Question

Which function has the greatest rate of change in the interval (−2,2)
?(1 point)
Responses

y=6x−8
y is equal to 6 x minus 8 - no response given

y=2x2+3x−1
y is equal to 2 x squared plus 3 x minus 1 - no response given

y=3⋅(12)x
y is equal to 3 times 1 half to the x th power - incorrect

y=x2−2

Answer 1

To identify which function has the greatest rate of change in the interval \((-2, 2)\), we can evaluate the derivative of each function on this interval. The derivative gives us the rate of change at any point.

1. **For \(y = 6x - 8\)**:
- Derivative: \(y' = 6\)
- The rate of change is constant at 6.

2. **For \(y = 2x^2 + 3x - 1\)**:
- Derivative: \(y' = 4x + 3\)
- We evaluate this at the endpoints of the interval:
- At \(x = -2\): \(y' = 4(-2) + 3 = -8 + 3 = -5\)
- At \(x = 2\): \(y' = 4(2) + 3 = 8 + 3 = 11\)
- The maximum rate of change is 11 at \(x = 2\).

3. **For \(y = 3 \cdot (1/2)^x\)**:
- Derivative: \(y' = 3 \cdot \ln(1/2) \cdot (1/2)^x\)
- Since \(\ln(1/2)\) is negative, the derivative will also be negative for all \(x\). Thus, there's no positive rate of change in this interval.

4. **For \(y = x^2 - 2\)**:
- Derivative: \(y' = 2x\)
- We evaluate this at the endpoints:
- At \(x = -2\): \(y' = 2(-2) = -4\)
- At \(x = 2\): \(y' = 2(2) = 4\)
- The maximum rate of change is 4 at \(x = 2\).

Now we can summarize the maximum rates of change:
- For \(y = 6x - 8\): 6
- For \(y = 2x^2 + 3x - 1\): 11 (at \(x = 2\))
- For \(y = 3 \cdot (1/2)^x\): Negative (no positive rate of change)
- For \(y = x^2 - 2\): 4 (at \(x = 2\))

Thus, the function with the greatest rate of change in the interval \((-2, 2)\) is:

**\(y = 2x^2 + 3x - 1\)**, with a maximum rate of change of **11**.