To compare the estimated average rate of change of the exponential function \( y = 9(13)^x \) and the quadratic function \( y = 9x^2 + 13x \) over the interval \([0.1, 0.6]\), we start by calculating the average rate of change for both functions over this interval 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 \( y = 9(13)^x \)
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Calculate \( f(0.1) \): \[ f(0.1) = 9(13)^{0.1} \]
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Calculate \( f(0.6) \): \[ f(0.6) = 9(13)^{0.6} \]
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Then compute the average rate of change: \[ \text{Average Rate of Change for Exponential} = \frac{f(0.6) - f(0.1)}{0.6 - 0.1} \]
For the Quadratic Function \( y = 9x^2 + 13x \)
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Calculate \( g(0.1) \): \[ g(0.1) = 9(0.1)^2 + 13(0.1) = 9(0.01) + 1.3 = 0.09 + 1.3 = 1.39 \]
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Calculate \( g(0.6) \): \[ g(0.6) = 9(0.6)^2 + 13(0.6) = 9(0.36) + 7.8 = 3.24 + 7.8 = 11.04 \]
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Then compute the average rate of change: \[ \text{Average Rate of Change for Quadratic} = \frac{g(0.6) - g(0.1)}{0.6 - 0.1} = \frac{11.04 - 1.39}{0.5} = \frac{9.65}{0.5} = 19.3 \]
Summary of Results
- The average rate of change for the exponential function is positive (since \( (13)^x \) grows rapidly as \( x \) increases).
- The average rate of change for the quadratic function is also positive (as it has a positive coefficient for \( x^2 \) and \( x \)).
Since both functions have a positive estimated average rate of change over the interval \([0.1, 0.6]\), the answer is:
neither function.
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