Question
Which statement about the average rate of change over the interval [a,b] for the function f(x) is true
The average rate of change can be calculated using the formula b-a/f(b)-f(a)
The average rate of change can not be zero
The average rate of change can be calculated using the formula f(b)-f(a)/a-b
The average rate of change is negative if f(b) < f(a)
The average rate of change can be calculated using the formula b-a/f(b)-f(a)
The average rate of change can not be zero
The average rate of change can be calculated using the formula f(b)-f(a)/a-b
The average rate of change is negative if f(b) < f(a)
Answers
GPT-4o mini
The correct statement about the average rate of change over the interval \([a,b]\) for the function \(f(x)\) is:
**The average rate of change is negative if \(f(b) < f(a)\).**
Let's clarify the other statements:
1. **The average rate of change can be calculated using the formula \(\frac{b-a}{f(b)-f(a)}\):** This statement is incorrect. The correct formula for the average rate of change of \(f\) from \(a\) to \(b\) is given by:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
2. **The average rate of change cannot be zero:** This statement is incorrect. The average rate of change can be zero if \(f(b) = f(a)\), meaning there is no change in the function values over the interval.
3. **The average rate of change can be calculated using the formula \(\frac{f(b)-f(a)}{a-b}\):** This statement is incorrect in its form. The average rate of change does use \(f(b) - f(a)\), but it divides by \(b - a\), not \(a - b\). A division by \(a - b\) would just flip the sign of the result.
Therefore, the only accurate statement here is that the average rate of change is negative if \(f(b) < f(a)\).
**The average rate of change is negative if \(f(b) < f(a)\).**
Let's clarify the other statements:
1. **The average rate of change can be calculated using the formula \(\frac{b-a}{f(b)-f(a)}\):** This statement is incorrect. The correct formula for the average rate of change of \(f\) from \(a\) to \(b\) is given by:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
2. **The average rate of change cannot be zero:** This statement is incorrect. The average rate of change can be zero if \(f(b) = f(a)\), meaning there is no change in the function values over the interval.
3. **The average rate of change can be calculated using the formula \(\frac{f(b)-f(a)}{a-b}\):** This statement is incorrect in its form. The average rate of change does use \(f(b) - f(a)\), but it divides by \(b - a\), not \(a - b\). A division by \(a - b\) would just flip the sign of the result.
Therefore, the only accurate statement here is that the average rate of change is negative if \(f(b) < f(a)\).