DHS Numerical Approximation of Roots (9649) Topical Revision

Numerical Methods – Approximation of Roots of Equations 1 H2 Double Math (TOPICAL REVISION) NUMERICAL METHODS (PART 1) APPROXIMATION OF ROOTS OF EQUATIONS Note: Questions with * may require concepts and techniques you may not have learnt yet. 1 Show that the equation 3 5 1 0,xx− + = has exactly one root in (0,1). Use two iterations of linear interpolation between 0x= and 1x= to yield a fractional approximation of this root. [4] Three possible rearrangements of the given equation in the form ()x F x= are 3 5 1,xx=− 31 ( 1),5xx=+ 3 4 1.x x x= − + Only one of these rearrangements will provide an iterative method, of the form 1 ()nnx F x+ = , 0 2,x = which converges to the root between 2 and 3. Use this rearrangement to find this root correct to 3 significant figures. [4] It is known the correct rearrangement above provides a convergent iterative method by analysing the derivative of F . For this purpose, show that 0 '( ) 1,Fx whenever 2x . [2] [EJC/FM/2018/P1/9] 2 (a) The curve with equation 2 2e 3xyx= + − has exactly one stationary point in the interval  1, 0− . Use the Newton -Raphson method to find the x-coordinate of the stationary point, correct to 4 decimal places. [5] (b) (i) Show that the equation 3 7 2 0xx − + = has a root,  , in the interval [0, 1]. [1] (ii) Student A uses the recurrence relation ( ) 3 1 1 14 27 nnnx x x+ =− − for finding  . Explain why Student A will fail to find  . [2] (iii) Student B uses the recurrence relation ( ) 3 1 1 27 nnxx+ =+ for finding  . Find the approximate value of  to 5 decimal places. [2] [NJC/FM/2019/P1/8] 3 Let 2f( ) sec 2 e xxx=− . The equation f( ) 0x = has a root  in the interval [0.1, 0.3]. (a) Use Newton-Raphson method with initial approximation 0 0.1x = to find the first four approximations 1 2 3 4, , , x x x x to  . Deduce the behaviour of nx for large n and use a graph of the function f to help you explain why the sequence is not converging to  .[5] (b) Determine, with explanation, which one of these iterative formula e is more suitable in approximating the root  . (I) ( )( ) 2 1 ln sec 2nnxx+ = (II) 1 2 1 1 cos e2 nx nx −− + =   Using an initial approximation of 0 0.3x = and the chosen iterative for mula, find the approximation to the root  , correct to three decimal places. [4] [RVHS/FM/2018/P1/Q10]

Numerical Methods – Approximation of Roots of Equations 2 4 (i) The function f is such that f(a)f(b) < 0, where a < b. A student concludes that the equation f(x) = 0 has exactly one root in the interval (a, b). Illustrate with a sketch, two possible scenarios in which the student could be wrong. [2] Given now that the equation 2 230x x − − = has exactly one root  in the interval (3,4). (ii) Derive the iterative formula 1 4 43n n x x + =+ using the fixed point iteration method. Using 3 as the initial value, apply this iter