Final Fall 2013 Solutions - Northwestern University NetID...

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Northwestern UniversityNetID:Math 230 Final ExamFall Quarter 2013December 11, 2013Instructions:Read each problem carefully.Write legibly.Show all your work on these sheets.Make sure that your final answer isclearlyindicated.If two answers arepresented then the average of the points for each answer will be given!This exam has 17 pages, and 9 problems.Before starting the exam, pleasecheck that your copy contains all of them and obtain a new copy of the examimmediately if it does not.You may not use books, notes or calculators.Good luck!(2 points)Put a check mark next to your section number, and write your NetIDin the upper right corner of this page! Do not write your name on this exam!Sec. #TimeInstructor218:00Chen, T-H319:00Bohmann339:00Bode379:00Chen, T-H399:00Wang4110:00Bode4710:00Bohmann5711:00Naber5911:00Wunsch6112:00Chen, X6712:00Wang711:00Zhu812:00Xia872:00Zhu892:00JuschenkoProb.PointsScorepossible02136220320420520620730820912TOTAL200
Math 230 Final ExamFall Quarter 2013Page 2 of 17Question 1(36 points, 3 points each).True or False? Circle the correct answer.PART AConsider the following statements forthree-dimensional space.(a) Two vectors which are parallel must have their dot product equal tozero.
(b) Two lines which are not parallel must intersect.
(c) Two curves which are tangent to each other at a point may have dif-ferent curvatures at that point.
(d) A circle has constant curvature.
(e) If a particle moves at constant speed then the tangential component ofthe acceleration is zero.
(f) The domains off(x, y) =px2-y2andg(x, y) = ln(x2-y2)aredifferent.
Math 230 Final ExamFall Quarter 2013Page 3 of 17PART BSuppose thatf:R2Rhas continuous second partial derivatives.(a) The mixed partial derivativesfxy(a, b)andfyx(a, b)are equal.
(b) There exists such a functionfwhose linear approximation at(0,0)isL(x, y) = 4x+ 2y-3and whose quadratic approximation at(0,0)isQ(x, y) = 4x2+ 2y2-3.
(c) Iffx(0,0) =fy(0,0) = 0andfxx(0,0)is positive, thenfhas a localminimum at(0,0).
(d) Iffx(0,0) =fy(0,0) = 0andfxx(0,0)·fyy(0,0)<[fxy(0,0)]2thenfhas a local minimum at(0,0).
For the next two statements, consider the following table.(x, y)ffxfyfxxfxyfyy(2,3)unknown001-36(-7,8)-600-4unknown7(e) A local minimum offoccurs at(2,3).

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