To find the estimated average rates of change of the exponential function and the quadratic function \( f(x) = x^2 - 20 \) over the interval \([-9, 2]\), we will follow these steps:
- Calculate the interval endpoints for both functions.
- Use the formula for average rate of change.
The average rate of change of a function \( f \) over the interval \([a, b]\) is given by the formula:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
For the Quadratic Function \( f(x) = x^2 - 20 \)
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Evaluate \( f(-9) \) and \( f(2) \):
\[ f(-9) = (-9)^2 - 20 = 81 - 20 = 61 \]
\[ f(2) = (2)^2 - 20 = 4 - 20 = -16 \]
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Substitute into the average rate of change formula:
\[ \text{Average Rate of Change} = \frac{f(2) - f(-9)}{2 - (-9)} = \frac{-16 - 61}{2 + 9} = \frac{-77}{11} = -7 \]
For the Exponential Function
Since the exponential function is not explicitly given, I'll assume a general form \( g(x) = a \cdot b^x \). You need the specific function to evaluate it properly. Let’s say we are working with an example function, such as \( g(x) = e^x \) (or another common base).
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Evaluate \( g(-9) \) and \( g(2) \) (if it's \( g(x) = e^x \)):
\[ g(-9) = e^{-9} \quad \text{(Use a calculator to get a decimal approximation)} \] \[ g(-9) \approx 0.0001234 \]
\[ g(2) = e^2 \approx 7.3891 \]
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Substitute into the average rate of change formula:
To find the average rate of change:
\[ \text{Average Rate of Change} = \frac{g(2) - g(-9)}{2 - (-9)} = \frac{7.3891 - 0.0001234}{2 + 9} = \frac{7.3891 - 0.0001234}{11} \approx \frac{7.389}{11} \approx 0.672 \]
Summary of Results
- Average rate of change for \( f(x) = x^2 - 20 \) over \([-9, 2]\): \(-7\)
- Average rate of change for the exponential function \( g(x) = e^x \) over \([-9, 2]\) (if \( g(x) = e^x \)): \(\approx 0.672\)
Please replace \( g(x) \) with the specific exponential function you have to get correct values.
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