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Question 1/ Vraag 1 Given: f(x) = e* - 31 (1.1) (a) Deduce that equation (1) has at least one root in the interval [a;b]=[0:1]. (b) Which theorem did you apply? (1.2) Use (1) and apply the bisection method (two iteration steps) to determine approximations P1 and p2 for the solution for f(1) = 0 on the interval [0:1]. Compleate the table below. Table 1. Bisection method (1) [3 marks [1 mark] [7 marks] n an 0 br 1 flan) f(bn) Pn 1 2 (1.3) (a) Use two iterations of the secant method to estimate the root p of (1). Make use of table 2 to present your numerical results. [7 marks] Table 2. Secant method n 1 Pn-1 0 Pn 1 f (Pra-1 f (Pn) Pn+1 2
Question 2 / Vraag 2 [4 marks (2.1) Use Newton's method to determine the value of 24. Use po = 2.8 as initial value and do two iteration steps. f(2)=.......... f'(I)=..... Table 3. Newton's method 72 f(pr) f'(Pn) Pn 2.8 0 1 2 (2.2) The polinomial f(3) = 3.1.23 - 5.955.02 + 10.7351 - 8.295 has a zero point i = 1.05. Use Newton's method to determine the other zero point. (5 marks
Question 3 / Vraag 3 Given: 2.1 + 3cos(z) - e" = 0; 1 € [1;1.5) (3.1) Find the minimum number of bisection method iteration needed to achieve an approzimation with accuracy 105. [5 marks) (3.2) Use Taylor's theorem and show that the newton Raphson method has quadratic convergence if the initial value Po is close enouph to the root p and if p has a multiplicity of one. [5 marks) (3.3) Given the non-linear system 4.12 + 4.12 + 5201 = 19 1692 + 3.1ż +111.01 – 10.12 = 10 Suppose one step on Newton's method gives the first approximation (11; c) = (0.24; 1.61) to solve the non-linear system. Apply another step on Newton's metod to compute second approximation. [7 marks) TOTAL / TOTAAL: 45