Regent College Maths Department. Further Pure 1 Numerical Solutions of Equations

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1 Regent College Maths Department Further Pure 1 Numerical Solutions of Equations

2 Further Pure 1 Numerical Solutions of Equations You should: Be able use interval bisection, linear interpolation and the Newton-Raphson method to find approximations to the roots of equations which cannot be solved by any analytical method. Further Pure 1 Numerical solutions of Equations Page 2

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10 Edexcel Past Examination Questions 1. f(x) = x 3 + x 3. The equation f(x) = 0 has a root,, between 1 and 2. (a) By considering f (x), show that is the only real root of the equation f(x) = 0. (b) Taking 1.2 as your first approximation to, apply the Newton-Raphson procedure once to f(x) to obtain a second approximation to. Give your answer to 3 significant figures. (c) Prove that your answer to part (b) gives the value of correct to 3 significant figures. 2. f(x) = 2 sin 2x + x 2. [P4 January 2002 Qn 4] The root of the equation f(x) = 0 lies in the interval [2, ]. Using the end points of this interval find, by linear interpolation, an approximation to. (4) [*P4 January 2003 Qn 4] Further Pure 1 Numerical solutions of Equations Page 10

11 3. f(x) = 3 x x 6. (a) Show that f(x) = 0 has a root between x = 1 and x = 2. (b) Starting with the interval (1, 2), use interval bisection three times to find an interval of width which contains. 4. f(x) = 2 x + x 4. [*P4 June 2003 Qn 4] The equation f(x) = 0 has a root in the interval [1, 2]. Use linear interpolation on the values at the end points of this interval to find an approximation to. 5. f(x) = 1 e x + 3 sin 2x [*P4 June 2004 Qn 2] The equation f(x) = 0 has a root in the interval 1.0 < x < 1.4. Starting with the interval (1.0, 1.4), use interval bisection three times to find the value of to one decimal place. [*FP1/P4 June 2005 Qn 4] Further Pure 1 Numerical solutions of Equations Page 11

12 6. The temperature C of a room t hours after a heating system has been turned on is given by = t e 0.5t, t 0. The heating system switches off when = 20. The time t =, when the heating system switches off, is the solution of the equation 20 = 0, where lies in the interval [1.8, 2]. (a) Using the end points of the interval [1.8, 2], find, by linear interpolation, an approximation to. Give your answer to 2 decimal places. (4) (b) Use your answer to part (a) to estimate, giving your answer to the nearest minute, the time for which the heating system was on. (1) 7. f(x) = 0.25x sin x. [#*FP1/P4 January 2006 Qn 5] (a) Show that the equation f(x) = 0 has a root between x = 0.24 and x = (b) Starting with the interval [0.24, 0.28], use interval bisection three times to find an interval of width which contains. [*FP1 June 2006 Qn 6] Further Pure 1 Numerical solutions of Equations Page 12

13 8. f (x) = x 3 + 8x 19. (a) Show that the equation f(x) = 0 has only one real root. (b) Show that the real root of f(x) = 0 lies between 1 and 2. (c) Obtain an approximation to the real root of f(x) = 0 by performing two applications of the Newton-Raphson procedure to f(x), using x = 2 as the first approximation. Give your answer to 3 decimal places. (d) By considering the change of sign of f(x) over an appropriate interval, show that your answer to part (c) is accurate to 3 decimal places. (4) [FP1 June 2007 Qn 4] 9. f(x) = 3x 2 x + x tan 2 2, < x <. The equation f(x) = 0 has a root in the interval [0.7, 0.8]. Use linear interpolation, on the values at the end points of this interval, to obtain an approximation to. Give your answer to 3 decimal places. (4) [*FP1 January 2008 Qn 4] Further Pure 1 Numerical solutions of Equations Page 13

14 10. f(x) = 4 cos x + e x. (a) Show that the equation f(x) = 0 has a root α between 1.6 and 1.7 (b) Taking 1.6 as your first approximation to α, apply the Newton-Raphson procedure once to f(x) to obtain a second approximation to α. Give your answer to 3 significant figures. (4) [FP1 June 2008Qn 2] 11. f(x) = 3 x x (a) Show that the equation f(x) = 0 has a root α in the interval [1.1, 1.2]. (b) Find f (x). (c) Using x 0 = 1.1 as a first approximation to α, apply the Newton-Raphson procedure once to f(x) to find a second approximation to α, giving your answer to 3 significant figures. (4) [FP1 January 2009 Qn 5] Further Pure 1 Numerical solutions of Equations Page 14

15 12. Given that α is the only real root of the equation x 3 x 2 6 = 0, (a) show that 2.2 < α < 2.3 (b) Taking 2.2 as a first approximation to α, apply the Newton-Raphson procedure once to f(x) = x 3 x 2 6 to obtain a second approximation to α, giving your answer to 3 decimal places. (c) Use linear interpolation once on the interval [2.2, 2.3] to find another approximation to α, giving your answer to 3 decimal places. 13. f(x) = 3x x. (a) Write down, to 3 decimal places, the value of f(1.3) and the value of f(1.4). (5) [FP1 June 2009 Qn 4] (1) The equation f(x) = 0 has a root α between 1.3 and 1.4 (b) Starting with the interval [1.3, 1.4], use interval bisection to find an interval of width which contains α. (c) Taking 1.4 as a first approximation to α, apply the Newton-Raphson procedure once to f(x) to obtain a second approximation to α, giving your answer to 3 decimal places. (5) [FP1 January 2010 Qn 2] Further Pure 1 Numerical solutions of Equations Page 15

16 14. f(x) = x 3 x 7 + 2, x > 0. (a) Show that f (x) = 0 has a root between 1.4 and 1.5. (b) Starting with the interval [1.4, 1.5], use interval bisection twice to find an interval of width that contains. (c) Taking 1.45 as a first approximation to, apply the Newton-Raphson procedure once to f(x) = x , x > 0 to obtain a second approximation to, giving your answer to 3 decimal places. (5) 15. f(x) = 5x x 6, x 0. The root of the equation f (x) = 0 lies in the interval [1.6,1.8]. [FP1 June 2010 Qn 3] (a) Use linear interpolation once on the interval [1.6, 1.8] to find an approximation to. Give your answer to 3 decimal places. (b) Differentiate f(x) to find f (x). (c) Taking 1.7 as a first approximation to, apply the Newton-Raphson process once to f(x) to obtain a second approximation to. Give your answer to 3 decimal places. (4) (4) [FP1 January 2011 Qn 3] Further Pure 1 Numerical solutions of Equations Page 16

17 16. f (x) = 3 x + 3x 7 (a) Show that the equation f (x) = 0 has a root between x =1 and x = 2. (b) Starting with the interval [1, 2], use interval bisection twice to find an interval of width 0.25 which contains. [FP1 June 2011 Qn 1] 17. f(x) = x x 1, x 0. 2x (a) Use differentiation to find f (x). The root of the equation f(x) = 0 lies in the interval [0.7, 0.9]. (b) Taking 0.8 as a first approximation to, apply the Newton-Raphson process once to f(x) to obtain a second approximation to. Give your answer to 3 decimal places. (4) [FP1 June 2011 Qn 4] Further Pure 1 Numerical solutions of Equations Page 17

18 18. (a) Show that f(x) = x 4 + x 1 has a real root in the interval [0.5, 1.0]. (b) Starting with the interval [0.5, 1.0], use interval bisection twice to find an interval of width which contains. (c) Taking 0.75 as a first approximation, apply the Newton Raphson process twice to f(x) to obtain an approximate value of. Give your answer to 3 decimal places. (5) [FP1 January 2012 Qn 3] 19. f(x) = x x 3x 7, x > 0. A root of the equation f(x) = 0 lies in the interval [3, 5]. Taking 4 as a first approximation to, apply the Newton-Raphson process once to f(x) to obtain a second approximation to. Give your answer to 2 decimal places. (6) [FP1 June 2012 Qn 3] f(x) = x + x 5, x > 0. (a) Find f (x). The equation f(x) = 0 has a root in the interval [4.5, 5.5]. (b) Using x 0 = 5 as a first approximation to, apply the Newton-Raphson procedure once to f(x) to find a second approximation to, giving your answer to 3 significant figures. (4) [FP1 Jan 2013 Qn 3] Further Pure 1 Numerical solutions of Equations Page 18

19 21. f(x) = cos(x 2 ) x + 3, 0 < x < π (a) Show that the equation f(x) = 0 has a root α in the interval [2.5, 3]. (b) Use linear interpolation once on the interval [2.5, 3] to find an approximation for α, giving your answer to 2 decimal places. [FP1 June 2013 Qn 2] 22. f(x) = x x x (a) Show that the equation f(x) = 0 has a root α between x = 2 and x = 2.5. (b) Starting with the interval [2, 2.5] use interval bisection twice to find an interval of width which contains α. The equation f(x) = 0 has a root β in the interval [ 2, 1]. (c) Taking 1.5 as a first approximation to β, apply the Newton-Raphson process once to f(x) to obtain a second approximation to β. Give your answer to 2 decimal places. (5) [FP1_R June 2013 Qn 3] Further Pure 1 Numerical solutions of Equations Page 19