To determine which statements about the average rate of change over the interval \([a,b]\) for a function \(f(x)\) are true, let's analyze the provided responses:
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The average rate of change cannot be zero.
- False. The average rate of change can be zero if \(f(b) = f(a)\), meaning the function has the same value at both endpoints.
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The average rate of change can be calculated using the formula \(b−af(b)−f(a)\).
- False. This formula is incorrect.
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The average rate of change can be calculated using the formula \(\frac{b−a}{f(b)−f(a)}\).
- False. This formula is also incorrect; the numerator should be \(f(b)−f(a)\) rather than \(b−a\).
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The average rate of change can be calculated using the formula \(\frac{f(b)−f(a)}{a−b}\).
- True. This formula correctly represents the average rate of change, but it is often expressed with a negative sign as \(\frac{f(b)−f(a)}{b−a}\) for conventional interpretation.
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The average rate of change is negative if \(f(b)<f(a)\).
- True. If \(f(b)\) is less than \(f(a)\), it indicates that the function is decreasing over that interval, resulting in a negative rate of change.
Summary of True Statements:
- The average rate of change can be calculated using \(\frac{f(b)−f(a)}{a−b}\) (based on the standard formula).
- The average rate of change is negative if \(f(b)<f(a)\).
Thus, the last two statements are true.