By C. B. Gupta

ISBN-10: 8122426859

ISBN-13: 9788122426854

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Example text

N – 1 ux – n ∴ ux = ux– 1 + ∆ux– 2 + ∆2ux– 3 + ..... + ∆n–1 ux – n + ∆n ux – n. Hence proved. Example 20. Prove that u1 x + u2 x2 + u3x3 + ..... x2 x3 x = u1 + ∆2 u1 + ..... and 0 < x < 1. 2 ∆ u1 + (1 − x ) (1 − x ) 3 1− x Solution. S. = x x3 x2 u1 + ∆u + ∆2 u1 + ..... 1 (1 − x ) 3 1− x (1 − x ) 2 x x2 x3 u E 1 u + ( − ) + ( E − 1) 2 u1 + ...... = 1 1 1− x (1 − x ) 2 (1 − x )3 FG x − x H 1 − x (1 − x) 2 = 2 + IJ K FG H FG x H (1 − x) IJ K 3 + = FG H x x 1+ (1 − x ) 1− x IJ K IJ K 2x3 x3 x2 u1 + + ......

X 0 1 2 3 4 f (x) 1 3 9 — 81 3. 0170, find log 102. 4. Estimate the production of cotton in the year 1935 from the data given below (in millions of pales). 2 5. Obtain the missing terms in the following table. 024 6. The values of x and y are given as below: x 5 6 9 11 y 12 13 14 16 Find the value of y when x = 10. 40 ADVANCED MATHEMATICS 7. Find u3 given u0 = 580, u1 = 556, u2 = 520 and u4 = 385. 8. 25 15 16 18 21 Estimate the weight of the body at the age of 7 months. 9. 50, using the following values of the function.

4) × (– 3) 4! 8368. Example 9. From the following table, find the form of the function f (x). x 0 1 2 3 4 f(x) 3 6 11 18 27 Solution. The difference table of the given data is as under: x f(x) 0 3 1 6 ∆f(x) ∆2f(x) ∆3f(x) ∆4f(x) 3 2 5 2 11 0 2 7 3 18 2 9 4 27 0 0 34 ADVANCED MATHEMATICS Here a = 0, h = 1, u = x−0 =x 1 By Newton-Gregory formula f (x) = f (a) + u ∆f (a) + u(u − 1) 2 u(u − 1) (u − 2) 3 ∆ f (a) + ∆ f (a) + ..... 2! 3! = f (0) + x ∆f (0) + x( x − 1) 2 x ( x − 1) ( x − 2) 3 ∆ f (0) + ∆ f (0) + .....