By C. B. Gupta

ISBN-10: 8122426859

ISBN-13: 9788122426854

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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. 0540 10.

4! 8) and Gauss’s backward difference formula is yu = y0 + u∆y– 1 + (u + 2) (u + 1) u (u − 1) u (u + 1) 2 (u + 1) u (u − 1) 3 ∆ y–1 + ∆ y–2 + ∆4y–2 + ...... 4! 2! 3! 9) is 3 3 u(u 2 − 1) ( ∆ y− 1 + ∆ y−2 ) u 2 (u 2 − 1) 4 ( ∆y0 + ∆y−1 ) u 2 2 ∆ y + + ∆ y–2 + ..... + –1 3! 2 4! 2 2! This formula is called the Stirling’s difference formula. yu = y0 + u SOLVED EXAMPLES Example 1. Given u0 = 580, u1 = 556, u2 = 520, u3 = —, u4 = 384, find u3. Solution. Let the missing term u3 = X 26 ADVANCED MATHEMATICS ∴ The forward difference table is x ux 0 580 1 556 ∆u x ∆2u x ∆ 3u x ∆ 4 ux – 24 – 12 – 36 2 X – 472 520 X – 484 1860 – 4X X – 520 3 1388 – X X 904 – 2X 384 – X 4 384 Here four values of ux are given.

N x + 5x + 6 Q 2 5. Evaluate F 1I H xK (i) ∆n (ii) ∆n [sin (ax + b)] (iv) ∆n [axn + bxn – 1] (iii) ∆6 (ax – 1) (bx2 – 1) (cx3 – 1) F∆ I e GH E JK 2 6. Prove that ex = x . Ee x ; the interval of differencing being h. ∆2e x 7. Prove that ∇n yx = ∆nyx–n. 8. Evaluate (i) (2∆2 + ∆ – 1) (x2 + 2x + 1) (ii) (∆ + 1) (2∆ – 1) (x2 + 2x +1) (iii) (E + 2) (E + 1) (2x + h + x) (iv) (E2 + 3E + 2) 2x+h + x 9. Write down the polynomial of lowest degree which satisfies the following set of number 0, 7, 26, 63, 124, 215, 342, 511.