Backward Difference Table Construction for Given Data Points

Math Problem Statement

Construct the backward difference table for the following set of values: (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 will use the given values:

$(9, 5.0), (10, 5.4), (11, 6.8), (12, 6.8), (13, 7.5), (14, 8.1)$

Here's a breakdown of how to construct the backward difference table:

Arrange the data: We will arrange the given points in a table and calculate the first, second, third, etc. backward differences.
Calculate backward differences: Each difference is calculated by subtracting the current value from the previous value.

Let me calculate this.The backward difference table for the given data is as follows:

\Delta^0 & \Delta^1 & \Delta^2 & \Delta^3 & \Delta^4 & \Delta^5 \\ \hline 5.0 & & & & & \\ 5.4 & 0.4 & & & & \\ 6.8 & 1.4 & 1.0 & & & \\ 6.8 & 0.0 & -1.4 & -2.4 & & \\ 7.5 & 0.7 & 0.7 & 2.1 & 4.5 & \\ 8.1 & 0.6 & -0.1 & -0.8 & -2.9 & -7.4 \\ \end{array}$$ ### Explanation: - **Δ^0**: The first column is the given set of y-values. - **Δ^1**: The first backward difference is calculated by subtracting the previous value from the current one. - **Δ^2, Δ^3, Δ^4**, etc.: Higher-order differences are calculated similarly, using the previous difference column. Let me know if you need further details or explanations on this! Here are some related questions: 1. How is the backward difference table used in numerical interpolation? 2. What is the significance of higher-order differences in a difference table? 3. How do forward and backward difference tables differ? 4. Can the backward difference method be used for extrapolation? If so, how? 5. How does the spacing of the x-values affect the backward difference table? **Tip**: If you notice alternating positive and negative values in higher-order differences, it might indicate that the data is fitting a polynomial or another smooth function.

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Numerical Analysis
Difference Table
Interpolation

Δy_i = y_i - y_{i-1}
Higher-order differences: Δ^n y_i = Δ^{n-1} y_i - Δ^{n-1} y_{i-1}

Newton's Backward Difference Formula

Undergraduate Level

Backward Difference Table for Data Points (9, 5.0), (10, 5.4), (11, 6.0), (12, 6.8), (13, 7.5), (14, 8.1)

Constructing a Backward Difference Table for Given Data Points

Constructing a Backward Difference Table for Given Values

Newton's Forward Difference Table and Interpolation for f(2.5)

Gauss Forward Interpolation: Solving for f(x) at x = 3.5