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. (15%) Find the largest root of the function214402155410559)(23−−+= xxxxf . Use 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.1. (14%) Find the largest positive root of )sin()(2xxxf = to a tolerance of 0.001 using Newton’s method. Note if )()( xgxfy=then )(')()()('' xgxfxgxfy +=2. (14%) If we had used )()( xfxxg−= instead of Newton’s method in the problem above where )sin()(2xxxf = would the series have converged? Explain why.3. (15%) Find a 3rd order Taylor’s series polynomial to approximate )cos()sin()( xxxf−= around 2π=a. Find an upper bound for the error. Note this error will be in terms of the input x. Note 12sin =πand 02cos =π4. (14%) Compute the number of terms needed in a Taylor’s series polynomial to approximate )cos()sin()(xxxf−= around 2π=a that gives an error of no more than 710− when using x where 443ππ<< x. Hint use the Taylor’s series error term. Hint 1<−ax .5. (14%) What is the range of values of ),,( zyxf for the following input ranges? 1.2±=x 5.05±=y 2π=z 5)sin(),,(223+−−= xxyzyxzyxf6. (14%) Find the rate of conversion for the following function.
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