To determine the estimated average rates of change for the exponential function and the quadratic function \( f(x) = x^2 - 20 \) over the interval \([-9, 2]\), we can use the formula for the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a = -9 \) and \( b = 2 \).
Calculating the Average Rate of Change for \( f(x) = x^2 - 20 \)
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Calculate \( f(-9) \): \[ f(-9) = (-9)^2 - 20 = 81 - 20 = 61 \]
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Calculate \( f(2) \): \[ f(2) = 2^2 - 20 = 4 - 20 = -16 \]
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Calculate the average rate of change: \[ \text{Average Rate of Change} = \frac{f(2) - f(-9)}{2 - (-9)} = \frac{-16 - 61}{2 + 9} = \frac{-77}{11} = -7 \]
Estimating the Average Rate of Change for the Exponential Function
To provide a meaningful comparison, we would need an exponential function. A common example would be \( g(x) = e^x \) or similar. Assuming we are comparing the same interval and we use \( g(x) = a^x \) for some base \( a \) (for instance, \( a = e \)), we could say:
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Calculate \( g(-9) \) and \( g(2) \) using, say, \( g(x) = e^x \):
\[ g(-9) \approx e^{-9} \quad \text{(which is a very small positive number)} \] \[ g(2) \approx e^2 \approx 7.39 \]
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Calculate the average rate of change: \[ \text{Average Rate of Change} = \frac{g(2) - g(-9)}{2 - (-9)} = \frac{7.39 - e^{-9}}{11} \] \( e^{-9} \) being negligible means \( \text{average rate} \approx \frac{7.39}{11} \approx 0.67 \)
Conclusion
- The average rate of change of \( f(x) = x^2 - 20 \) over the interval \([-9, 2]\) is \(-7\).
- The average rate of change of a common exponential function like \( g(x) = e^x \) tends to be a small positive value.
As a result, in terms of the provided options, the best conclusion is:
- “The estimated average rate of change of the exponential function over the interval \([-9, 2]\) is negative, while the estimated average rate of change of \( f(x) \) over the interval \([-9, 2]\) is positive.”
In summary, we can infer:
The estimated average rate of change of the exponential function is less than the rate of change of the quadratic function when evaluated over that interval.