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Maths A2 Coursework Change of Sign Method:- 3x3 - 6x + 1 = 0 Sketch: - Looking at the graph we find that the roots lie between these intervals: [-2,-1] [0,1] [1,2] Finding root interval between [0,1] using a Decimal Search Root [0 , 1] X1 = 0 ?f(0) = 3(0)3 - 6(0) + 1 = 1 X2 = 0.1 ?f(0.1) = 3(0.1) 3 - 6(0.1) + 1 = 0.403 X f(X) 0 1 0.1 0.403 0.2 -0.176 Root [0.1 , 0.2] X f(X) 0.1 0.403 0.11 0.34399 0.12 0.28518 0.13 0.22659 0.14 0.16823 0.15 0.11013 0.16 0.052288 0.17 -0.005261 Root [0.16 , 0.17] X f(X) 0.16 0.052298 0.161 0.046520 0.162 0.040755 0.163 0.034992 0.164 0.029233 0.165 0.023476 0.166 0.017723 0.167 0.011972 0.168 0.0062249 0.169 0.00048043 0.170 -0.005261 Root [0.169 , 0.170] X f(X) 0.1690 0.00048043 0.1691 -0.0000923852 Root [0.1690 , 0.1691] X f(X) 0.1690 0.00048043 0.16901 0.00042299 0.16902 0.00036557 0.16903 0.00030814 0.16904 0.00025071 0.16905 0.00019328 0.16906 0.00013586 0.16907 0.00078427 0.16908 0.00021 0.16909 -0.000036426 From this method we find that the root lies between [0.16908 , 0.16909] ? Root = 0.169085 ± 0.000005 test f (0.16908) = 0.00021 f (0.16909) = -0.000036426 There is a change of sign, hence the root does lie in the interval [0.16908 , 0.16909] Graphical Illustration Rearrangement Method (Fixed Point Iteration): - Solve 2x3 - 5x +1 = 0 Re-arrange this so that "X=..."> 2x3 - 5x + 1 = 0 2x3 + 1 = 5x 2x3 + 1 = x 5 g(x) = x Sketch: - Graph shows that...

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