Question

compare two functions which one has the greater rate of change on the interval [0,2] and [2,4]

-1,-1
0,3
1,7
2,11
4,19
5,23


0,2
1,3
2,5
3,9
4,17

Answer 1

To compare the rate of change for the two functions on the given intervals \([0, 2]\) and \([2, 4]\), we first need to determine the rate of change for each interval. The rate of change can be calculated as the change in the function's output values divided by the change in the input values. This is essentially the slope of the line connecting the two points on the graph.

### Function 1 Data Points:
- \((-1, -1)\)
- \((0, 3)\)
- \((1, 7)\)
- \((2, 11)\)
- \((4, 19)\)
- \((5, 23)\)

### Function 2 Data Points:
- \((0, 2)\)
- \((1, 3)\)
- \((2, 5)\)
- \((3, 9)\)
- \((4, 17)\)

### Calculate Rate of Change for Function 1:

**Interval \([0, 2]\)**:
- Points: \((0, 3)\) and \((2, 11)\)
- Rate of Change = \(\frac{11 - 3}{2 - 0} = \frac{8}{2} = 4\)

**Interval \([2, 4]\)**:
- Points: \((2, 11)\) and \((4, 19)\)
- Rate of Change = \(\frac{19 - 11}{4 - 2} = \frac{8}{2} = 4\)

### Calculate Rate of Change for Function 2:

**Interval \([0, 2]\)**:
- Points: \((0, 2)\) and \((2, 5)\)
- Rate of Change = \(\frac{5 - 2}{2 - 0} = \frac{3}{2} = 1.5\)

**Interval \([2, 4]\)**:
- Points: \((2, 5)\) and \((4, 17)\)
- Rate of Change = \(\frac{17 - 5}{4 - 2} = \frac{12}{2} = 6\)

### Summary:
- For **Function 1**:
- Rate of Change \([0, 2]\) = 4
- Rate of Change \([2, 4]\) = 4
- For **Function 2**:
- Rate of Change \([0, 2]\) = 1.5
- Rate of Change \([2, 4]\) = 6

### Comparison:
- In the first interval \([0, 2]\), Function 1 has a greater rate of change (4) compared to Function 2 (1.5).
- In the second interval \([2, 4]\), Function 2 has a greater rate of change (6) compared to Function 1 (4).

### Conclusion:
On the interval \([0, 2]\), Function 1 has the greater rate of change, while on the interval \([2, 4]\), Function 2 has the greater rate of change.