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By C. B. Gupta

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2! 3! 5 − 1) 2 ∆ f (40) + ∆ f (40) 3! 2! 5 − 3) 4 ∆ f (40) 4! 5) (0. 867. Example 7. 5 Solution. 4 By Newton-Gregory forward formula f (x) = f (a) + u ∆ f (a) + u(u − 1) 2 u(u − 1) (u − 2) 3 ∆ f (a) + ∆ f (a) + .... 2! 3! 5 − 1) 2 ∆ f (1) + ∆3 f (1) 3! 2! 41) 2! 3! 4 By Newton-Gregory backward formula f (a + nh + hu) = f (a + nh) + u∇f (a + nh) + u(u + 1) 2 ∇ f (a + nh) 2! 54 + u(u + 1) (u + 2) 3 ∇ f (a + nh) + ..... 3! 41) 3! 2! 439375. Example 8. The population of a town in the decennial census were as under estimate the population for the year 1895 and 1925.

16. Find the value of y(5) from the following table using Lagrange’s interpolation formula. x 1 2 3 4 7 y 2 4 8 16 128 17. Find f(3) by Lagrange’s interpolation formula from the following table. x 0 1 2 4 5 f (x) 0 16 48 88 0 18. Find the value of sin 52° from the given table. 8660 41 INTERPOLATION 19. Find the number of men getting wages between Rs 10 and Rs 15 from the following table. Wages in Rs Frequency 0–10 10–20 20–30 30–40 9 30 35 42 20. 25. 25 21. Find the value of the area of the circle of diameter 82 from the following data.

Bm 2 x 2 + ...... + bmn x n = bm′ Assuming a11 ≠ 0. The variable x1 eliminated from second equation by subtracting U| || V| || W Step 2. To eliminate x2 from the third, fourth ...... nth equation Assuming b22 ≠ 0. The variable x2 eliminated from third equation by subtracting b32 times the b22 second equation from the third equation, similarly we eliminate x2 from fourth equation by subtracting b42 times the second equation from the fourth equation etc. We get new system of equation as b22 44 ADVANCECED MATHEMATICS U| || V| | = b ′′ |W a11 x1 + a12 x 2 + a13 x 3 + ......

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