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 can use the formula for average rate of change over an interval \([a, b]\):
\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \]
Step 1: Calculate the average rate of change for \( f(x) \)
First, we need to evaluate \( f(2) \) and \( f(3) \):
\[ f(2) = \sqrt{3(2) - 4} = \sqrt{6 - 4} = \sqrt{2} \approx 1.414 \]
\[ f(3) = \sqrt{3(3) - 4} = \sqrt{9 - 4} = \sqrt{5} \approx 2.236 \]
Now we can calculate the average rate of change for \( f \):
\[ \text{Average rate of change of } f = \frac{f(3) - f(2)}{3 - 2} = \frac{\sqrt{5} - \sqrt{2}}{1} \approx 2.236 - 1.414 \approx 0.822 \]
Step 2: Calculate the average rate of change for \( g(x) \)
Now evaluate \( g(2) \) and \( g(3) \):
\[ g(2) = \sqrt[3]{2(2) - 4} = \sqrt[3]{4 - 4} = \sqrt[3]{0} = 0 \]
\[ g(3) = \sqrt[3]{2(3) - 4} = \sqrt[3]{6 - 4} = \sqrt[3]{2} \approx 1.260 \]
Now we can calculate the average rate of change for \( g \):
\[ \text{Average rate of change of } g = \frac{g(3) - g(2)}{3 - 2} = \frac{\sqrt[3]{2} - 0}{1} \approx \sqrt[3]{2} \approx 1.260 \]
Step 3: Compare the average rates of change
Now we compare the average rates of change:
- Average rate of change of \( f \): \( \approx 0.822 \)
- Average rate of change of \( g \): \( \approx 1.260 \)
Difference in average rates of change:
\[ \text{Difference} = g_{\text{avg}} - f_{\text{avg}} \approx 1.260 - 0.822 \approx 0.438 \]
Final result:
Rounding to the nearest tenth, the difference between the estimated average rates of change is approximately:
\[ \boxed{0.4} \]
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