Numerical Methods for Engine acin/About/Droid/uploads/Numerical%20Methods.pdf YouTube engineers 130/987 Use an initial guess of x = 0.5 and iterate until & ≤0.01%. Verify that the process is linearly convergent as described in Box 6.1. 6.2 Determine the highest real root of f(x) = 2x³- 11.7x² + 17.7x - 5 (a) Graphically. (b) Fixed-point iteration method (three iterations, x0 = 3). Note: Make certain that you develop a solution that converges on the root. (c) Newton-Raphson method (three iterations, x0 = 3). 286% 0.5 Use (a) mixed-point Iteration and (D) the Newton-Raphson method to determine a root of f(x) -0.9x² + 1.7x+ 2.5 using Xo -- = = 0.01%. 5. Perform the computation until & is less than ε, Also perform an error check of your final answer. 6.4 Determine the real roots of f(x) = -1 + 5.5x-4x² + 0.5x3: (a) graphically and (b) using the Newton-Raphson method to within & = 0.01%. 6.5 Employ the Newton-Raphson method to determine a real root for f(x) = −1 +5.5x-4x² + 0.5x³ using initial guesses of (a) 4.52 174 OPEN METHODS and (b) 4.54. Discuss and use graphical and analytical methods to ex- plain any peculiarities in your results. 6.6 Determine the lowest real root of f(x)=-12-21x+ 18x2 2.4x³: (a) graphically and (b) using the secant method to a value of ε, corresponding to three significant figures. 6.7 Locate the first positive root of f(x)= = sinx + cos(1 + x²) - 1 where x is in radians. Use four iterations of the secant method with Q Search Pick the best numerical technique, justify your choice and then use that technique to determine the root. Note that it is known that for positive initial guesses, all techniques except fixed-point iteration will eventually converge. Perform iterations until the approximate relative error falls below 2%. If you use a bracket- ing method, use initial guesses of x= O and x = 2. If you use the Newton-Raphson or the modified secant method, use an ini- tial guess of x = 0.7. If you use the secant method, use initial guesses of x-1= 0 and x = 2. 614 Ice (a) the Newton-Ranhson method and (h) the modified HAGESPRIES ENG (1) (2) Problem 6.5: Do the following (ignore textbook's instructions): i. Plot function using Matlab. Print this plot. ii. Find all three roots by Newton-Raphson Method. Write your own Matlab function, print the function, and indicate the answers. (Use error tolerance of 0.01%). iii. Use Matlab's roots function to verify your answer (no need to print). Use Newton-Raphson to estimate the root of f(x) = ex-x using x = 0 as an initial guess. Re-use the Matlab function from Prob. (1), print the code and indicate the answer. Also, use Excel Solver to verify your answer (print the cells that you used).