2(a) Derive the equations for the coefficients of a quadratic polynomial that fits a set of n data points, (xi, yi), i =

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2 A Derive The Equations For The Coefficients Of A Quadratic Polynomial That Fits A Set Of N Data Points Xi Yi I 1 (163.15 KiB) Viewed 129 times

2(a) Derive the equations for the coefficients of a quadratic polynomial that fits a set of n data points, (xi, yi), i = 1 ... n, using the least square method. (6 marks) (b) A robot is programmed to move in a quadratic path on level ground, which is the x-y plane. Due to imperfections in the motion mechanism, there are minor errors in the movement. Its positions, recorded at regular x-intervals, are as shown in the table below: x-1 0 1 2 3 y 7.2 4.1 2.9 3.8 6.8 Obtain the curve that best fits the measured positions. (7 marks) (c) What is the standard error in your result? (4 marks) (d) Comment on the suitability of using Lagrange Interpolation polynomial to fit the curve for the robot instead. (3 marks) 3. A second-order differential equation is given below day dy - 2 dt2 + y = tett dt (0<t<1) with initial conditions y(0) = 0, and y(0)' = 0. (a) Convert the above governing equation to an equivalent system, which consists of several first-order ordinary differential equations. (2 marks) (b) For an initial-value problem y'= f(x,y) with initial condition y(xo) = yo, the Modified Euler method is written as Yn+ = yn +0.5Ah(y".+y), with initial guess y = f(x+1 Yn+d) and you = y + Ah y'n. Using the Modified Euler method with a step size of 0.4, determine the numerical solution of the initial-value problem in (a). (8 marks) (c) To solve the differential equation = f(x,y) with initial condition y(x) = y, the dx second-order Runge-Kutta approximation is written as Yn+1 = y +ashf(x, yn)+bAhf (x, +cah, yn + dahf(x, y)). If a = 0, b = 1.0, and c = d = 0.5, it becomes the Midpoint method. Using a step size of 0.4, determine the numerical solution of the initial-value problem in (a).