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Instructor Hacker Course AME 302 Chapter 9 Homework Set Numerical Integration Name Instructions This homework is broke into three components Part 1 true false concept questions Part 2 basic computational skills and Part 3 derivation analysis and problem solving questions The rules for each component are listed below Part 1 There is no partial credit given for true false problems Since these are concept questions you do not need to show any work for these problems There are 10 problems worth 40 4 points each Part 2 There is no partial credit given for multiple choice problems Although there is no partial credit on this assignment you must show your work on all of the problems If you fail to show work you will receive a zero for the problem even if it is correct There are 4 problems worth 40 10 points each Part 3 On these problems you must show all of your work to receive any credit If in doubt write it out Show your work as clearly as you can if I can t understand how you got an answer I will not give you credit for it Remember I know how to solve the problem and to make matters worse I have a lot of training in following logical arguments There are 2 problems worth 20 10 points each Warning The de nition of little or no work will be determined by the instructor not the student Parts 1 3 Circle your answers here Do not detach this sheet from the homework 16 17 10 pts 10 pts 11 12 13 14 15 a a a a a b b b b b c c c c c d d d d d e e e e e 1 2 3 4 5 6 7 8 9 T F T F T F T F T F T F T F T F T F 10 T F Attention For all problems involving writing a MATLAB program you must turn in your MATLAB code with the output to receive any credit AME 302 chapter 9 hw set Copyright Wayne Hacker 2018 All rights reserved 2 Part 1 True or False Concept Questions Component Newton Cotes Numerical Integration Formulas Problem 1 True or False The composite trapezoidal rule uses linear functions to approximate the underlying function over each subinterval Problem 2 True or False The composite trapezoidal rule cannot be applied to a data set unless the underlying function that generated the data is known Problem 3 True or False Simpson s 1 3 rule uses quadratic polynomials on each subinterval to approximate the underlying function whereas Simpson s 3 8 rule uses cubic polynomials on each subinterval Problem 4 True or False For a small xed step size h cid 28 1 Simpson s 1 3 rule is more accurate than the trapezoidal rule Problem 5 True or False For a small xed step size h cid 28 1 Simpson s 3 8 rule is more accurate than Simpson s 1 3 rule Advanced Numerical Integration Formulas Involving Functions Problem 6 True or False In order for Richardson Extrapolation to be applied to a numerical integration procedure the error term in the procedure must have a predictable form in terms of the step size or some other parameter Problem 7 True or False Romberg Integration uses Richardson Extrapolation to improve the initial estimates that are made using the composite Simpson s 1 3 rule method Problem 8 True or False The two point Gaussian quadrature method f x dx c0f x0 c1f x1 uses a trapezoidal method where the two evaluation points x0 and x1 are the endpoints of the interval of integration cid 90 1 1 Problem 9 True or False Gaussian quadrature is ideally suited for experimental data because the underlying function that generated the data need not be known to apply the method Problem 10 True or False Adaptive quadrature methods are ideally suited for work ing with slow varying functions AME 302 chapter 9 hw set Copyright Wayne Hacker 2018 All rights reserved 3 Part 2 Basic Computational Skills Component Newton Cotes Numerical Integration Formulas Problem 11 Consider the following elementary integral I cid 90 4 0 cid 12 1 e x dx x e x cid 12 cid 12 4 0 3 e 4 3 018316 Evaluate the integral using a the composite trapezoidal rule with n 4 and compute the true percent relative error based on the exact solution to two decimal places a cid 15 t 1 67 b cid 15 t 1 27 c cid 15 t 2 67 d cid 15 t 2 27 e None of these Problem 12 Consider the following elementary integral I cid 90 4 0 cid 12 1 e x dx x e x cid 12 cid 12 4 0 3 e 4 3 018316 Evaluate the integral using Simpson s 1 3 rule with n 4 and compute the true percent relative error based on the exact solution to two decimal places a cid 15 t 0 16 b cid 15 t 0 26 c cid 15 t 0 36 d cid 15 t 0 46 e None of these Problem 13 Consider the following elementary integral I cid 90 4 0 cid 12 1 e x dx x e x cid 12 cid 12 4 0 3 e 4 3 018316 Evaluate the integral using a single application of Simpson s 3 8 rule and compute the true percent relative error based on the exact solution to two decimal places a cid 15 t 0 9 b cid 15 t 1 1 c cid 15 t 1 3 d cid 15 t 1 5 e None of these AME 302 chapter 9 hw set Copyright Wayne Hacker 2018 All rights reserved 4 Advanced Numerical Integration Formulas Involving Functions Problem 14 Consider the following elementary integral I cid 90 4 0 cid 12 1 e x dx x e x cid 12 cid 12 4 0 3 e 4 3 018316 Evaluate the integral using the two point Gaussian quadrature approximation and com pute the true percent relative error based on the exact solution to two decimal places a cid 15 t 0 9 b cid 15 t 1 2 c cid 15 t 1 5 d cid 15 t 1 8 e None of these Problem 15 Consider the following integral I 200 cid 90 30 0 cid 18 z cid 19 5 z e z 15 dz Use the Romberg integration algorithm to nd R3 3 rounded to four decimal places Keep all of the digits until the end For computational convenience the rst column of the trapezoidal approximations are included to a more than su cient level of accuracy in the table below That is you will not need to compute beyond the level of accuracy of what is given MATLAB s built in integration function gives the value of the integral as I 1480 5685 to four place accuracy What is the true absolute percent relative error based on the exact solution to two signi cant gures i 1 2 3 O h2 i O h4 …

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