EGN 3420 Midterm Spring 2006 NAME___________________________ Comp name _________________________ Write neatly. I cannot give you credit if I cannot read your answer. You can use a calculator. Write on test sheets. Use the back of the sheet if necessary. Show all work for partial credit and explain how you are going to solve the problem. 1. (16%) Find the largest root of the function131875013597513510)(23−+−= xxxxf . Use the Bisection method only to estimate the root using 3 iterations. Start with an interval of 18 and 22. You may use the chart below.1. (16%) Find the largest positive root (the one around 20) of the equation in problem #1, 131875013597513510)(23−+−= xxxxf using Newton’s method. Do only 3 iterations.2. (18%) Using Newton’s methods and the equation in problem #1, the left graph is of g(x) (Newton’s) and the graph on the right is of its derivative, g’(x). You can answer these questions by only looking at the graphs. Explain. )(xg dxxdg )( a. Will the algorithm converge if we use an initial guess of 30? b. How about if we use an initial guess of 9? c. Does the algorithm give quadratic convergence from 30 to the root?1. (17%) Find a 3rd order Taylor’s series polynomial to approximate )sin(10)(2xxxf += around π=a . Find an upper bound for the error. Note this error will be in terms of the input x. Note ()0sin=πand ()1cos−=π2. (17%) What is the range of values of ),,( zyxffor the following input ranges? 1.5±=x 5.020±=y 13±=z 37),,(32232++−= zyxxyyxezyxfx3. (16%) Find the rate of conversion for the following function.
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