To find the estimated average rates of change of the exponential function (let's denote it as \( g(x) \)) and the quadratic function \( f(x) = x^2 - 20 \) over the interval \([-9, 2]\), we will use the formula for the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
where \( a \) and \( b \) are the endpoints of the interval.
Step 1: Average Rate of Change for the Quadratic Function \( f(x) \)
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Identify the interval endpoints:
- \( a = -9 \)
- \( b = 2 \)
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Calculate \( f(-9) \):
\[ f(-9) = (-9)^2 - 20 = 81 - 20 = 61 \]
- Calculate \( f(2) \):
\[ f(2) = (2)^2 - 20 = 4 - 20 = -16 \]
- Calculate the average rate of change:
\[ \text{Average Rate of Change of } f = \frac{f(2) - f(-9)}{2 - (-9)} = \frac{-16 - 61}{2 + 9} = \frac{-77}{11} = -7 \]
Step 2: Average Rate of Change for the Exponential Function \( g(x) \)
Since you mentioned an exponential function but did not provide the function \( g(x) \), let's denote it generically as \( g(x) \). If you have specific \( g(x) \), please provide it for precise calculation. However, for the calculation of rates of change process, I'll illustrate it with an example exponential function, like \( g(x) = e^x \).
- Calculate \( g(-9) \): Assuming the exponential function is \( g(x) = e^x \):
\[ g(-9) = e^{-9} \quad (\text{calculate the numerical value}) \]
Using a calculator:
\[ g(-9) \approx 0.0001234 \]
- Calculate \( g(2) \):
\[ g(2) = e^2 \approx 7.38906 \]
- Calculate the average rate of change:
\[ \text{Average Rate of Change of } g = \frac{g(2) - g(-9)}{2 - (-9)} = \frac{7.38906 - 0.0001234}{2 + 9} = \frac{7.3889366}{11} \approx 0.6726 \]
Summary of Results
- Average Rate of Change of Quadratic Function \( f(x) \): \( -7 \)
- Average Rate of Change of Exponential Function \( g(x) \) (assuming \( g(x) = e^x \)): \( \approx 0.6726 \)
If you provide the specific exponential function, I can adjust the calculation accordingly!
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