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CHEMISTRY 440/540 9/29/08 EXERCISES ON Newton-Raphson and Numerical Integration. These problems need not be submitted for grading. They are for the student’s own evaluation of how well they understood Lecture #1 on 9/29/08 1 2 3 4 5 6 7 Set up a Newton iteration for computing the square root x of a positive number, c, and apply it to c = 2. How many iterations does it take, from an initial guess value of 1.00 to obtain the square root of this number to 6 decimal places? Use the NR technique to obtain the solution of a transcendental equation: 2 Sin x = x. such equations are analytically insoluble. Apply the NR technique to solve for f(x) = x3 + x – 1 = 0. Find all three solutions. An initial guess value will converge to the nearest solution to it. Make a back-of-the-envelope plot of the function to enable you to wisely choose the initial guess values that can allow you to isolate and converge to all three solutions. Without a rough plot, there is always the brute force method of trial and error. Also solve the following functions using the NR method: (a) e-x – tan x = 0; (b) sin x = cot x; (c) x + ln x = 2; (d) e-x2 - x = 0. Where, in chemistry, would you apply Numerical integration techniques? Why would one ever want to know the area under a curve for chemical analysis? The trapezoidal method/rule is never, ever, ever going to be exactly accurate. Why is this the case? How can one improve on the accuracy of the trapezoidal method? What is the limiting factor in extending this improvement? Use both the trapezoidal and Simpson methods to evaluate ∫32 e-0.05u by using n = 10, 100 and 106. Of course, use a computer for these calculations, or a smart calculator or good old Microsoft excel.