To find the average rate of change of the function \( F(x) = x^2 - 3 \) over a given interval \([a, b]\), we use the formula:
\[ \text{Average Rate of Change} = \frac{F(b) - F(a)}{b - a} \]
Let's calculate the average rate of change for each interval.
1. Interval \([-3, 0]\)
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Calculate \( F(-3) \): \[ F(-3) = (-3)^2 - 3 = 9 - 3 = 6 \]
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Calculate \( F(0) \): \[ F(0) = (0)^2 - 3 = 0 - 3 = -3 \]
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Now, use the average rate of change formula: \[ \text{Average Rate of Change} = \frac{F(0) - F(-3)}{0 - (-3)} = \frac{-3 - 6}{0 + 3} = \frac{-9}{3} = -3 \]
2. Interval \([2, 3]\)
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Calculate \( F(2) \): \[ F(2) = (2)^2 - 3 = 4 - 3 = 1 \]
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Calculate \( F(3) \): \[ F(3) = (3)^2 - 3 = 9 - 3 = 6 \]
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Now, use the average rate of change formula: \[ \text{Average Rate of Change} = \frac{F(3) - F(2)}{3 - 2} = \frac{6 - 1}{1} = 5 \]
3. Interval \([5, 6]\)
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Calculate \( F(5) \): \[ F(5) = (5)^2 - 3 = 25 - 3 = 22 \]
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Calculate \( F(6) \): \[ F(6) = (6)^2 - 3 = 36 - 3 = 33 \]
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Now, use the average rate of change formula: \[ \text{Average Rate of Change} = \frac{F(6) - F(5)}{6 - 5} = \frac{33 - 22}{1} = 11 \]
Summary of Results
- Average Rate of Change on \([-3, 0]\) is \( -3 \)
- Average Rate of Change on \([2, 3]\) is \( 5 \)
- Average Rate of Change on \([5, 6]\) is \( 11 \)