15A/188 CLASS: B.Stat. St. JOSEPH’S COLLEGE (AUTONOMOUS) TIRUCHIRAPPALLI – 620 002 SEMESTER EXAMINATIONS – APRIL 2015 TIME: 3 Hrs. MAXIMUM MARKS: 100 SEM SET PAPER CODE TITLE OF THE PAPER IV 2013 11UST430301A NUMERICAL MATHEMATICS SECTION – A Answer all the questions: Choose the correct answer: 20 x 1 = 20 1. The backward difference operator is defined by b) —y n = y n -1 - y n a) —y n = y n - y n -1 c) —y n = y n - y n +1 d) —y n = y n - y0 2. Bessel’s formula is obtained from some modifications of a) Newton’s forward b) Gauss’s forward c) Newton’s backward d) Gauss’s backward 3. Successive approximation’s method starts with a) Newton’s forward difference formula b) Newton’s divided difference formula c) Newton’s backward difference formula d) Lagrange’s interpolation formula 4. Regula falsi method is also known as a) Method of Newton Raphson b) Method of chords c) Method of false position d) b and c 5. In general quadrature formula if we put n = 2 we get b) Weddle’s rule a) Simpson’s 1 rule 3 th d) None of these Simpson’s 3 8 rule Fill in the blanks: 6. The process of computing the value of a function inside the given range is called ______. c) 7. The mean of Gauss forward and Gauss backward formula is _______. 8. Lagrange’s interpolation formula is ______ . 9. If f(a) is –ve and f(b) is +ve, then a root lies between ______ and ______. 10. th Simpson’s 3 8 rule is ______. State True or False: 11. E = 1 + D. 12. Laplase-Everett’s formula is derived from Gauss’s backward interpolation formula. 13. Inverse interpolation is in general, finding the value of x for a given value of f(x). 14. Newton’s Raphson method is also referred to as the method of tangent. 15. Simpson’s 3/8 rule can be applied only if the number of sub intervals is a multiple of 4. Match the following: 16. D log f(x) - a) d 17. Central difference operator - b) n = 2 18. Elimination of third order differences 19. x n +1 20. Simpson’s 1/3 rule È Df ( x ) ˘ log c) Í1 + f ( x ) ˙ Î ˚ - d) Inverse interpolation - f (x n ) e) x n f ¢( x n ) SECTION – B Answer all the questions: 21. a. Derive Newton divided difference formula. 5 x 4 = 20 OR b. Using the following table find f(656) x: 654 658 659 661 y : 2.8156 2.8182 2.8189 2.8202 22. a. Derive Stirling’s formula for interpolation. OR b. Using Bessel’s formula find f(25) given f(20) = 2854, f(24) = 3162, f(28) = 3544, f(32) = 3992. 23. a. Explain the method of successive approximation. OR b. Tabulate y = x3 for x = 2, 3, 4, 5 and calculate the cube root of 10 correct to three decimal places. 24. a. Find the smallest positive root of the equation x3 – 2x + 0.5 = 0 using Newton’s Raphson method. OR b. Find the approximate value of real root of xlog10x = 1.2 by regular falsi method. 25. a. d2y From the following table, obtain at the point x = 0.96. 2 dx x : 0.96 0.98 1.00 1.02 1.04 f ( x : 0.7825 0.7739 0.7651 0.7563 0.7473 OR 1 dx b. using Trapezoidal rule with h = 0.2. Evaluate Ú 2 01 + x SECTION – C Answer any FOUR questions: 4 x 15 = 60 26. a) Derive Newton-Gregory formula for forward interpolation. b) The following table gives certain corresponding values of x and log10x. Compute the value of log10323.5, by using Lagrange’s interpolation formula. x : 321 .0 322 .8 324 .2 325 .0 log10 x : 2.50651 2.50893 2.51081 2.51188 27. a) Using Stirling’s formula to evaluate f(1.22) given: x : 1.0 1.1 1.2 1.3 1.4 f ( x ) : 0.841 0.891 0.932 0.963 0.985 b) Drive Everett’s formula. 28. Find the value of x when y =85 using Lagrange’s formula for inverse interpolation from the following table. x: 2 5 8 14 y : 94 .8 87 .9 81 .3 68 .7 29. Find a real root of the equation x3 – 2x – 5 = 0, using the Bisection method. 30. Derive Weddle’s formula for numerical integration. ************