Question

[20 Marks] Question 2 a) Given f(x)= x - 7x2 +14x-6 i) Show that there is a root a in interval [0,1] (1 mark) ii) Find the minimum number of iterations needed by the bisection method to approximate the root, a of f(x) = 0 on [0,1] with accuracy of 2 decimal points. (3 marks) iii) Find the root (a) of f(x)= x - 7x² +14x6 on [0,1] using the bisection method with accuracy of 2 decimal points. (6 marks) b) Given f(x) = cos xx i) Write the iterative formula for f(x) using Newton Method. (2 marks) ii) Determine which initial starting point x, (x) = 0 and Xo = 0.5) is appropriate to be used for finding root using Newton method. Give your justification. (3 marks) iii) Use the formula in (i) and the initial value chosen in (ii) to find a positive real root of f(x). Use ε = 0.0005 (5 marks)

Answer 1

a) Emor alan Given f(x)= x? - 7x2 + 14x-6 i) Show that there is a root a in interval [0,1] me have fros=-6 and fcl)=2. There fore, by ritermediate value theoreom 7 on a such that f(x)=0 in[0,1]. ii) Find the minimum number of iterations needed by the bisection method to approximate the root, a of f(x) = 0 on [0,1] with accuracy of 2 decimal points. In bisection where in is the number of stevodons. < .005 (for two decimal correct accuracy) 2 - 2 > 200 →n=8] [:28 - 256, 23= 128] iii) Find the root (a) of f(x)=x*-7x' +14x-6 on [0,1] using the bisection method with accuracy of 2 decimal points. Bisection Method The following table f(a) f(b) f(c) fires the zero using Bisection method with 1 0.75 0.984375 0.259766 & stessisans 20.585938 0.5625 0.59375 0.054047 7 0.59 Conect to two 0.5625 0.59375 0.054047 0.578125 -0.05262 decimos ploos. 0.578125 -0.05262 0.59375 0.054047 0.585938 0.001031 8 0.578125 -0.05262 0.585938 0.001031 0.582031 b) Given f(x)=cosx-x i) Write the iterative formula for f(x) using Newton Method. n a b C = Xm 1 0 1 0.5 -0.625 2 -6 -0.625 -0.625 0.5 0.5 0.5 3 0.75 0.625 2 2 0.984375 0.259766 0.259766 0.625 0.5625 4 -0.16187 5 -0.625 -0.16187 -0.16187 0.625 6 -0.02572 * In or method, the (9+13th iteration is heren by: Xunt = Xn- foan) (f'Can); =0,1,2) n ii) Determine which initial starting point x, (x, = 0 and x, = 0.5) is appropriate to be used for finding root using Newton method. Give your justification. Since, interval for zero as: [0, 1]. So, o + = 0.5 is an appropriate initial starting point. iii) Use the formula in (i) and the initial value chosen in (ii) to find a positive real root of f(x). Use ε = 0.0005 Xn f(x) f'(x) Xn+1 0.5 -0.625 7.75 0.5806451613 The zero from the 0.580645 -0.03525 6.882414 0.5857662632 table 3 0.585766 -0.00014 6.828639 0.5857864373 0.585786 -2.1E-09 6.828427 0.5857864376 0.585786 6.828427 0.5857864376 0.585786 6.828427 0.5857864376 0.585786 6.828427 0.5857864376 0.585786 6.828427 0.5857864376 0.585786 6.828427 0.5857864376 0 2 3 4 0 = 0.586 5 0 6 0 7 0 8 0