To determine which function has the greater average rate of change over the interval \([0, 1.1]\), we can calculate the average rate of change for both functions.
The average rate of change of a function \(f\) over an interval \([a, b]\) is given by the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Step 1: Calculate for the exponential function \(y = 4x\)
Let \(f(x) = 4x\).
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Evaluate \(f(0)\): \[ f(0) = 4 \cdot 0 = 0 \]
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Evaluate \(f(1.1)\): \[ f(1.1) = 4 \cdot 1.1 = 4.4 \]
Now, calculate the average rate of change: \[ \text{Average Rate of Change for } 4x = \frac{f(1.1) - f(0)}{1.1 - 0} = \frac{4.4 - 0}{1.1} = \frac{4.4}{1.1} = 4 \]
Step 2: Calculate for the quadratic function \(y = 4x^2\)
Let \(g(x) = 4x^2\).
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Evaluate \(g(0)\): \[ g(0) = 4 \cdot 0^2 = 0 \]
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Evaluate \(g(1.1)\): \[ g(1.1) = 4 \cdot (1.1)^2 = 4 \cdot 1.21 = 4.84 \]
Now, calculate the average rate of change: \[ \text{Average Rate of Change for } 4x^2 = \frac{g(1.1) - g(0)}{1.1 - 0} = \frac{4.84 - 0}{1.1} = \frac{4.84}{1.1} \approx 4.4 \]
Conclusion
- Average rate of change for \(y = 4x\) over \([0, 1.1]\): \(4\)
- Average rate of change for \(y = 4x^2\) over \([0, 1.1]\): \(\approx 4.4\)
Thus, the function with the greater estimated average rate of change on the interval is the quadratic function.
The answer is: Option 2: quadratic function.
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