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 1 510 2 5975 18750 x x 13 13 13 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 16 Find the largest root of the function f x x 3 1 16 Find the largest positive root the one around 20 of the equation in problem 1 510 2 5975 18750 f x x 3 x x 13 13 13 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 g x a dg x dx 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 f x x 2 10 sin x around a Find an upper bound for the error Note this error will be in terms of the input x Note sin 0 and cos 1 2 17 What is the range of values of f x y z for the following input ranges x 5 1 y 20 0 5 z 3 1 f x y z e2x x3 y 2 x x 3 zy 37 2 y 3 16 Find the rate of conversion for the following function f n n 2 2 5n 2 6n 3 n 3n 4 37n 5 n 1 5
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