MAE 384. Advanced Mathematical Methods for Engineers.

| May 20, 2015

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MAE 384. Advanced Mathematical Methods for Engineers. Homework Assignment 1. Due May 21. 1. Write the following numbers in 32-bit single-precision standard form (as stored in a computer). (a) 66.25 (b) .06298828125 (c) -30952 2. Convert the following to decimal format: (a) 1100101 (b) 10001101.001 (c) 10110001110001.01010111 3. Consider the function f(x) = 1 cos x sin x Notice that, for small x, the numerator consists of a subtraction of two numbers close to each other in size (both near 1), and the denominator is also a small number. (a) Find the solution for x = 0:001 to six digits, including zeros. Be sure to round your result to 6 digits at each calculation step. (b) Use MATLAB (format long) to calculate f(0:001). Consider this to be the true value. Find the true relative error between cases (a) and (b). (c) Multiply f(x) by 1+cos x 1+cos x and be sure to simplify your expression. (Recall that sin2 x + cos2 x = 1.) Then Önd the value of the new f(x) for x = 0:001 to 6 digits. (Round at each step in the calculation.) Find the true relative error, and compare with your results in (b). Explain your result. 4. Write a MATLAB program to Önd cos x to a prescribed error input by the user. Use the series solution, given by cos x = X1 n=0 (1)n (2n)! x (2n) (Recall that 0! = 1.) Find cos( 3 ) to within a relative error of 0.0001. Use an estimated relative approximation error given by “a = Sn Sn1 Sn Note that Sn is the value of the summation for n terms in the series. You should present your solution to this problem formally: (a) Statement of the problem (1 sentence) (b) How you solved the problem (a few sentences and, possibly, a áow chart) (c) Results (value of cos( 3 ), error, number of terms required) (d) Discussion (anything you may want to add) (e) MATLAB code

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