 # 1. Derive the following formula for approximating the derivative an...

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1. Derive the following formula for approximating the derivative and show that is o(h2) by establishing its error term: 3f(x)-4f(x-h)+j(x-2h) - - j'(x) 2 2h 2. Consider the boundary value problem IS" =0, 0Show that the problem has no solution if O # 8. and infinite many solutions when O = 8. 3. (a) Write a MATLAB function that implements the finite difference method for solving the boundary value problem (b) Use the function written in part (a) to solve the boundary value problem 2.x 2 = u(0) =0, u(1) = In(2). The true solution is u(x) Solve the boundary value problem for h = 0.1.0.05,0.025,0.0125, and print the errors of the numerical solution at T = 0.2,0.4.0.6.0.8. Comment on how errors decrease when h is halved. Plot the true solution and the numerical solution for h = 0.05 on the same plot. 4. The general solution of the equation xu" - (2x + 1)z/ + (x+ 1)u = 0 is 21(I) = GIE* + Find the solution of the equation with the boundary conditions u '(0) 1, u'(1) =0 Write down a formula for a discrete approximation of the boundary conditions. Implement the method by modifying the function in Prob- lem 3, and solve the problem with h = 0.1,0.05,0.025,0.0125. Print the errors in a loglog plot and comment on how the error decreases when h is halved.

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