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 the average rate of change over the interval \([a, b]\):
\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \]
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 for \( f(x) \)
Using the values obtained: \[ \text{Average rate of change of } f = \frac{f(3) - f(2)}{3 - 2} = \frac{\sqrt{5} - \sqrt{2}}{1} = \sqrt{5} - \sqrt{2} \]
Step 3: Calculate \( g(2) \) and \( g(3) \)
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Calculate \( g(2) \): \[ g(2) = \sqrt[3]{2(2) - 4} = \sqrt[3]{4 - 4} = \sqrt[3]{0} = 0 \]
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Calculate \( g(3) \): \[ g(3) = \sqrt[3]{2(3) - 4} = \sqrt[3]{6 - 4} = \sqrt[3]{2} \]
Step 4: Calculate the average rate of change for \( g(x) \)
Using the values obtained: \[ \text{Average rate of change of } g = \frac{g(3) - g(2)}{3 - 2} = \frac{\sqrt[3]{2} - 0}{1} = \sqrt[3]{2} \]
Step 5: Calculate the difference in estimated average rates of change
Now we need to calculate the difference: \[ \text{Difference} = (\sqrt{5} - \sqrt{2}) - \sqrt[3]{2} \]
Step 6: Numerical Approximation
We can approximate these values:
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt[3]{2} \approx 1.260 \)
Substituting these approximations into the expression for the difference: \[ \text{Difference} \approx (2.236 - 1.414) - 1.260 \] Calculating \( 2.236 - 1.414 \) gives approximately \( 0.822 \).
Now we find: \[ 0.822 - 1.260 \approx -0.438 \]
Final step: Rounding to the nearest tenth
The difference to the nearest tenth is: \[ \text{Difference} \approx -0.4 \]
Thus, the final answer is: \[ \text{The difference between the estimated average rates of change is approximately } -0.4. \]