The statement that is true about the average rate of change over the interval \([a, b]\) for the function \(f(x)\) is:
The average rate of change can be calculated using the formula \(\frac{f(b) - f(a)}{b - a}\).
This formula represents how much the function \(f\) changes on average for each unit increase in \(x\) over the interval \([a,b]\).
To clarify, here are the evaluations of the other statements:
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The average rate of change can be calculated using the formula \(b - a \over f(b) - f(a)\). (False) - This is incorrect since the correct formula is \(\frac{f(b) - f(a)}{b - a}\).
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The average rate of change can be calculated using the formula \(f(b) - f(a) \over a - b\). (False) - This would yield the negative of the correct average rate of change.
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The average rate of change can be calculated using the formula \( \frac{f(b) - f(a)}{a - b}\). (False) - While it’s mathematically valid, it represents the negative average rate of change.
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The average rate of change is negative if \(f(b) < f(a)\). (True) - If the value of the function at \(b\) is less than at \(a\), the average rate of change will be negative.
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The average rate of change cannot be zero. (False) - The average rate of change can be zero in cases where \(f(a) = f(b)\).
Thus, the most accurate and complete answer is that the average rate of change can be calculated using the formula:
\(\frac{f(b) - f(a)}{b - a}\).
1 answer