To determine 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 average rate of change:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = 1 \) and \( b = 2.39 \).
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 \( f(2.39) \): \[ f(2.39) = 3.5(2.39^2) = 3.5(5.7121) \approx 20.49235 \]
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Now plug these values into the average rate of change formula: \[ \text{Average Rate of Change for } f(x) = \frac{20.49235 - 3.5}{2.39 - 1} \approx \frac{16.99235}{1.39} \approx 12.21 \]
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 plug these values into the average rate of change formula: \[ \text{Average Rate of Change for } g(x) = \frac{8.365 - 3.5}{2.39 - 1} \approx \frac{4.865}{1.39} \approx 3.50 \]
Comparison:
- The estimated average rate of change of \( f(x) \) is approximately \( 12.21 \).
- The estimated average rate of change of \( g(x) \) is approximately \( 3.50 \).
Conclusion:
The estimated average rate of change of \( f(x) \) is greater than the estimated average 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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