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By C. B. Gupta

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From the following table, find the form of the function f (x). x 3 5 7 9 11 f (x) 6 24 58 108 174 24. If lx represents the number of persons living at age x in a life table, find as accurately as the data will permit the value of l47. Given that l20 = 512, l30 = 439, l40 = 346, l50 = 243. ANSWERS 1. 14 2. 31 3. 0086 4. 609 6. 67 7. 25 8. 7 lbs 9. 1295 11. x3 – 9x2 + 17x + 6 14. 47692 15. 395 16. 32 18. 788003 19. 15 20. 875 21. 5280 22. 87 23. 2x2 – 7x + 9 24. 274. 5. 090 10. x2 + 2x + 3 17. 84 3 CHAPTER Solution of Linear Simultaneous Equations INTRODUCTION Simultaneous linear equations occur in the field of science and engineering like as analysis of a network under sinusoidal steady-state condition, determination of the output of a chemical plant and finding the cost of reaction, the analysis of electronic circuits having a number of invariant element etc.

5), we get an = f (x) = f (a + nh) + (x – a – nh) ∇ 2 f ( a + nh) ∇f ( a + nh) + (x – a – nh) (x – a – (n – 1)h) 2 ! h2 h n + ...... + (x – a – nh) (x – a – (n – 1)h) ...... (x – a – h) ∇ f ( a + nh) n ! 6), we get f (a + nh + hu) = f (a + nh) + u∇f (a + nh) + + ..... + u(u + 1) 2 ∇ f (a + nh) 2! u(u + 1) (u + 2) ...... (u + n − 1) n ∇ f (a + nh). n! 4. , xn not necessarily equally spaced. Since there are (n + 1) values of f (x) so (n + 1)th difference is zero. Thus f (x) is supposed to be polynomial in x of degree n.

On solving we get X1 and X2. 2. , a + nh respectively where h is the difference of the arguments. Let f (x) be a polynomial in x of degree n. So f (x) can be written as f (x) = a0 + a1(x – a) + a2 (x – a) (x – a – h) + a3 (x – a) (x – a – h) (x – a – 2h) + ...... + an (x – a) (x – a – h) ..... h 2h2 a2 = f (a + 2h) – 2ha1 – a0 a2 = = f ( a + 2h) − 2( f ( a + h) − f ( a)) − f ( a) 2 ! h2 f ( a + 2h) − 2 f ( a + h) + f ( a) ∆2 f ( a) = 2 ! h2 2 ! h2 23 INTERPOLATION Proceeding in the same way, we get a3 = ∆3 f ( a) ......

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