To find the estimated average rate of change for each function over the interval \([0.1, 0.6]\), we can use the formula for average rate of change:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \(f(x)\) is the function, and \(a = 0.1\) and \(b = 0.6\).
For the exponential function: \(y = 9(13^x)\)
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Calculate \(f(0.1)\): \[ f(0.1) = 9(13^{0.1}) \approx 9(1.347) \approx 12.123 \]
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Calculate \(f(0.6)\): \[ f(0.6) = 9(13^{0.6}) \approx 9(5.395) \approx 48.555 \]
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Calculate the average rate of change: \[ \text{Average Rate of Change} = \frac{48.555 - 12.123}{0.6 - 0.1} = \frac{36.432}{0.5} = 72.864 \]
For the quadratic function: \(y = 9x^2 + 13x\)
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Calculate \(f(0.1)\): \[ f(0.1) = 9(0.1^2) + 13(0.1) = 9(0.01) + 1.3 = 0.09 + 1.3 = 1.39 \]
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Calculate \(f(0.6)\): \[ f(0.6) = 9(0.6^2) + 13(0.6) = 9(0.36) + 7.8 = 3.24 + 7.8 = 11.04 \]
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Calculate the average rate of change: \[ \text{Average Rate of Change} = \frac{11.04 - 1.39}{0.6 - 0.1} = \frac{9.65}{0.5} = 19.3 \]
Comparison of rates of change:
- Average rate of change of the exponential function = \(72.864\)
- Average rate of change of the quadratic function = \(19.3\)
Both rates of change are positive. Therefore, the conclusion is:
Response: neither function.