To compare the estimated average rates of change of the functions \( f(x) = 3.5x^2 \) and \( g(x) = 3.5x \) over the interval \([1, 2.39]\), we can use the formula for the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = 1 \) and \( b = 2.39 \).
Step 1: Calculate the average rate of change for \( f(x) \)
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Calculate \( f(1) \): \[ f(1) = 3.5(1^2) = 3.5 \]
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Calculate \( f(2.39) \): \[ f(2.39) = 3.5(2.39^2) = 3.5 \cdot 5.7121 \approx 20.03635 \]
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Now calculate the average rate of change: \[ \text{Average Rate of Change of } f(x) = \frac{f(2.39) - f(1)}{2.39 - 1} = \frac{20.03635 - 3.5}{2.39 - 1} \approx \frac{16.53635}{1.39} \approx 11.88 \]
Step 2: Calculate the average rate of change for \( g(x) \)
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Calculate \( g(1) \): \[ g(1) = 3.5(1) = 3.5 \]
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Calculate \( g(2.39) \): \[ g(2.39) = 3.5(2.39) \approx 8.365 \]
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Now calculate the average rate of change: \[ \text{Average Rate of Change of } g(x) = \frac{g(2.39) - g(1)}{2.39 - 1} = \frac{8.365 - 3.5}{2.39 - 1} \approx \frac{4.865}{1.39} \approx 3.5 \]
Comparison:
- Average rate of change of \( f(x) \) is approximately \( 11.88 \).
- Average rate of change of \( g(x) \) is approximately \( 3.5 \).
From this calculation, we see that the estimated rate of change of \( f(x) \) is greater than the estimated rate of change of \( g(x) \) over the interval \([1, 2.39]\).
Thus, the correct response is:
- The estimated rate of change of \( f(x) \) is greater than the estimated rate of change of \( g(x) \) over the interval [1, 2.39].
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