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2009 Test 2 solution

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Department
Mathematics
Course
MAT137Y1
Professor
N/ A
Semester
Winter

Description
MAT137Y20082009WinterSessionTermTest2Solutions1Evaluatethefollowinglimitsorshowthelimitdoesnotexist18ilimcotxx0xLetLbethelimitThenxcosxsinxcosxxsinxcosxcosx1HlimlimLlimh0h0h0sinxxxsinxsinxxcosxxsinxsinxxcosx0Hlimlim0h0h0sinxxcosxcosxcosxxsinx22xcosx8iilim2xx2009222xcosxx1x1Since1cosx1wehave222x2009x2009x20092xcosxsobythesqueezetheoremlim12xx20092cosxwhichdoesnotexistbutNoteapplyingLHopitalsRuletwiceyieldslimx2LHopitalsRuledoesnotapplyinthissituation12152RecallthatthevolumeofarightcircularconeisVprhwhereristheradiusofthe3circularbaseandhistheheightoftheconeFindthelargestpossiblevolumeofarightcircularconethatisinscribedinasphereofradiusRMakesurethatyouransweryieldsanabsolutemaximum12prhThisquestionisposedinSHE4543whichprovidesadiagramWemaximizeV32222SincerhRRbythePythagoreanTheoremwesolveforrtoget1112222pRhRhpRhphRhVh333whereh202RWesolvefortheabsolutemaximumdifferentiatinghi1102222VhhR2hhR0RhR2hhR0pR334R22222Rh2hRR2h2hR03h4hR0h34RSincelimVhlimVh0thecriticalpointhmustyieldmaximumvolume3h0h2RThereforethemaximumvolumeis23R4R32pR124RpRV3393811
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