To calculate the average rate of change of the function \( f(x) = 3^x + 2 \) over the interval \([-2, -1]\), we can use the formula for the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Here, \( a = -2 \) and \( b = -1 \). We first need to find \( f(-2) \) and \( f(-1) \).
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Calculate \( f(-2) \): \[ f(-2) = 3^{-2} + 2 = \frac{1}{3^2} + 2 = \frac{1}{9} + 2 = \frac{1}{9} + \frac{18}{9} = \frac{19}{9} \]
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Calculate \( f(-1) \): \[ f(-1) = 3^{-1} + 2 = \frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{7}{3} \]
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Now plug \( f(-2) \) and \( f(-1) \) into the average rate of change formula: \[ \text{Average Rate of Change} = \frac{f(-1) - f(-2)}{-1 - (-2)} = \frac{\frac{7}{3} - \frac{19}{9}}{-1 + 2} \]
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Simplify the denominator: \[ -1 + 2 = 1 \]
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Now simplify the numerator (\( \frac{7}{3} - \frac{19}{9} \)): To subtract these fractions, we need a common denominator. The least common denominator between 3 and 9 is 9. \[ \frac{7}{3} = \frac{7 \times 3}{3 \times 3} = \frac{21}{9} \] Thus, \[ \frac{7}{3} - \frac{19}{9} = \frac{21}{9} - \frac{19}{9} = \frac{21 - 19}{9} = \frac{2}{9} \]
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Now, substitute back: \[ \text{Average Rate of Change} = \frac{\frac{2}{9}}{1} = \frac{2}{9} \]
Therefore, the average rate of change of the function \( f(x) = 3^x + 2 \) over the interval \([-2, -1]\) is \(\frac{2}{9}\).