To find the average rates of change of the two functions over the specified interval, we will need to use the formula for average rate of change between two points, which is given by:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Where \( f(x) \) is the function in question, \( a \) and \( b \) are the x-values of the two points.
Since the specific points of the exponential function and the interval are not provided, let's assume some general points for the analysis. For example, if the exponential function is \( f(x) = e^x \), then choose points \( a = 0 \) and \( b = 2 \):
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For the Exponential Function (Example: \( f(x) = e^x \)):
- Calculate \( f(0) = e^0 = 1 \)
- Calculate \( f(2) = e^2 \approx 7.39 \)
\[ \text{Average Rate of Change} = \frac{e^2 - e^0}{2 - 0} = \frac{7.39 - 1}{2} \approx \frac{6.39}{2} \approx 3.195 \]
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For the Quadratic Function \( f(x) = (x - 0.1)^2 - 24.01 \):
- Calculate \( f(0) = (0 - 0.1)^2 - 24.01 = 0.01 - 24.01 = -24 \)
- Calculate \( f(2) = (2 - 0.1)^2 - 24.01 = (1.9)^2 - 24.01 \approx 3.61 - 24.01 = -20.4 \)
\[ \text{Average Rate of Change} = \frac{-20.4 - (-24)}{2 - 0} = \frac{-20.4 + 24}{2} = \frac{3.6}{2} = 1.8 \]
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Comparison of Average Rates of Change:
- Average rate of change of the exponential function: \( \approx 3.195 \)
- Average rate of change of the quadratic function: \( 1.8 \)
From these calculations, we observe that:
The average rate of change of the exponential function is significantly greater than the average rate of change of the quadratic function.
Hence, the correct response would be:
The average rate of change of the exponential function is significantly greater than the average rate of change of the quadratic function.
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