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 will use the formula for the average rate of change of a function \( f \) over the interval \([a, b]\):
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Step 1: Calculate the average rate of change for \( y = 16 \cdot 4^x \)
Let \( f(x) = 16 \cdot 4^x \).
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Calculate \( f(1.1) \): \[ f(1.1) = 16 \cdot 4^{1.1} \approx 16 \cdot 4^{1.1} \approx 16 \cdot 4.6416 \approx 74.2656 \]
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Calculate \( f(1.6) \): \[ f(1.6) = 16 \cdot 4^{1.6} \approx 16 \cdot 4^{1.6} \approx 16 \cdot 10.0794 \approx 161.271 \]
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Compute the average rate of change: \[ \text{Average Rate of Change} = \frac{f(1.6) - f(1.1)}{1.6 - 1.1} = \frac{161.271 - 74.2656}{0.5} = \frac{87.0054}{0.5} \approx 174.0108 \]
Step 2: Calculate the average rate of change for \( y = 16x^2 + 4x \)
Let \( g(x) = 16x^2 + 4x \).
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Calculate \( g(1.1) \): \[ g(1.1) = 16 \cdot (1.1)^2 + 4 \cdot (1.1) = 16 \cdot 1.21 + 4.4 = 19.36 + 4.4 = 23.76 \]
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Calculate \( g(1.6) \): \[ g(1.6) = 16 \cdot (1.6)^2 + 4 \cdot (1.6) = 16 \cdot 2.56 + 6.4 = 40.96 + 6.4 = 47.36 \]
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Compute the average rate of change: \[ \text{Average Rate of Change} = \frac{g(1.6) - g(1.1)}{1.6 - 1.1} = \frac{47.36 - 23.76}{0.5} = \frac{23.6}{0.5} = 47.2 \]
Step 3: Calculate the difference
Now we can find how much greater the estimated average rate of change of the first function is than that of the second function:
\[ \text{Difference} = 174.0108 - 47.2 \approx 126.8108 \]
Conclusion
The difference is approximately 127, which suggests that it is closer to 100 than to 10 or 1. Thus, the estimated average rate of change of the function \( y=16 \cdot 4^x \) is approximately 100 units greater than that of \( y=16x^2 + 4x \) over the interval \([1.1, 1.6]\).