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A2 Pure Maths Coursework Solving Equations using Numerical Methods There are three methods with which you can solve an equation. 1. Change of Sign Method 2. Newton-Raphson Method 3. Rearranging the equation f(x)=0 in the form x= g(x) Hardware and Software Used For the coursework I have used a computer for attaining more accurate results and to avoid errors. The software I have used are * Microsoft Word * GraphCalc (A software used for drawing graphs) Change of Sign Method x4- 4x +2 We see that the roots lie in between [0, 1] and [1, 2] We confirm this by looking for a change of sign x 0 1 2 f(x) 2 -1 10 f(x) is found by substituting the values of x in the equation. We look for roots in [0, 1] x 0 0.1 0.2 0.3 0.4 0.5 0.6 f(x) 2 1.6001 1.2016 0.8081 0.4256 0.0625 -0.2704 We look for roots in [0.5, 0.6] x 0.5 0.51 0.52 f(x) 0.0625 0.02765 -0.00688 We look for roots in [0.51, 0.52] x 0.51 0.511 0.512 0.513 0.514 0.515 0.516 0.517 0.518 f(x) 0.02765 0.02418 0.02071 0.01725 0.01380 0.01034 0.00689 0.00344 -0.000002 We look for roots in [0.517, 0.518] x 0.517 0.5171 0.5172 0.5173 0.5174 f(x) 0.00344 0.00309 0.00275 0.0024 0.00206 x 0.5175 0.5176 0.5177 0.5178 0.5179 0.518 f(x) 0.00172 0.00137 0.00103 0.00068 0.00034 -0.000002 Root is in between [0.5179, 0.5180] Failure Case There are some cases where this method cannot be used to solve the equation. Newton-Raphson Method f `(x)...

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