To find the estimated average rates of change of the functions \( p(x) \) and \( q(x) \) over the interval \([0.1, 8.9]\), we need to calculate the average rate of change using the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( f(x) \) is the function, \( a = 0.1 \), and \( b = 8.9 \).
For \( p(x) = \sqrt{x} - 5 \):
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Calculate \( p(0.1) \): \[ p(0.1) = \sqrt{0.1} - 5 \approx 0.316 - 5 \approx -4.684 \]
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Calculate \( p(8.9) \): \[ p(8.9) = \sqrt{8.9} - 5 \approx 2.983 - 5 \approx -2.017 \]
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Now, calculate the average rate of change: \[ \text{Average Rate of Change of } p = \frac{p(8.9) - p(0.1)}{8.9 - 0.1} = \frac{-2.017 - (-4.684)}{8.8} \approx \frac{2.667}{8.8} \approx 0.303 \]
For \( q(x) = 5\sqrt[3]{x} - 1 \):
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Calculate \( q(0.1) \): \[ q(0.1) = 5\sqrt[3]{0.1} - 1 \approx 5 \cdot 0.464 - 1 \approx 2.32 - 1 = 1.32 \]
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Calculate \( q(8.9) \): \[ q(8.9) = 5\sqrt[3]{8.9} - 1 \approx 5 \cdot 2.080 - 1 \approx 10.4 - 1 = 9.4 \]
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Now, calculate the average rate of change: \[ \text{Average Rate of Change of } q = \frac{q(8.9) - q(0.1)}{8.9 - 0.1} = \frac{9.4 - 1.32}{8.8} = \frac{8.08}{8.8} \approx 0.918 \]
Comparison:
- The average rate of change for \( p(x) \) is approximately \( 0.303 \).
- The average rate of change for \( q(x) \) is approximately \( 0.918 \).
Based on these calculations, we can conclude:
- The estimated average rate of change of \( q(x) \) is greater than that of \( p(x) \) over \([0.1, 8.9]\).
Thus, the appropriate response is:
The estimated average rate of change of \( q(x) \) is greater than the estimated average rate of change of \( p(x) \) over \([0.1, 8.9]\).