To find the estimated average rate of change of both functions over the interval \([0.1, 0.6]\), we will calculate the average rate of change using the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \(a = 0.1\) and \(b = 0.6\).
For the exponential function \(y = 9(1/3)^x\):
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Calculate \(f(0.1)\): \[ f(0.1) = 9\left(\frac{1}{3}\right)^{0.1} \approx 9 \times 0.933 = 8.397 \]
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Calculate \(f(0.6)\): \[ f(0.6) = 9\left(\frac{1}{3}\right)^{0.6} \approx 9 \times 0.481 = 4.329 \]
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Compute the average rate of change: \[ \text{Average Rate of Change} = \frac{4.329 - 8.397}{0.6 - 0.1} = \frac{-4.068}{0.5} = -8.136 \]
For the quadratic function \(y = 9x^2 + \frac{1}{3}x\):
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Calculate \(f(0.1)\): \[ f(0.1) = 9(0.1)^2 + \frac{1}{3}(0.1) = 9(0.01) + \frac{0.1}{3} = 0.09 + 0.0333 \approx 0.1233 \]
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Calculate \(f(0.6)\): \[ f(0.6) = 9(0.6)^2 + \frac{1}{3}(0.6) = 9(0.36) + 0.2 = 3.24 + 0.2 = 3.44 \]
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Compute the average rate of change: \[ \text{Average Rate of Change} = \frac{3.44 - 0.1233}{0.6 - 0.1} = \frac{3.3167}{0.5} \approx 6.6334 \]
Conclusion:
- The average rate of change for the exponential function is approximately \(-8.136\) (negative).
- The average rate of change for the quadratic function is approximately \(6.6334\) (positive).
Thus, the function with a negative estimated average rate of change over the interval \([0.1, 0.6]\) is:
the exponential function.