Business Mathematics and Statistics · Ch 5 — Numerical Methods (Finite Differences, Interpolation)

Finite Differences — the Forward and Backward Difference Operators

This Tamil Nadu HSC Class 12 Business Mathematics and Statistics chapter introduces numerical methods for estimating the value of a function at a point that lies BETWEEN known, equally-spaced data values — a genuinely practical business-mathematics tool for reading between the lines of a table (sales figures, price indices, census counts) recorded only at fixed intervals.

The forward difference operator $\Delta$

For values $y_0,y_1,y_2,\dots$ of a function recorded at equally spaced points $x_0,x_0+h,x_0+2h,\dots$ (common spacing $h$ ), the forward difference operator is

\Delta y_i = y_{i+1}-y_i

Higher-order differences are formed by differencing the differences themselves: $\Delta^2y_i=\Delta y_{i+1}-\Delta y_i$ , and so on. These are laid out in a forward difference table.

The backward difference operator $\nabla$

The backward difference operator, used when interpolating near the END of a table, is

\nabla y_i = y_i-y_{i-1}

with higher-order backward differences formed the same way: $\nabla^2y_i=\nabla y_i-\nabla y_{i-1}$ .

Worked reasoning: building a forward difference table

For $x=0,1,2,3,4$ with $y=1,3,7,13,21$ : $\Delta y_0=3-1=2$ , $\Delta y_1=7-3=4$ , $\Delta y_2=13-7=6$ , $\Delta y_3=21-13=8$ . Then $\Delta^2y_0=4-2=2$ , $\Delta^2y_1=6-4=2$ , $\Delta^2y_2=8-6=2$ — CONSTANT second differences. Then $\Delta^3y_0=2-2=0$ , confirming the data fits a quadratic polynomial exactly (a degree-2 polynomial always has constant, non-zero second differences and zero third differences).

$x$	$y$	$\Delta y$	$\Delta^2y$	$\Delta^3y$
0	1	2	2	0
1	3	4	2
2	7	6
3	13	8
4	21

Note

Constant $n$ -th differences signal a degree- $n$ polynomial

This is a genuinely useful diagnostic: if a table's differences become constant at order $n$ (and zero beyond), the underlying data fits a degree- $n$ polynomial exactly — the very fact this chapter's interpolation formulas rely on.

Definition 1Forward Difference Operator (Δ)

$\Delta y_i=y_{i+1}-y_i$ , the difference between consecutive tabulated values; higher-order forward differences are formed by repeating this on the previous order's differences.

Definition 2Backward Difference Operator (∇)

$\nabla y_i=y_i-y_{i-1}$ , used when interpolating near the end of a table; higher-order backward differences are formed the same way.

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Finite Differences — the Forward and Backward Difference Operators