Question
Which statement about the average rate of change over the interval [a,b]
for the function f(x)
is true?(1 point)
Responses
The average rate of change can be calculated using the formula b−af(b)−f(a)
.
The average rate of change can be calculated using the formula Start Fraction b minus a over f left parenthesis b right parenthesis minus f left parenthesis a right parenthesis End Fraction .
The average rate of change can be calculated using the formula f(b)−f(a)a−b
.
The average rate of change can be calculated using the formula Start Fraction f left parenthesis b right parenthesis minus f left parenthesis a right parenthesis over a minus b End Fraction .
The average rate of change is negative if f(b)<f(a)
.
The average rate of change is negative if f left parenthesis b right parenthesis less than f left parenthesis a right parenthesis .
The average rate of change cannot be zero.
for the function f(x)
is true?(1 point)
Responses
The average rate of change can be calculated using the formula b−af(b)−f(a)
.
The average rate of change can be calculated using the formula Start Fraction b minus a over f left parenthesis b right parenthesis minus f left parenthesis a right parenthesis End Fraction .
The average rate of change can be calculated using the formula f(b)−f(a)a−b
.
The average rate of change can be calculated using the formula Start Fraction f left parenthesis b right parenthesis minus f left parenthesis a right parenthesis over a minus b End Fraction .
The average rate of change is negative if f(b)<f(a)
.
The average rate of change is negative if f left parenthesis b right parenthesis less than f left parenthesis a right parenthesis .
The average rate of change cannot be zero.
Answers
GPT-4o mini
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:
1. **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 \(\frac{f(b) - f(a)}{b - a}\).
2. **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.
3. **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.
4. **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.
5. **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:
\(\frac{f(b) - f(a)}{b - a}\).
**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:
1. **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 \(\frac{f(b) - f(a)}{b - a}\).
2. **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.
3. **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.
4. **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.
5. **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:
\(\frac{f(b) - f(a)}{b - a}\).