrt[3]{0} = 0 \]
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Calculate \( g(3) \): \[ g(3) = \sqrt[3]{2(3) - 4} = \sqrt[3]{6 - 4} = \sqrt[3]{2} \]
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Calculate the average rate of change: \[ \text{Average Rate of Change of } g = \frac{g(3) - g(2)}{3 - 2} = \frac{\sqrt[3]{2} - 0}{1} = \sqrt[3]{2} \]
Now we can find the estimated average rates of change:
- Average rate of change of \( f \) is \( \sqrt{5} - \sqrt{2} \).
- Average rate of change of \( g \) is \( \sqrt[3]{2} \).
Convert the results into decimal approximations:
- Approximately,
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{5} \approx 2.236 \)
- Hence, \[ \sqrt{5} - \sqrt{2} \approx 2.236 - 1.414 \approx 0.822 \]
- For \( g \),
- \( \sqrt[3]{2} \approx 1.26 \)
Finally, state the difference in the estimated average rates of change:
\[ \text{Difference} = |\text{Average Rate of Change of } f - \text{Average Rate of Change of } g| \approx |0.822 - 1.26| \approx | -0.438 | \approx 0.438 \]
Now, rounding \( 0.438 \) to the nearest tenth gives us \( 0.4 \).
Therefore, the difference in the estimated average rates of change to the nearest tenth is:
\[ \boxed{0.4} \]