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 iterative formula to find an approximation for , correct to 3 decimal places. You are required to check the accuracy of your answer in this question. [3] (iii) By using linear interpolation once, obtain, correct to 3 decimal places, a first approximation 1 to  . Using 1 as the initial value, apply the Newton-Raphson method once to obtain 2 , leaving your answer to 3 decimal places. [3] (iv) Explain why the Newton -Raphson method in this case fails to give an approximation to . [1] (v) Illustrate, on a single diagram, how 1 and 2 are obtained. [2] [CJC/FM/2018/P1/Q5] 5 (i) By sketching the graphs of 1tan 5 xy − =  and 1sin 6 xy − =  on the same diagram, show that the equation 11tan sin56 xx−−   =       has a positive real root,  . [2] (ii) Find the integer N such that 1NN   + . [2] (iii) Illustrate graphically, using 0xN= , why the iterative procedure 1 1 5 tan sin 6 n n xx − +  =   will not give a good approximate value of  . [2] (iv) Taking the initial approximation to be N, use the Newton-Raphson method to find  to 2 decimal places. [3] [RI/FM/2017/P2/Q1] 6 (i) Show, with the aid of a sketch graph, that the equation ln 0x k x+= has exactly one real root,  , in the interval 1 12 x if 1 2ln 2k  . [3] (ii) In the case 1k = , a student tries to use fixed point iteration in the form of F( )xx= to find the value of  . (a) In his first attempt, the student uses F( ) lnxx=− and 0 0.5x = . Calculate the value of 1x and 2x , correct to 4 decimal places. Explain why this method will fail to find the value of  . [2]

Numerical Methods – Approximation of Roots of Equations 3 (b) Suggest a possible F( )x for the student and using 0 0.5x = , find the value of , correct to 3 decimal places. [2] (c) Use a diagram to explain how the iteration in (b) converges to  , showing clearly the position of 0 1 2, and x x x . [2] [HCI/FM/2017/P2/Q2] 7 (i) Show algebraically that the equation 32 60xx− − = has exactly one real root in the interval ( )2, 3 . [2] (ii) Find an estimate of the root using 2 iterations of linear interpolation, correct to two decimal places. [2] (iii) Determine algebraically if the estimate in (ii) is an underestimate or overestimate of the root. [3] The equation 32 60xx− − = , can be rearranged in the following ways (you are not required to verify): (A) ( ) 1 2 36xx=+ (B) 2 6xx x=− (C) 1 26xx x =+ (iv) Determine with reasons which of the above expressions (A), (B) and (C) converges under the fixed-point iteration method using the initial value 0 2x = . If it converges, use a graph to demonstrate 3 iterations of its convergence to the root, labelling the points 0x , 1x and 2x clearly. [5] [TJC/FM/2019/P1/7] 8 By considering the graphs of 2tanyx= and 3yx= , show that the equation 23tan 0xx−= has exactly two roots in the interval 3 .22 x Denoting the smaller root by  , where 12  , use linear interpolation once on the interval  1, 2 to estimate the value of  , giving your answer correct to 3 decimal places. Comment on the suitability of the method used in this case. [3] Taking 1 1.8x = , where 1x is the initial approximation of  , use the Newton-Raphson method to obtain a sequence of approximations for  , giving your answer correct to 3 decimal places. You should demonstrate that the root is found to the required degree of accuracy. [3] With the aid of a sketch, explain why any initial approximation 1x such that 1 2 x  will produce a sequence converging to  whereas some approximations 1x such that 1 3 2x   will not converge to . [3] [HCI et al/FM/2018/P1/3] 9 The function f is such that ( )f 2cos e xxx −=− . (i) Matthias concludes that the equation ( )f0 x = has no roots in the interval  1,5 . Explain how Matthias may have arrived at this conclusion and draw a sketch to illustrate why he is wrong. [2]

Numerical Methods – Approximation of Roots of Equations 4 The equation 2cos e 0 xx −−= has a root  in the interval  1, 2 . (ii) Alice uses linear interpolation twice on the interval  1, 2 to find an approximation to  . Find the approximation to  given by this method, giving your answer to 2 decimal places. Without any further interpolation, demonstrate how Alice can verify its correctness to 2 decimal places. [4] (iii) Tabitha decides to use the Newton -Raphson method to find  with an initial approximation of 1 1 = , terminating the process when she has found two successive iterates that are equal when rounded to 5 decimal places. State the value of each of the iterates calculated correct to 5 decimal places. [3] Buoyed by her success, Tabitha decided to try a