Question
the main points Are (0,-2) (1,1) (3,0) (4,-1) (5,-1) (6,0)
Given the graph of f(x)
, on which interval is the average rate of change the greatest?
(1 point)
Responses
The function has the greatest average rate of change over the interval [5,6]
.
The function has the greatest average rate of change over the interval left bracket 5 comma 6 right bracket .
The function has the greatest average rate of change over the interval [4,5]
.
The function has the greatest average rate of change over the interval left bracket 4 comma 5 right bracket .
The function has the greatest average rate of change over the interval [1,4]
.
The function has the greatest average rate of change over the interval left bracket 1 comma 4 right bracket .
The function has the greatest average rate of change over the interval [0,1]
.
Given the graph of f(x)
, on which interval is the average rate of change the greatest?
(1 point)
Responses
The function has the greatest average rate of change over the interval [5,6]
.
The function has the greatest average rate of change over the interval left bracket 5 comma 6 right bracket .
The function has the greatest average rate of change over the interval [4,5]
.
The function has the greatest average rate of change over the interval left bracket 4 comma 5 right bracket .
The function has the greatest average rate of change over the interval [1,4]
.
The function has the greatest average rate of change over the interval left bracket 1 comma 4 right bracket .
The function has the greatest average rate of change over the interval [0,1]
.
Answers
Answered by
GPT-4o mini
To determine where the average rate of change is greatest, we can calculate the average rate of change between each pair of points.
The average rate of change of a function \( f(x) \) between two points \( (x_1, f(x_1)) \) and \( (x_2, f(x_2)) \) is given by the formula:
\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]
Let's calculate the average rate of change for each of the relevant intervals:
1. **Interval [0, 1]**
- Points: \( (0, -2) \) and \( (1, 1) \)
- Average Rate of Change: \( \frac{1 - (-2)}{1 - 0} = \frac{3}{1} = 3 \)
2. **Interval [1, 4]**
- Points: \( (1, 1) \) and \( (4, -1) \)
- Average Rate of Change: \( \frac{-1 - 1}{4 - 1} = \frac{-2}{3} \approx -0.67 \)
3. **Interval [4, 5]**
- Points: \( (4, -1) \) and \( (5, -1) \)
- Average Rate of Change: \( \frac{-1 - (-1)}{5 - 4} = \frac{0}{1} = 0 \)
4. **Interval [5, 6]**
- Points: \( (5, -1) \) and \( (6, 0) \)
- Average Rate of Change: \( \frac{0 - (-1)}{6 - 5} = \frac{1}{1} = 1 \)
5. **Interval [3, 4]**
- Points: \( (3, 0) \) and \( (4, -1) \)
- Average Rate of Change: \( \frac{-1 - 0}{4 - 3} = \frac{-1}{1} = -1 \)
6. **Interval [1, 3]**
- Points: \( (1, 1) \) and \( (3, 0) \)
- Average Rate of Change: \( \frac{0 - 1}{3 - 1} = \frac{-1}{2} = -0.5 \)
Now we can summarize the average rates of change calculated:
- [0, 1]: 3
- [1, 4]: -0.67
- [4, 5]: 0
- [5, 6]: 1
- [3, 4]: -1
- [1, 3]: -0.5
From these calculations, the greatest average rate of change occurs over the interval **[0, 1]** with a rate of 3.
Based on the choices you've listed, none explicitly match our calculation for the interval [0, 1], but if we had to choose an option that indicates a positive change, informally [5, 6] has the highest rate after [0, 1]. If that's not a concern and strictly looking for the highest calculated value, it’s [0, 1].
The average rate of change of a function \( f(x) \) between two points \( (x_1, f(x_1)) \) and \( (x_2, f(x_2)) \) is given by the formula:
\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]
Let's calculate the average rate of change for each of the relevant intervals:
1. **Interval [0, 1]**
- Points: \( (0, -2) \) and \( (1, 1) \)
- Average Rate of Change: \( \frac{1 - (-2)}{1 - 0} = \frac{3}{1} = 3 \)
2. **Interval [1, 4]**
- Points: \( (1, 1) \) and \( (4, -1) \)
- Average Rate of Change: \( \frac{-1 - 1}{4 - 1} = \frac{-2}{3} \approx -0.67 \)
3. **Interval [4, 5]**
- Points: \( (4, -1) \) and \( (5, -1) \)
- Average Rate of Change: \( \frac{-1 - (-1)}{5 - 4} = \frac{0}{1} = 0 \)
4. **Interval [5, 6]**
- Points: \( (5, -1) \) and \( (6, 0) \)
- Average Rate of Change: \( \frac{0 - (-1)}{6 - 5} = \frac{1}{1} = 1 \)
5. **Interval [3, 4]**
- Points: \( (3, 0) \) and \( (4, -1) \)
- Average Rate of Change: \( \frac{-1 - 0}{4 - 3} = \frac{-1}{1} = -1 \)
6. **Interval [1, 3]**
- Points: \( (1, 1) \) and \( (3, 0) \)
- Average Rate of Change: \( \frac{0 - 1}{3 - 1} = \frac{-1}{2} = -0.5 \)
Now we can summarize the average rates of change calculated:
- [0, 1]: 3
- [1, 4]: -0.67
- [4, 5]: 0
- [5, 6]: 1
- [3, 4]: -1
- [1, 3]: -0.5
From these calculations, the greatest average rate of change occurs over the interval **[0, 1]** with a rate of 3.
Based on the choices you've listed, none explicitly match our calculation for the interval [0, 1], but if we had to choose an option that indicates a positive change, informally [5, 6] has the highest rate after [0, 1]. If that's not a concern and strictly looking for the highest calculated value, it’s [0, 1].