For the following set of values Construct the backward diference table (9,5.0), (10, 5.4) (11,6.0), (12, 6.8), (13,7.5), (14.8.1).

To construct the **backward difference table**, we follow these steps:

1. **Arrange the data** in a table format.
2. **Calculate backward differences** for each order (first, second, etc.).

The given data points are:
\[
(x, y) = (9, 5.0), (10, 5.4), (11, 6.0), (12, 6.8), (13, 7.5), (14, 8.1)
\]

### Step 1: Organize the data
We organize the \( x \) and \( y \) values into the table.

| \( x \)   | \( y \)   |
|-----------|-----------|
| 9         | 5.0       |
| 10        | 5.4       |
| 11        | 6.0       |
| 12        | 6.8       |
| 13        | 7.5       |
| 14        | 8.1       |

### Step 2: Calculate backward differences
We calculate the differences step by step.

#### First-order backward differences (\( \Delta^1 y \))
\[
\Delta^1 y = y_n - y_{n-1}
\]
For example:
\[
\Delta^1 y_5 = 8.1 - 7.5 = 0.6
\]
Now we calculate all first-order differences:

| \( x \)   | \( y \)   | \( \Delta^1 y \) |
|-----------|-----------|------------------|
| 9         | 5.0       |                  |
| 10        | 5.4       |                  |
| 11        | 6.0       |                  |
| 12        | 6.8       |                  |
| 13        | 7.5       | 0.6              |
| 14        | 8.1       | 0.6              |

#### Second-order backward differences (\( \Delta^2 y \))
\[
\Delta^2 y = \Delta^1 y_n - \Delta^1 y_{n-1}
\]
For example:
\[
\Delta^2 y_4 = 0.6 - 0.6 = 0.0
\]
Now calculate all second-order differences:

| \( x \)   | \( y \)   | \( \Delta^1 y \) | \( \Delta^2 y \) |
|-----------|-----------|------------------|------------------|
| 9         | 5.0       |                  |                  |
| 10        | 5.4       |                  |                  |
| 11        | 6.0       |                  |                  |
| 12        | 6.8       | 0.6              |                  |
| 13        | 7.5       | 0.6              | 0.0              |
| 14        | 8.1       | 0.6              | 0.0              |

#### Third-order backward differences (\( \Delta^3 y \))
\[
\Delta^3 y = \Delta^2 y_n - \Delta^2 y_{n-1}
\]
For example:
\[
\Delta^3 y_3 = 0.0 - 0.0 = 0.0
\]
Now calculate all third-order differences:

| \( x \)   | \( y \)   | \( \Delta^1 y \) | \( \Delta^2 y \) | \( \Delta^3 y \) |
|-----------|-----------|------------------|------------------|------------------|
| 9         | 5.0       |                  |                  |                  |
| 10        | 5.4       |                  |                  |                  |
| 11        | 6.0       |                  |                  |                  |
| 12        | 6.8       | 0.6              | 0.0              |                  |
| 13        | 7.5       | 0.6              | 0.0              | 0.0              |
| 14        | 8.1       | 0.6              | 0.0              | 0.0              |

This completes the backward difference table. Since the differences stabilize at 0 after the second order, higher-order differences will also be 0.

Would you like more details on any step?

Here are 5 related questions to expand this topic:
1. How do you interpolate a value using a backward difference table?
2. What is the significance of the order of backward differences?
3. How would you construct a forward difference table?
4. Can you use Newton's backward interpolation formula with this table?
5. How do backward differences differ from finite differences?

**Tip:** When the differences stabilize at a constant value (like here), it suggests that the underlying function could be well-approximated by a polynomial of degree corresponding to the last non-zero difference.

For the following set of values Construct the backward difference table (9,5.0), (10, 5.4) (11,6.0), (12, 6.8), (13,7.5), (14.8.1).

Learn how to construct the backward difference table for the data points (9, 5.0), (10, 5.4), (11, 6.0), (12, 6.8), (13, 7.5), (14, 8.1). This problem covers key numerical analysis concepts, including the calculation of first, second, and third-order backward differences. Suitable for students in Grades 10-12.

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