ME2016.MT2.F04_Solution - George W We ocl ruff School of...

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Unformatted text preview: George W. We ocl ruff School of Mechanical Engin ee rin g Georgia Institute of Technology ME 2016 MeCord 3.: Paredis Fall 2004 Midterm Examination #2 Friday, November 12, lilfldl Tltis exam is a titty-minute exam {the entire class period}. The exam is closed hook and closed notes. You may use one double-sided sheet of formulae [3.5”x11”}. You are also allowed to use a non—graphing calculator. You are on your honor not to cottsult with your classmates. Cheating will not be tolerated and will result in immediate and strict action. Please t" eel Free to use any pictures, graphs, Flow charts, code fragments, or written comments in describing your Understanding of the problems and their solutions. Be sure to clearly state any assumptions or conditions regarding the solution. Please write all of your answers on these sheets. You may use the back of the paper it" necessary. A totai of ltlU points are awarded. Honor Pledge My signature below indicates that I ltave read the above text. On my honor, t pledge that l have neither given rtor received inappropriate aid in the preparation ol‘this examination. I also pledge that l have not spoken with any students who have previously taken the ME 2U] ti eaani today {if you have, please come and we will make arrangements for an alternate exam — no penalty}. Further, 1 will not discuss this exam with any other student taking tvIE Zfild this semester who has not yet taken the exam until alter all the exams have been given [SPTvl on the day of the exam}. inn—h Fl 1' II. it"— . ',—-I if f ’1' _,_.-Ir_ l"_'_,_ I”: I, 1‘ ILJ'K ’1'” :59— \FJ Printed Name r” Signature George W. Woodrnfl' School of Mechanical Engineering Georgia Institute of Teohnolog}F Question 1: Solving Linear Equations lQuestion 1.1 [5 points): What is pivoting in Gauss elimination? (Pr'lr'ob'aj rd 1&1; (fracas: {oz/J “w”: (fl’hd My FMP'W' Eff?“ “519;; ; 'Elfjdfi Cz/haQuss Law'mf’hh- Question 1.3 (5 points): Why is sealing necessaryr in Gauss elimination? _ 6k 'L’vé W (ago not? fié :5,“th *v’c 1‘5 mg; (“GM/M George W. Woodmif School of Merhanical Engineering Georgia Institute of Technologyr Question 2: Interpolation [15 points] Lag-Mtge polynomials are defined by: so) = i Ail-{stifle} with Lax} = an 1;. w 'J‘ Clo the graph below, please, provide a graphical interpretation of ffllxl through the three given data points. Also sketch each ofthe Lagtaoge coefficient polynomials. Lita} . Clearlyr label all the clata points and curves. I 3 George W. Woodrufl' School of Mechanical Engineering Georgia Institute of Technology Question 3: Numerical Integration Given beiow is a graph that depicts the total absolute error as a function of the segment a'u'i'. size for the numerical computation of the integral 1" = J. smkafi total error ".5 E3 " .-. .L Dan M GI -1—l—I— - - - - - - - - - - - - - - - - - - - - . . . . . . . . . . . . . . . . . . . . . . . _ u . . _ _ _ . _ . . . - - - - - . - - i.- aegm eni size Question 3.1 {15 points): Provide an interpretation of this graph. George W. Woodmff Sehool of Mechanical Engineering Georgia Institute of Technology Question 3.2 [5 points): Which numerical integration method was used to generate this graph? Justify yoLIr answer. (Hint: it was one of the methods discussed in class). jig Law—oi was Va firm. 5 ‘ em 5/3 . Beige an .8er i3 005;) . Fats Wetatecfi i; ii; George W. Woodrufl' Sehooi of Mechanical Engineering Georgia Institute oi'Teehnologyr Question 4: Solving UDEs Question 4.] [ED points]: Provide a graphical niterpretation ofRalston’s method. This is a seeond—order Rungenliutta method given by the following equations: J1f+1=Jif+[é~k-l+%k1]fi Where the it's are given by: J J‘ r - f,— {jb thigh 15!.“ Question 4.2 (5 points}: Provide and example of a mathematical equation that can he solved using Raisin-11’s method. George W. 1|tl'r'oodruff School of Mechanical Engineering Georgia Institute of Technology Question 5: Curve Fitting (25 points} Context: Consider the problem of fitting a circle through a set ot'data points, {.tlhvfl. as is illustrated in the figure below. This problem is vcrjir similar to linear regression. and can be solved using the generic approach for solving curve fitting problems discussed in class. The he}.r to solving this problem is to parameterize the circle in the following itnplicit i 'av: A" + _t" -|‘ ox + by + c = U. The radius and center point of the circle can 3+5: then be found as Radius =fl——-e and Carder =[-—ol2,—bll] (hint: notice that the expression tbr the circle is linear in the coefficients u, b. and e) #1 as e es 1 Task: Describe the computational process for obtaining the least squares solution for or b, and e, given a set of data points, fxgi'r}. lfveu decide to use a matrix formulation in your computations, then you are allowed to describe the solution in matrix tbrrn {Don‘t waste your time computing a matrix inverse or solving a set oflinear equations}. Partial credit alternative: lfvou have difficulties solving the problem above, then you are allowed to answer the following question instead. You will receive partial credit. Describe the generic process for curve fitting. Clearlv identifyr and explain euclt of the steps you go through. George W. Wendrnfl Selma! of Mechanical Engineering Genrgia Institute of Teehnniflg}r @Enrnwfiafi 6&6 GM??— ?LEJraL + m1+éfi7+ C3-C9 cm: {121, +293, + {3.2JL will aw:- 11:3 2-3,:- (an) --. 7mm .* Y fig. : Cl 2 :72 Zn (Kia‘i'af4’flz’i +‘éa1'+c>‘= XL 3‘61 “ :33: :9 .: c2 ifxfigf+m +33% we)? 3-43 I: gr 3 a : 52 (1:34qu mar +53; +9) Genrge W. Wendrnfi School of Mechanical Engineering Genrgia Institute of Technology ' ...
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