To find the average rate of change of each function over the interval [1.1, 1.6], we can use the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( f(x) \) is the function, \( a = 1.1 \), and \( b = 1.6 \).
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Calculating for \( y = 16 \cdot 4^x \):
First, we need to evaluate \( y \) at \( x = 1.1 \) and \( x = 1.6 \).
\[ f(1.1) = 16 \cdot 4^{1.1} \]
\[ f(1.6) = 16 \cdot 4^{1.6} \]
Now calculating these values:
- \( 4^{1.1} \approx 4.64 \) (approximately)
- \( 4^{1.6} \approx 9.24 \) (approximately)
Then,
\[ f(1.1) \approx 16 \cdot 4.64 \approx 74.24 \]
\[ f(1.6) \approx 16 \cdot 9.24 \approx 147.84 \]
Now we compute the average rate of change:
\[ \text{Average Rate of Change} = \frac{147.84 - 74.24}{1.6 - 1.1} = \frac{73.6}{0.5} = 147.2 \]
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Calculating for \( y = 16x^2 + 4x \):
We do the same for this function:
\[ g(1.1) = 16(1.1^2) + 4(1.1) = 16(1.21) + 4.4 \approx 19.36 + 4.4 \approx 23.76 \]
\[ g(1.6) = 16(1.6^2) + 4(1.6) = 16(2.56) + 6.4 \approx 40.96 + 6.4 \approx 47.36 \]
Now, compute the average rate of change:
\[ \text{Average Rate of Change} = \frac{47.36 - 23.76}{1.6 - 1.1} = \frac{23.6}{0.5} = 47.2 \]
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Finding the difference:
Now we find the difference between the two average rates of change:
\[ \text{Difference} = 147.2 - 47.2 = 100 \]
Thus, the estimated average rate of change of the function \( y = 16 \cdot 4^x \) is approximately 100 greater than that of the function \( y = 16x^2 + 4x \) over the interval [1.1, 1.6].