To find the estimated average rate of change of the functions \( y = 16 \cdot 4^x \) and \( y = 16x^2 + 4x \) over the interval \([1.1, 1.6]\), we can use the formula for average rate of change, which is:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a \) and \( b \) are the endpoints of the interval.
For \( y = 16 \cdot 4^x \):
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Calculate \( f(1.1) \): \[ f(1.1) = 16 \cdot 4^{1.1} \approx 16 \cdot 4^{1.1} \approx 16 \cdot 4.57 \approx 73.12 \]
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Calculate \( f(1.6) \): \[ f(1.6) = 16 \cdot 4^{1.6} \approx 16 \cdot 4^{1.6} \approx 16 \cdot 10.08 \approx 161.28 \]
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Now, apply the average rate of change formula: \[ \text{Average Rate of Change} = \frac{f(1.6) - f(1.1)}{1.6 - 1.1} = \frac{161.28 - 73.12}{0.5} = \frac{88.16}{0.5} \approx 176.32 \]
For \( y = 16x^2 + 4x \):
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Calculate \( f(1.1) \): \[ f(1.1) = 16 \cdot (1.1)^2 + 4 \cdot (1.1) = 16 \cdot 1.21 + 4 \cdot 1.1 = 19.36 + 4.4 = 23.76 \]
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Calculate \( f(1.6) \): \[ f(1.6) = 16 \cdot (1.6)^2 + 4 \cdot (1.6) = 16 \cdot 2.56 + 4 \cdot 1.6 = 40.96 + 6.4 = 47.36 \]
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Now, apply the average rate of change formula: \[ \text{Average Rate of Change} = \frac{f(1.6) - f(1.1)}{1.6 - 1.1} = \frac{47.36 - 23.76}{0.5} = \frac{23.6}{0.5} \approx 47.2 \]
Compare the Average Rates of Change:
- Average rate of change for \( y = 16 \cdot 4^x \): \( \approx 176.32 \)
- Average rate of change for \( y = 16x^2 + 4x \): \( \approx 47.2 \)
Finding the Difference:
\[ \text{Difference} = 176.32 - 47.2 \approx 129.12 \]
Conclusion:
The estimated average rate of change of the function \( y = 16 \cdot 4^x \) is approximately \( 129.12 \) greater than that of \( y = 16x^2 + 4x \).
Since we need to determine if this difference is approximately 1, 10, or 100:
- Clearly, \( 129.12 \) is greater than 100.
Thus, the answer is 100 (approximately).