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 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 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 cannot be zero.
The average rate of change cannot be zero.
The average rate of change can be calculated using the formula f(b)−f(a)a−b
.
for the function f(x)
is true?(1 point)
Responses
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 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 cannot be zero.
The average rate of change cannot be zero.
The average rate of change can be calculated using the formula f(b)−f(a)a−b
.
Answers
GPT-4o mini
To determine the correct statement about the average rate of change of a function \( f(x) \) over the interval \([a,b]\), let's go through the options:
1. **The average rate of change is negative if \( f(b) < f(a) \).**
- This statement is true. If \( f(b) \) is less than \( f(a) \), it implies that the output of the function decreases from \( a \) to \( b\), resulting in a negative average rate of change.
2. **The average rate of change is negative if \( f(b) < f(a) \).**
- This statement is redundant since it is the same as the first statement.
3. **The average rate of change can be calculated using the formula \( \frac{b-a}{f(b)-f(a)} \).**
- This statement is incorrect. The average rate of change is defined as \( \frac{f(b) - f(a)}{b - a} \), not the other way around.
4. **The average rate of change cannot be zero.**
- This statement is false. The average rate of change can be zero if \( f(a) = f(b) \), meaning there is no change in the function's value over the interval.
5. **The average rate of change can be calculated using the formula \( f(b) - f(a) \, a - b \).**
- This statement is incorrect as presented; the correct formula should be \( \frac{f(b) - f(a)}{b - a} \).
6. **The average rate of change can be calculated using the formula \( \frac{f(b)-f(a)}{a-b} \).**
- This statement is also incorrect; while \( f(b) - f(a) \) is appropriate, the denominator should be \( b - a \), not \( a - b \).
The correct and relevant statements from those provided are the first one and its duplicate:
**The average rate of change is negative if \( f(b) < f(a) \).**
1. **The average rate of change is negative if \( f(b) < f(a) \).**
- This statement is true. If \( f(b) \) is less than \( f(a) \), it implies that the output of the function decreases from \( a \) to \( b\), resulting in a negative average rate of change.
2. **The average rate of change is negative if \( f(b) < f(a) \).**
- This statement is redundant since it is the same as the first statement.
3. **The average rate of change can be calculated using the formula \( \frac{b-a}{f(b)-f(a)} \).**
- This statement is incorrect. The average rate of change is defined as \( \frac{f(b) - f(a)}{b - a} \), not the other way around.
4. **The average rate of change cannot be zero.**
- This statement is false. The average rate of change can be zero if \( f(a) = f(b) \), meaning there is no change in the function's value over the interval.
5. **The average rate of change can be calculated using the formula \( f(b) - f(a) \, a - b \).**
- This statement is incorrect as presented; the correct formula should be \( \frac{f(b) - f(a)}{b - a} \).
6. **The average rate of change can be calculated using the formula \( \frac{f(b)-f(a)}{a-b} \).**
- This statement is also incorrect; while \( f(b) - f(a) \) is appropriate, the denominator should be \( b - a \), not \( a - b \).
The correct and relevant statements from those provided are the first one and its duplicate:
**The average rate of change is negative if \( f(b) < f(a) \).**