Question

a function is shown in the table below on which interval of x is the average rate of change of the function the greatest
left and right are side by side
x
1.0
2.2
3.11
4.26
5.32
y
1.3
2.12
3.24
4.36
5.40

Answer 1

To determine the interval on which the average rate of change of the function represented in the table is the greatest, we will calculate the average rate of change between each pair of points (intervals) in the table. The average rate of change between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[
\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's calculate the average rate of change for each interval:

1. **Interval between \(x = 1.0\) and \(x = 2.2\)**:
- \(y_1 = 1.3\), \(y_2 = 2.12\)
- \(\frac{2.12 - 1.3}{2.2 - 1.0} = \frac{0.82}{1.2} \approx 0.6833\)

2. **Interval between \(x = 2.2\) and \(x = 3.11\)**:
- \(y_1 = 2.12\), \(y_2 = 3.24\)
- \(\frac{3.24 - 2.12}{3.11 - 2.2} = \frac{1.12}{0.91} \approx 1.2308\)

3. **Interval between \(x = 3.11\) and \(x = 4.26\)**:
- \(y_1 = 3.24\), \(y_2 = 4.36\)
- \(\frac{4.36 - 3.24}{4.26 - 3.11} = \frac{1.12}{1.15} \approx 0.9739\)

4. **Interval between \(x = 4.26\) and \(x = 5.32\)**:
- \(y_1 = 4.36\), \(y_2 = 5.40\)
- \(\frac{5.40 - 4.36}{5.32 - 4.26} = \frac{1.04}{1.06} \approx 0.9811\)

Now, let's summarize the average rates of change we calculated:

- \(1.0\) to \(2.2\): \(\approx 0.6833\)
- \(2.2\) to \(3.11\): \(\approx 1.2308\)
- \(3.11\) to \(4.26\): \(\approx 0.9739\)
- \(4.26\) to \(5.32\): \(\approx 0.9811\)

The greatest average rate of change occurs in the interval from \(x = 2.2\) to \(x = 3.11\) with an average rate of change of approximately \(1.2308\).

Therefore, the interval where the average rate of change of the function is the greatest is:

**From \(x = 2.2\) to \(x = 3.11\)**.