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Business Mathematics and Statistics · Class 12 Commerce

Ch 5Numerical Methods (Finite Differences, Interpolation) — Class 12 Business Mathematics and Statistics, concept-first.

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 ind…

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Key concepts

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Finite Differences and the Difference Table

The forward difference and backward difference (with higher orders formed by repeated differencing) organize equally-spaced tabulated data into a difference table; constant -th differences signal the data fits a degree-…

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Chapter contents

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1

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 val…

2

Newton's Forward Interpolation Formula

Newton's forward formula estimates for an -value near the BEGINNING of an equally-spaced table, using the first row's own forward differences.

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Newton's Backward Interpolation Formula

Newton's backward formula estimates for an -value near the END of an equally-spaced table, using the LAST row's own backward differences.

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Lagrange's Interpolation Formula

Newton's forward/backward formulas both require EQUALLY spaced -values. Lagrange's interpolation formula removes that restriction entirely — it works for ANY set of distinct data points, equally space…

Exercises

+Show 5 questions5 questions
  1. Q6Construct the forward difference table for the data $x=0,1,2,3$; $y=2,5,10,17$, and find $\Delta^2y_0$.Free
  2. Q7Using Newton's forward interpolation formula on $x=0,1,2,3$; $y=2,5,10,17$, estimate $y$ at $x=0.5$.Free
  3. Q8Using Newton's backward interpolation formula on the same table ($x=0,1,2,3$; $y=2,5,10,17$), estimate $y$ at $x=2.5$.Preview
  4. Q9Using Lagrange's interpolation formula, find $y$ at $x=1$ given the points $(0,1),(2,9),(3,19)$.Preview
  5. Q10Using Newton's backward interpolation formula on $x=1,2,3,4,5$; $y=2,5,10,17,26$, estimate $y$ at $x=4.5$.Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 27 questions27 questions
  1. Q1$\Delta^2 y_0 =$ (a) $y_2 + y_1 + 2y_0$ (b) $y_2 - 2y_1 + y_0$ (c) $y_2 + 2y_1 - y_0$ (d) $y_2 + 2y_1 + y_0$Preview
  2. Q2Find an equation of the parabolic form $y = ax^2 + bx + c$ passing through $(0, 0)$, $(1, 1)$ and $(2, 20)$ using Lagranges Interpolation.Preview
  3. Q3From the following table, find the missing value. | $x$ | $2$ | $3$ | $4$ | $5$ | $6$ | | --- | --- | --- | --- | --- | --- | | $f(x)$ | $45…Preview
  4. Q4$\Delta f(x) =$ ______. (a) $f(x+h) - f(x)$ (b) $f(x+h)$ (c) $f(x) - f(x-h)$ (d) $f(x) - f(x+h)$Preview
  5. Q5$\nabla f(a) =$ ______. (a) $f(a) - f(a-h)$ (b) $f(a) + f(a-h)$ (c) $f(a)$ (d) $f(a) - f(a+h)$Preview
  6. Q6If $h = 1$ then prove that $(E^{-1}\Delta)x^3 = 3x^2 - 3x + 1$.Preview
  7. Q7Construct a forward difference table for $y = f(x) = x^3 + 3x$ for $x = 1, 2, 3, 4, 5$.Preview
  8. Q8Find the missing entry in the following table. | $x$ | $0$ | $1$ | $2$ | $3$ | $4$ | | --- | --- | --- | --- | --- | --- | | $y_x$ | $1$ | $…Preview
  9. Q9$E\equiv$ (a) $1+\nabla$ (b) $1+\Delta$ (c) $1-\nabla$ (d) $1-\Delta$Preview
  10. Q10$E(Ey_0)=$ (a) $y_2$ (b) $y_0$ (c) $y_3$ (d) $y_1$Preview
  11. Q11Evaluate : $\Delta(\log ax)$Preview
  12. Q12Find the missing entry in the following table | $x$ | 0 | 1 | 2 | 3 | 4 | | --- | --- | --- | --- | --- | --- | | $y_x$ | 1 | 3 | 9 | - | 81…Preview
  13. Q13(a) Using Lagrange's interpolation formula find $y(10)$ from the following table. | X | 5 | 6 | 9 | 11 | | --- | --- | --- | --- | --- | | Y…Preview
  14. Q14If $f(x)=x^2+2x+2$ and the interval of differencing is unity, then $\Delta f(x)$ is : (a) $x-3$ (b) $x+3$ (c) $2x+3$ (d) $2x-3$Preview
  15. Q15$(1+\Delta)(1-\nabla)=$ (a) $-1$ (b) $0$ (c) $\Delta$ (d) $1$Preview
  16. Q16Evaluate : $\Delta^2 e^x$Preview
  17. Q17Given $U_0=1,\ U_1=11,\ U_2=21,\ U_3=28$ and $U_4=29$, find $\Delta^4 U_0$.Preview
  18. Q18Lagrange's interpolation formula can be used for : (a) unequal intervals only (b) both equal and unequal intervals (c) equal intervals only…Preview
  19. Q19$\nabla \equiv$ (a) $1-E^{-1}$ (b) $1+E$ (c) $1+E^{-1}$ (d) $1-E$Preview
  20. Q20If $y=x^{3}-x^{2}+x-1$, calculate the values of $y$ for $x=0,1,2,3,4,5$ and form the forward differences table.Preview
  21. Q21From the following table, find the missing value. | $x$ | 2 | 3 | 4 | 5 | 6 | | --- | --- | --- | --- | --- | --- | | $f(x)$ | 45.0 | 49.2 |…Preview
  22. Q22(a) If $h=1$, Evaluate $\Delta\left[\dfrac{5x+12}{x^{2}+5x+6}\right]$ OR (b) Construct the cost of living index number for 2011 on the basis…Preview
  23. Q23If $h = 1$ then $\Delta(x^2) =$ (a) $2x + 1$ (b) $2x$ (c) $1$ (d) $2x - 1$Preview
  24. Q24$\nabla f(a) =$ (a) $f(a) - f(a - h)$ (b) $f(a) + f(a - h)$ (c) $f(a)$ (d) $f(a) - f(a + h)$Preview
  25. Q25If $f(x) = x^2 + 3x$ and $h = 1$ then show that $\Delta f(x) = 2x + 4$Preview
  26. Q26Given $y_3 = 2$, $y_4 = -6$, $y_5 = 8$, $y_6 = 9$ and $y_7 = 17$, calculate $\Delta^4 y_3$.Preview
  27. Q27(a) Using interpolation method estimate the output of a factory in 1986 from the following data. | Year | 1974 | 1978 | 1982 | 1990 | | ---…Preview

More questions

+Show 5 questions5 questions
  1. Example 1Construct the forward difference table for the data $x=0,1,2,3,4$; $y=1,3,7,13,21$, and state $\Delta^3y_0$.Free
  2. Example 2Using Newton's forward interpolation formula on the table $x=0,1,2,3,4$; $y=1,3,7,13,21$, estimate $y$ at $x=0.5$.Free
  3. Example 3Using Newton's backward interpolation formula on the same table ($x=0,1,2,3,4$; $y=1,3,7,13,21$), estimate $y$ at $x=3.5$.Preview
  4. Example 4Using Lagrange's interpolation formula, find $y$ at $x=2$ given the points $(1,2),(3,10),(4,17)$.Preview
  5. Example 5Construct the backward difference table for $x=1,2,3,4,5$; $y=2,5,10,17,26$, and state $\nabla^2y_5$.Preview