To compare the average rates of change of the functions \( p(x) = \sqrt{x + 4} \) (square root function) and \( f(x) = \sqrt[3]{x + 4} \) (cube root function) over the interval \([-4, -3]\), we first calculate the values of the functions at the endpoints of the interval.
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Calculating \( p(x) = \sqrt{x + 4} \):
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At \( x = -4 \): \[ p(-4) = \sqrt{-4 + 4} = \sqrt{0} = 0 \]
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At \( x = -3 \): \[ p(-3) = \sqrt{-3 + 4} = \sqrt{1} = 1 \]
Therefore, the average rate of change of \( p(x) \) over the interval \([-4, -3]\) is calculated as follows: \[ \text{Average Rate of Change of } p(x) = \frac{p(-3) - p(-4)}{-3 - (-4)} = \frac{1 - 0}{-3 + 4} = \frac{1}{1} = 1 \]
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Calculating \( f(x) = \sqrt[3]{x + 4} \):
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At \( x = -4 \): \[ f(-4) = \sqrt[3]{-4 + 4} = \sqrt[3]{0} = 0 \]
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At \( x = -3 \): \[ f(-3) = \sqrt[3]{-3 + 4} = \sqrt[3]{1} = 1 \]
Therefore, the average rate of change of \( f(x) \) over the interval \([-4, -3]\) is calculated as follows: \[ \text{Average Rate of Change of } f(x) = \frac{f(-3) - f(-4)}{-3 - (-4)} = \frac{1 - 0}{-3 + 4} = \frac{1}{1} = 1 \]
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Both functions have an average rate of change of 1 over the interval \([-4, -3]\).
Conclusion: The functions have the same average rate of change of 1.