To analyze the estimated average rates of change of the functions \( b(x) \) and \( d(x) \) over the interval \([-12, -3]\), we first need to compute the rates of change for both functions.
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Finding \( b(x) \): \[ b(x) = \sqrt[3]{3x + 9} \] We will evaluate \( b(-12) \) and \( b(-3) \):
\[ b(-12) = \sqrt[3]{3(-12) + 9} = \sqrt[3]{-36 + 9} = \sqrt[3]{-27} = -3 \]
\[ b(-3) = \sqrt[3]{3(-3) + 9} = \sqrt[3]{-9 + 9} = \sqrt[3]{0} = 0 \]
The average rate of change of \( b(x) \) over the interval \([-12, -3]\) is: \[ \text{Average Rate of Change of } b = \frac{b(-3) - b(-12)}{-3 - (-12)} = \frac{0 - (-3)}{-3 + 12} = \frac{3}{9} = \frac{1}{3} \]
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Finding \( d(x) \): \[ d(x) = \sqrt{-3x + 9} \] We will evaluate \( d(-12) \) and \( d(-3) \):
\[ d(-12) = \sqrt{-3(-12) + 9} = \sqrt{36 + 9} = \sqrt{45} \approx 6.71 \]
\[ d(-3) = \sqrt{-3(-3) + 9} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24 \]
The average rate of change of \( d(x) \) over the interval \([-12, -3]\) is: \[ \text{Average Rate of Change of } d = \frac{d(-3) - d(-12)}{-3 - (-12)} = \frac{4.24 - 6.71}{-3 + 12} \approx \frac{-2.47}{9} \approx -0.27 \]
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Comparison of Average Rates of Change:
- The average rate of change for \( b(x) \approx \frac{1}{3} \) (positive).
- The average rate of change for \( d(x) \approx -0.27 \) (negative).
Based on this analysis, the appropriate response regarding the estimated average rates of change is:
The estimated average rate of change of \( b(x) \) is greater than the estimated average rate of change of \( d(x) \) because \( b(x) \) is increasing over the interval, but \( d(x) \) is decreasing.