EGN 3420 Midterm Fall 2005 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 59 2 554 440 x x Use 105 21 21 the Bisection method only to estimate the root using 3 iterations Start with an interval of 5 and 6 You may use the chart below 15 Find the largest root of the function f x x 3 1 14 Find the largest positive root of f x x 2 sin x to a tolerance of 0 001 using Newton s method Note if y f x g x then y f x g x f x g x 2 14 If we had used g x x f x instead of Newton s method in the problem above where f x x 2 sin x would the series have converged Explain why 3 15 Find a 3rd order Taylor s series polynomial to approximate f x sin x cos x around a 2 Find an upper bound for the error Note this error will be in terms of the input x Note sin 1 and cos 0 2 2 4 14 Compute the number of terms needed in a Taylor s series polynomial to approximate f x sin x cos x around a that gives an error of no more than 10 7 when using x where 2 3 x Hint use the Taylor s series error term Hint x a 1 4 4 5 14 What is the range of values of f x y z for the following input ranges x 2 1 y 5 0 5 z 2 f x y z x 3 y sin z y 2 x x 2 5 6 14 Find the rate of conversion for the following function f n n 2 3n 4 n 3n 4 37n 5 5
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