1. Using a computer program, find the positive root of the equation x = sin x using bisection method with a tolerance of 10-8. Submit your source code and program output. State your numerical answer and the number of iterations used. 2. In a diffraction and interference experiment, light of wavelength goes through a single slit of width d. The diffracted light intensity I at angle is given by I = Io 2 (sina) where a = (d/) sin and I is a constant. (a) Plot a graph of I/Io against a which includes 3 maxima on each side of the y-axis. They are the first, second, and third order maxima. (b) From your graph, roughly estimate the values of a; > 0 (i=1,2 and 3) at which the first, second and third order maxima occur. (c) To evaluate a, accurately, the roots of a nonlinear equation F(a) calculated. Find F(a). = 0 should be (d) Hence, write a computer program for finding a root of your nonlinear equation using the Newton-Raphson method up to a tolerance of 10-8. Run your program 3 times with appropriate initial guesses to locate the 3 maxima. In each case, write down your answer, initial guess, and the number of iterations used. Submit only the version of your program for finding the first order maximum. (e) What are the values of sin for the 3 maxima? (10) (20) 1. Using a computer program, find the positive root of the equation x = sin x using bisection method with a tolerance of 10-8. Submit your source code and program output. State your numerical answer and the number of iterations used. 2. In a diffraction and interference experiment, light of wavelength goes through a single slit of width d. The diffracted light intensity I at angle is given by I = Io 2 (sina) where a = (7d/A) sin and I is a constant. (a) Plot a graph of I/Io against a which includes 3 maxima on each side of the y-axis. They are the first, second, and third order maxima. (b) From your graph, roughly estimate the values of a; > 0 (i=1,2 and 3) at which the first, second and third order maxima occur. (c) To evaluate a, accurately, the roots of a nonlinear equation F(a) calculated. Find F(a). = 0 should be (d) Hence, write a computer program for finding a root of your nonlinear equation using the Newton-Raphson method up to a tolerance of 10-8. Run your program 3 times with appropriate initial guesses to locate the 3 maxima. In each case, write down your answer, initial guess, and the number of iterations used. Submit only the version of your program for finding the first order maximum. (e) What are the values of sin for the 3 maxima? (10) (20)