F06 - Numerical Methods - Tutorial

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Numerical Methods (Tutorial) Page 1 of 5 National Junior College 2016 – 2017 H2 Further Mathematics Topic F6: Numerical Methods (Tutorial) Basic Mastery Questions Interval Bisection, Linear Interpolation and Newton-Raphson’s Method 1 Estimate the positive root of the equation  ln 2x x  , giving your answer to four significant figures. J73/P1/14 2 Use linear interpolation to find an approximation to the root of equation  ln 4sin 0x x  which lies between 1 and 2. Give your answer correct to three decimal places. 3 Show that the equation e 2 0x x   has one and only one real root. Use the Newton - Raphson process to find this root correct to three decimal places. J75/P1/14 4 Sketch the following graphs on a single diagram, stating the x-coordinates of all intersections with the x-axis and the equations of any asymptotes. (i)      2 4 2 2y x x x x x     (ii)   2 1 1 xy x   Use linear interpolation once on the interval  1,0 to obtain an approximation to a root of the equation     2 2 14 . 1 xx x x    The Newton-Raphson method is to be used to find an approximation to another root of the equation. Use the method, with 2x as a first approximation, to obtain a second approximation to this root, giving your answer correct to 2 places of decimal. N2000/P2/12 Iterations involving recurrence relations of the form  1 Fn nx x  5 The equation 3 12 1 0x x   has two positive roots,  and  ,    and one negative root. (i) Prove that 0 1   and 3 4   . (ii) Use the iterative formula   1 31 12 1n nx x   , 1n , with 3.5 as a starting value to approximate  correct to two decimal places. (iii) With the aid of a graph, show that the iterative formula in (ii) will converge to  for some starting values more than  . www.KiasuExamPaper.com 423

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Numerical Methods (Tutorial) Page 2 of 5 Trapezium Rule and Simpson’s Rule 6 Estimate the values of the following definite integrals, taking the number of ordinates in each case, using (a) the trapezium rule, (b) Simpson’s Rule. (i) 2 0 1 d1 cos xx   , 3 ordinates (ii) 0.4 2 0 1 dx x , 5 ordinates Euler’s Method and Improved Euler’s Method 7 Apply Euler’s method with step size of 0.1 to obtain an approximation to the given initial value problems at y(0.5). (a) 2d 1d y yt   ,  0 0.5y  (b) d d y t tyt   ,  0 1y  8 Apply improved Euler’s method with step size of 0.1 to the initial value problems given in Question 7 to obtain an approximation to the value at y(0.5). Practice Questions 9 By considering the graphs of cosy x and 1 4y x , show that the equation 4cos 0x x  has one