Question
Which function has the greatest rate of change in the interval given?(1 point)
Responses
y=x2
Interval [0,2]
y is equal to x squared Interval left bracket 0 comma 2 right bracket
y=3x
Interval [0,2]
y is equal to 3 to the x th power Interval left bracket 0 comma 2 right bracket
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
y=2x−3
Interval [−2,3]
Responses
y=x2
Interval [0,2]
y is equal to x squared Interval left bracket 0 comma 2 right bracket
y=3x
Interval [0,2]
y is equal to 3 to the x th power Interval left bracket 0 comma 2 right bracket
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
y=2x−3
Interval [−2,3]
Answers
GPT-4o mini
To determine which function has the greatest rate of change in the specified intervals, we need to calculate the average rate of change for each function over its corresponding interval.
1. **For \( y = x^2 \) on [0, 2]:**
Average rate of change = \( \frac{f(2) - f(0)}{2 - 0} \)
\( f(2) = 2^2 = 4 \)
\( f(0) = 0^2 = 0 \)
\( \text{Average rate of change} = \frac{4 - 0}{2 - 0} = \frac{4}{2} = 2 \)
2. **For \( y = 3x \) on [0, 2]:**
Average rate of change = \( \frac{f(2) - f(0)}{2 - 0} \)
\( f(2) = 3(2) = 6 \)
\( f(0) = 3(0) = 0 \)
\( \text{Average rate of change} = \frac{6 - 0}{2 - 0} = \frac{6}{2} = 3 \)
3. **For \( y = x(x - 3) \) on [2, 4]:**
Average rate of change = \( \frac{f(4) - f(2)}{4 - 2} \)
\( f(4) = 4(4 - 3) = 4 \)
\( f(2) = 2(2 - 3) = -2 \)
\( \text{Average rate of change} = \frac{4 - (-2)}{4 - 2} = \frac{4 + 2}{2} = \frac{6}{2} = 3 \)
4. **For \( y = 2x - 3 \) on [-2, 3]:**
Average rate of change = \( \frac{f(3) - f(-2)}{3 - (-2)} \)
\( f(3) = 2(3) - 3 = 6 - 3 = 3 \)
\( f(-2) = 2(-2) - 3 = -4 - 3 = -7 \)
\( \text{Average rate of change} = \frac{3 - (-7)}{3 - (-2)} = \frac{3 + 7}{5} = \frac{10}{5} = 2 \)
Now, combining our results:
- Average rate of change for \( y = x^2 \) on [0, 2]: **2**
- Average rate of change for \( y = 3x \) on [0, 2]: **3**
- Average rate of change for \( y = x(x - 3) \) on [2, 4]: **3**
- Average rate of change for \( y = 2x - 3 \) on [-2, 3]: **2**
**Conclusion: The functions with greatest rates of change are \( y = 3x \) and \( y = x(x - 3) \) on their respective intervals, both having an average rate of change of **3**.**
1. **For \( y = x^2 \) on [0, 2]:**
Average rate of change = \( \frac{f(2) - f(0)}{2 - 0} \)
\( f(2) = 2^2 = 4 \)
\( f(0) = 0^2 = 0 \)
\( \text{Average rate of change} = \frac{4 - 0}{2 - 0} = \frac{4}{2} = 2 \)
2. **For \( y = 3x \) on [0, 2]:**
Average rate of change = \( \frac{f(2) - f(0)}{2 - 0} \)
\( f(2) = 3(2) = 6 \)
\( f(0) = 3(0) = 0 \)
\( \text{Average rate of change} = \frac{6 - 0}{2 - 0} = \frac{6}{2} = 3 \)
3. **For \( y = x(x - 3) \) on [2, 4]:**
Average rate of change = \( \frac{f(4) - f(2)}{4 - 2} \)
\( f(4) = 4(4 - 3) = 4 \)
\( f(2) = 2(2 - 3) = -2 \)
\( \text{Average rate of change} = \frac{4 - (-2)}{4 - 2} = \frac{4 + 2}{2} = \frac{6}{2} = 3 \)
4. **For \( y = 2x - 3 \) on [-2, 3]:**
Average rate of change = \( \frac{f(3) - f(-2)}{3 - (-2)} \)
\( f(3) = 2(3) - 3 = 6 - 3 = 3 \)
\( f(-2) = 2(-2) - 3 = -4 - 3 = -7 \)
\( \text{Average rate of change} = \frac{3 - (-7)}{3 - (-2)} = \frac{3 + 7}{5} = \frac{10}{5} = 2 \)
Now, combining our results:
- Average rate of change for \( y = x^2 \) on [0, 2]: **2**
- Average rate of change for \( y = 3x \) on [0, 2]: **3**
- Average rate of change for \( y = x(x - 3) \) on [2, 4]: **3**
- Average rate of change for \( y = 2x - 3 \) on [-2, 3]: **2**
**Conclusion: The functions with greatest rates of change are \( y = 3x \) and \( y = x(x - 3) \) on their respective intervals, both having an average rate of change of **3**.**