To compare 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 compute the average rate of change for both functions over that interval.
The average rate of change of a function \( h(x) \) over the interval \([a, b]\) is given by:
\[ \text{Average Rate of Change} = \frac{h(b) - h(a)}{b - a} \]
Step 1: Compute \( f(2) \) and \( f(3) \)
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For \( f(2) \): \[ f(2) = \sqrt{3(2) - 4} = \sqrt{6 - 4} = \sqrt{2} \]
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For \( f(3) \): \[ f(3) = \sqrt{3(3) - 4} = \sqrt{9 - 4} = \sqrt{5} \]
Average rate of change for \( f(x) \):
\[ \text{Average Rate of Change for } f = \frac{\sqrt{5} - \sqrt{2}}{3 - 2} = \sqrt{5} - \sqrt{2} \]
Step 2: Compute \( g(2) \) and \( g(3) \)
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For \( g(2) \): \[ g(2) = \sqrt[3]{2(2) - 43} = \sqrt[3]{4 - 43} = \sqrt[3]{-39} \]
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For \( g(3) \): \[ g(3) = \sqrt[3]{2(3) - 43} = \sqrt[3]{6 - 43} = \sqrt[3]{-37} \]
Average rate of change for \( g(x) \):
\[ \text{Average Rate of Change for } g = \frac{\sqrt[3]{-37} - \sqrt[3]{-39}}{3 - 2} = \sqrt[3]{-37} - \sqrt[3]{-39} \]
Step 3: Calculate the difference
We need to find the difference in the average rates of change:
\[ \text{Difference} = \left( \sqrt{5} - \sqrt{2} \right) - \left( \sqrt[3]{-37} - \sqrt[3]{-39} \right) \]
Let's calculate the values numerically:
- \( \sqrt{5} \approx 2.236 \)
- \( \sqrt{2} \approx 1.414 \)
So, \[ \sqrt{5} - \sqrt{2} \approx 2.236 - 1.414 \approx 0.822 \]
Now, for the cube roots:
- \( \sqrt[3]{-37} \approx -3.332 \)
- \( \sqrt[3]{-39} \approx -3.386 \)
So, \[ \sqrt[3]{-37} - \sqrt[3]{-39} \approx -3.332 + 3.386 \approx 0.054 \]
Final Calculation of the Difference
Now, let's compute the difference:
\[ \text{Difference} \approx 0.822 - 0.054 = 0.768 \]
Rounding to the nearest tenth, the difference between the estimated average rates of change is approximately:
\[ \boxed{0.8} \]
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