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C3 : NUMERICAL METHODS Score: 1) Percentage: Grade: Effort: A sequence is defined by the recurrence relation un + 1 = un a , n = 1, 2, 3, ..., 2 un where a is a constant. (a) Given that a = 20 and u1 = 3, find the values of u2, u3 and u4, giving your answers to 2 decimal places. (3) …………………………………………………………………………………………………… …………………………………………………………………………………………………… …………………………………………………………………………………………………… …………………………………………………………………………………………………… …………………………………………………………………………………………………...... (b) Given instead that u1 = u2 = 3, (i) calculate the value of a, (3) …………………………………………………………………………………………………… …………………………………………………………………………………………………… …………………………………………………………………………………………………… …………………………………………………………………………………………………...... .......................................................................................................................................................... (ii) write down the value of u5. (1) …………………………………………………………………………………………………… …………………………………………………………………………………………………… [Edexcel P2 Jan 04 Qu 2] Produced for GlosMaths by S Milthorp & S Lomax 2) f(x) = x3 + x2 4x 1. The equation f(x) = 0 has only one positive root, . (a) Show that f(x) = 0 can be rearranged as x= 4x 1 , x 1. x 1 (2) ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………….. The iterative formula xn + 1 = 4 xn 1 is used to find an approximation to . xn 1 (b) Taking x1 = 1, find, to 2 decimal places, the values of x2, x3 and x4. (3) ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………….. (c) By choosing values of x in a suitable interval, prove that = 1.70, correct to 2 decimal places. (3) ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………….. Produced for GlosMaths by S Milthorp & S Lomax (d) Write down a value of x1 for which the iteration formula xn + 1 = 4 xn 1 does not produce a xn 1 valid value for x2. Justify your answer. (2) …………………………………………………………………………………………………… …………………………………………………………………………………………………… …………………………………………………………………………………………………… …………………………………………………………………………………………………….. [Edexcel P2 June 04 Qu 5] 3) Given that f(x) = ex + x – 2, x 0, (a) show that the solution of f(x) = 0 lies between x = 3 and x = 4. (2) ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… …………………………………………………………………………………….. The iterative formula xn + 1 = (2 – e xn )2 is used to solve the equation f(x) = 0. (d) Taking x0 = 4, write down the values of x1, x2, x3 and x4, and hence find an approximation to the solution of f(x) = 0, giving your answer to 3 decimal places. (4) ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ……………………………………………………………………………………..…………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………...... [Edexcel P2 June 03 Qu 5] Overall, for this assessment I think my success level is: Produced for GlosMaths by S Milthorp & S Lomax Low O High O O O Did your score match your estimated ‘success level’ ? Skill Qu I can locate the roots of f(x)=0 by considering changes of sign of f(x) in an interval of x in which f(x) is continuous. I can find terms of a recurrence relation I can approximate solutions of equations using simple iterative methods, including recurrence relations of the form xn 1 f ( xn ) . Yippee!! - I got all the questions correct. I made mistakes and need to practise this topic more. Top 3 topics I need to study further are: ☺ ☺ ☺ Produced for GlosMaths by S Milthorp & S Lomax