Question
Which function has the greatest rate of change in the interval given?(1 point)
Responses
y=3x
Interval [0,2]
y is equal to 3 to the x th power Interval left bracket 0 comma 2 right bracket - no response given
y=x2
Interval [0,2]
y is equal to x squared Interval left bracket 0 comma 2 right bracket - incorrect
y=2x−3
Interval [−2,3]
y is equal to 2 x minus 3 Interval left bracket negative 2 comma 3 right bracket - no response given
y=x(x−3)
Interval [2,4]
y is equal to x times open paren x minus 3 close paren Interval left bracket 2 comma 4 right bracket - no response given
Remaining Attempts : 1
Responses
y=3x
Interval [0,2]
y is equal to 3 to the x th power Interval left bracket 0 comma 2 right bracket - no response given
y=x2
Interval [0,2]
y is equal to x squared Interval left bracket 0 comma 2 right bracket - incorrect
y=2x−3
Interval [−2,3]
y is equal to 2 x minus 3 Interval left bracket negative 2 comma 3 right bracket - no response given
y=x(x−3)
Interval [2,4]
y is equal to x times open paren x minus 3 close paren Interval left bracket 2 comma 4 right bracket - no response given
Remaining Attempts : 1
Answers
GPT-4o mini
To determine which function has the greatest rate of change over the given intervals, we can calculate the average rate of change for each function between the specified bounds of their intervals. The average rate of change is calculated using the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \(f(a)\) and \(f(b)\) are the function values at the endpoints of the interval \([a, b]\).
### Let's calculate the average rate of change for each function:
1. **For \(y = 3x\) on \([0, 2]\)**:
- \(f(0) = 3(0) = 0\)
- \(f(2) = 3(2) = 6\)
- Average rate of change = \(\frac{6 - 0}{2 - 0} = \frac{6}{2} = 3\)
2. **For \(y = 3^x\) on \((0, 2)\)**:
- \(f(0) = 3^0 = 1\)
- \(f(2) = 3^2 = 9\)
- Average rate of change = \(\frac{9 - 1}{2 - 0} = \frac{8}{2} = 4\)
3. **For \(y = x^2\) on \([0, 2]\)**:
- \(f(0) = 0^2 = 0\)
- \(f(2) = 2^2 = 4\)
- Average rate of change = \(\frac{4 - 0}{2 - 0} = \frac{4}{2} = 2\)
4. **For \(y = 2x - 3\) on \([-2, 3]\)**:
- \(f(-2) = 2(-2) - 3 = -4 - 3 = -7\)
- \(f(3) = 2(3) - 3 = 6 - 3 = 3\)
- Average rate of change = \(\frac{3 - (-7)}{3 - (-2)} = \frac{3 + 7}{3 + 2} = \frac{10}{5} = 2\)
5. **For \(y = x(x - 3)\) on \([2, 4]\)**:
- \(f(2) = 2(2 - 3) = 2(-1) = -2\)
- \(f(4) = 4(4 - 3) = 4(1) = 4\)
- Average rate of change = \(\frac{4 - (-2)}{4 - 2} = \frac{4 + 2}{2} = \frac{6}{2} = 3\)
### Summary of Average Rates of Change:
- \(y = 3x\): 3
- \(y = 3^x\): 4
- \(y = x^2\): 2
- \(y = 2x - 3\): 2
- \(y = x(x - 3)\): 3
### Conclusion:
The function that has the greatest rate of change in the provided intervals is \(y = 3^x\) on the interval \((0, 2)\) with an average rate of change of **4**.
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \(f(a)\) and \(f(b)\) are the function values at the endpoints of the interval \([a, b]\).
### Let's calculate the average rate of change for each function:
1. **For \(y = 3x\) on \([0, 2]\)**:
- \(f(0) = 3(0) = 0\)
- \(f(2) = 3(2) = 6\)
- Average rate of change = \(\frac{6 - 0}{2 - 0} = \frac{6}{2} = 3\)
2. **For \(y = 3^x\) on \((0, 2)\)**:
- \(f(0) = 3^0 = 1\)
- \(f(2) = 3^2 = 9\)
- Average rate of change = \(\frac{9 - 1}{2 - 0} = \frac{8}{2} = 4\)
3. **For \(y = x^2\) on \([0, 2]\)**:
- \(f(0) = 0^2 = 0\)
- \(f(2) = 2^2 = 4\)
- Average rate of change = \(\frac{4 - 0}{2 - 0} = \frac{4}{2} = 2\)
4. **For \(y = 2x - 3\) on \([-2, 3]\)**:
- \(f(-2) = 2(-2) - 3 = -4 - 3 = -7\)
- \(f(3) = 2(3) - 3 = 6 - 3 = 3\)
- Average rate of change = \(\frac{3 - (-7)}{3 - (-2)} = \frac{3 + 7}{3 + 2} = \frac{10}{5} = 2\)
5. **For \(y = x(x - 3)\) on \([2, 4]\)**:
- \(f(2) = 2(2 - 3) = 2(-1) = -2\)
- \(f(4) = 4(4 - 3) = 4(1) = 4\)
- Average rate of change = \(\frac{4 - (-2)}{4 - 2} = \frac{4 + 2}{2} = \frac{6}{2} = 3\)
### Summary of Average Rates of Change:
- \(y = 3x\): 3
- \(y = 3^x\): 4
- \(y = x^2\): 2
- \(y = 2x - 3\): 2
- \(y = x(x - 3)\): 3
### Conclusion:
The function that has the greatest rate of change in the provided intervals is \(y = 3^x\) on the interval \((0, 2)\) with an average rate of change of **4**.