Final Winter 2013 Solutions - Northwestern University NetID...

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Northwestern UniversityNetID:ID#:Math 230 Final ExamSolutions!Winter Quarter 2013Monday, March 18, 2013Complete the front page!!!! [2 points!]Instructions:ShowALLof your work on the pages provided! Explain your reasoning usingcomplete sentences and correct grammar, spelling, and punctuation. All numerical answersMUSTbeexact; e.g., you should writeπinstead of 3.14...,2 instead of 1.414..., and13instead of 0.3333...Makesure that your final answer isclearlyindicated. This test has12questions onthirteenpages. Checkto make sure you have pages 1 -??. Calculators, course notes, textbooks, etc. areNOTallowed.Do not write your name on this exam!Write your NetID and your Student ID Number in the spaces provided atthe top of this page.Mark your section!!XSec. #TimeInstructor218:00Yang319:00Lior4110:00TiM5111:00Broderick6112:00Garton6312:00Lior711:00Goldstein812:00BroderickQuestionPossible PointsScoreFront Page2113218318420514620714816912102211181213Total200
Math 230 Final ExamSolutions!!!Winter Quarter 2013Page 2 of??Question 1.Each of the following equations (in either spherical or cylindrical coordinates) describesa sphere inR3. For each sphere, give an equation in rectangular coordinates that describes the surface.Find the radius and center of each sphere.(a)(6 points)r2+z2= 4 + 4rcos(θ)-2rsin(θ)2+y2+z2= 4 + 4x-2y=x2-4x+y2+ 2y+z2= 4
(b)(7 points)ρ= 6 cos(θ) sin(ϕ) + 24 sin(θ) sin(ϕ)-8 cos(ϕ)
Math 230 Final ExamSolutions!!!Winter Quarter 2013Page 3 of??Question 2(6 points each).For each of the following, calculate the limit or explain why the limitdoes not exist.(a)lim(x,y,z)(-1,0,0)3x3+x2y2-z4x2+y2+z4+ 2

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