To find the estimated average rate of change of each function over the interval \([1.1, 1.6]\), we can apply the formula for the average rate of change of a function \(f(x)\) over the interval \([a, b]\):
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Step 1: Calculate for \(y = 16 \cdot 4^x\)
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Evaluate the function at the endpoints:
- \(f(1.1) = 16 \cdot 4^{1.1}\)
- \(f(1.6) = 16 \cdot 4^{1.6}\)
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Calculate \(f(1.1)\): \[ f(1.1) = 16 \cdot 4^{1.1} \approx 16 \cdot 4.57 \approx 73.12 \]
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Calculate \(f(1.6)\): \[ f(1.6) = 16 \cdot 4^{1.6} \approx 16 \cdot 6.86 \approx 109.76 \]
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Average Rate of Change for \(y = 16 \cdot 4^x\): \[ \text{Average Rate of Change} = \frac{109.76 - 73.12}{1.6 - 1.1} \approx \frac{36.64}{0.5} \approx 73.28 \]
Step 2: Calculate for \(y = 16x^2 + 4x\)
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Evaluate the function at the endpoints:
- \(g(1.1) = 16 \cdot (1.1)^2 + 4 \cdot (1.1)\)
- \(g(1.6) = 16 \cdot (1.6)^2 + 4 \cdot (1.6)\)
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Calculate \(g(1.1)\): \[ g(1.1) = 16 \cdot (1.21) + 4 \cdot (1.1) = 16 \cdot 1.21 + 4.4 \approx 19.36 + 4.4 = 23.76 \]
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Calculate \(g(1.6)\): \[ g(1.6) = 16 \cdot (2.56) + 4 \cdot (1.6) = 16 \cdot 2.56 + 6.4 \approx 40.96 + 6.4 = 47.36 \]
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Average Rate of Change for \(y = 16x^2 + 4x\): \[ \text{Average Rate of Change} = \frac{47.36 - 23.76}{1.6 - 1.1} \approx \frac{23.6}{0.5} \approx 47.2 \]
Step 3: Compare the Average Rates of Change
Now, we compare the two average rates of change:
- Average Rate of Change for \(y = 16 \cdot 4^x\) is approximately \(73.28\).
- Average Rate of Change for \(y = 16x^2 + 4x\) is approximately \(47.2\).
Step 4: Calculate the Difference
\[ \text{Difference} = 73.28 - 47.2 \approx 26.08 \]
Conclusion
The estimated average rate of change of the function \(y = 16 \cdot 4^x\) is approximately \(26.08\) greater than the estimated average rate of change of the function \(y = 16x^2 + 4x\) over the interval \([1.1, 1.6]\).
Since \(26.08\) is closest to 10 among the options provided (1, 10, or 100), we conclude:
The estimated average rate of change of the function \(y = 16 \cdot 4^x\) is approximately 10 greater than the estimated average rate of change of the function \(y = 16x^2 + 4x\) over the interval \([1.1, 1.6]\).