Solutions - How to Think Like a Mathematician.pdf

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1、HowtoThinkLikeaMathematicianSolutionstoExercisesSeptember17,2009ThefollowingaresolutionstoexercisesinmybookHowtoThinkLikeaMathematician.Chapter1Exercises1.10(i)5(ii)3(iii)4(iv)0(v)Infinite(vi)2(vii)2(viii)3.Exercises1.20(i){−1,1,2,3,4,5,7}(ii)Z.Exercises1.23(i)(a){0,1,2,3,4,5}

2、,(b)∅,(c){0,1,5}.(ii)Z,∅,Z.Exercises1.33MyintentionwasthatthedomainwouldbethelargestinR.√√(i)R{(5−13)/2,(5+13)/2},(ii)R,(iii)Manyanswersarepossible.Hint:Wecanalwaysfindalinebetweentwopointsintheplanesowecouldusealinearfunctionsuchasf(x)=ax+bwhereaandbaredeterminedbyusingtheeq

3、uationsf(−1)=5andf(3)=−2.Oneyouhavemasteredtheideabehindthistryfindingaquadraticwhosegraphpassesthroughthepoints.Thenmoveontofunctionsinvolvinghigherordersorsines,cosinesandtheexponentialfunction.Exercises1.34(i)A∩B={2},A∪B={0,2,3,4,5,6,7,8,10}(orX{1,9}ifyouprefer),AB={0,4,6

4、,8,10},BA={3,5,7},A×Bhas24elements(whichisalottotype),X×Ahas66elements,Ac={1,3,5,7,9},Bc={0,1,4,6,8,9,10}.(ii)Thequadratichasroots2and7sotheunionis{2,3,4,5,6,7,8,9,10}andtheintersectionis{7}.(iii)Thepreciseanswerswilldependonthesetsyoutake.Youshouldalwaysgetequalitiesofsetse

5、xceptin(d)and(f).(Thisdoesnotexcludethesituationwhereyourexamplesgiveequalitiesin(d)and(f).However,thereexistexampleswheretheydon’thaveequality.)(iv)(a)AandBdisjoint.1(b)A∪BAC(A∩B)c(c)Anexampleoffrom(i):(A∩B)∪(A∩B)orA∩(B∪C).(v)Notapplicable.Chapter2Solutionsnotapplicabletothi

6、schapter.Chapters3and4Thesolutionstothesechapterswilldependonyourpersonalanswersandwrit-ingstyle.Exercises3.2(i)Don’tforgettoexplainwhata,b,c,α,β,γandhareandexplainwhatareassumptionsandwhatareconclusions.(ii)Therearesomemathematicalmistakes.Thesecondlineisactuallyf0(x)yetthes

7、tudenthasequatedittof.Alsotheysay(takingintoaccountthepreviouscomment)thatf0(x)=6x2−24x+18impliesthatx=1and3.Thisisnottrue,itistheequationf0(x)=0thatgivesusthevaluesforx.Equalitysignsaremissinginthecalculationofx.Somenotationismissing.Whenfindingmaximaandminimaweputthevaluesof

8、xintotheexpressionforthesecondderivativeandsoweshouldreallyhaved2yd2