Open and Closed Sets - Ximera

A set is open if every point in is an interior point. A set is closed if it contains all of its boundary points. Determine if the following sets ...  Statistics GetHelpContactmyinstructorReporterrortoauthorsRequesthelpusingXimeraReportbugtoprogrammers Another MathEditor Failed Saved! Saving… Reconnecting… Save Update Erase EditMeProfileSuperviseLogout SignInSignInwithGoogleSignInwithTwitterSignInwithGitHubSignInWarning×Youareabouttoeraseyourworkonthisactivity.Areyousureyouwanttodothis? No,keepmywork. Yes,deletemywork.UpdatedVersionAvailable×Thereisanupdatedversionofthisactivity.Ifyouupdatetothemostrecentversionofthisactivity,thenyourcurrentprogressonthisactivitywillbeerased.Regardless,yourrecordofcompletionwillremain.Howwouldyouliketoproceed? Keeptheoldversion. Deletemyworkandupdatetothenewversion.MathematicalExpressionEditor×+–×÷xⁿ√ⁿ√πθφρ( )| |sincostanarcsinarccosarctaneˣlnlog Cancel OKXimeratutorial HowtouseXimeraThiscourseisbuiltinXimera. Howismyworkscored?Weexplainhowyourworkisscored. Areviewofintegration AreviewofdifferentiationWereviewdifferentiationandintegration. AreviewofintegrationWereviewdifferentiationandintegration. AreviewofintegrationtechniquesWereviewcommontechniquestocomputeindefiniteanddefiniteintegrals. Areasbetweencurves AreabetweencurvesWeintroducetheprocedureof“Slice,Approximate,Integrate”anduseitstudythe areaofaregionbetweentwocurvesusingthedefiniteintegral. Accumulatedcross-sections Accumulatedcross-sectionsWecanalsousetheprocedureof“Slice,Approximate,Integrate”tosetupintegrals tocomputevolumes. Solidsofrevolution Whatisasolidofrevolution?Wedefineasolidofrevolutionanddiscusshowtofindthevolumeofoneintwo differentways. ThewashermethodWeusetheprocedureof“Slice,Approximate,Integrate”todevelopthewasher methodtocomputevolumesofsolidsofrevolution. TheshellmethodWeusetheprocedureof“Slice,Approximate,Integrate”todeveloptheshellmethod tocomputevolumesofsolidsofrevolution. ComparingwasherandshellmethodWecompareandcontrastthewasherandshellmethod. Lengthofcurves LengthofcurvesWecanusetheprocedureof“Slice,Approximate,Integrate”tofindthelengthof curves. Applicationsofintegration PhysicalapplicationsWeapplytheprocedureof“Slice,Approximate,Integrate”tomodelphysical situations. Integrationbyparts IntegrationbypartsWelearnanewtechnique,calledintegrationbyparts,tohelpfindantiderivativesof certaintypesofproductsbyreexaminingtheproductrulefordifferentiation. Trigonometricintegrals TrigonometricintegralsWecanusesubstitutionandtrigonometricidentitiestofindantiderivativesofcertain typesoftrigonometricfunctions. Trigonometricsubstitution TrigonometricsubstitutionWeintegratebysubstitutionwiththeappropriatetrigonometricfunction. Partialfractions RationalfunctionsWediscussanapproachthatallowsustointegraterationalfunctions. Improperintegrals ImproperIntegralsWecanuselimitstointegratefunctionsonunboundeddomainsorfunctionswith unboundedrange. 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Interpretingthegradient InterpretingthegradientvectorThegradientisthefundamentalnotionofaderivativeforafunctionofseveral variables. mooculusCalculus eContinuityoffunctionsofseveral variablesOpenandClosedSetsBartSnappandJim Talamo Wegeneralizethenotionofopenandclosedintervalstoopenandclosedsetsin . Whenwemakedefinitionsanddiscussseveralimportanttheoremsforfunctionsofa singlevariable,weneedthenotionofanopenintervaloraclosedinterval.A typicalexampleofanopenintervalis,whichrepresentsthesetofallsuch that,andanexampleofaclosedintervalis,whichrepresentsthesetof allsuchthat.Weneedanalogousdefinitionsforopenandclosedsetsin . Asetisacollectionofdistinctobjects. Givenaset,wesaythatisanelementofifisoneofthedistinctobjectsin,and wewritetodenotethis. Giventwosetsand,wesaythatisasubsetofifeveryelementofisalsoan elementofwritetodenotethis. Withtheseideasinmind,wenowdiscussspecialtypesofsubsets. (OpenBalls) Wegivethesedefinitionsingeneral,forwhenoneisworkinginsincetheyarereally notallthatdifferenttodefineinthanin. Anopenballincenteredatwithradiusisthesetofallpointssuchthatthe distancebetweenandislessthan. Inanopenballisoftencalledanopendisk. (InteriorandBoundaryPoints)Supposethat. Apointisaninteriorpointofifthereexistsanopenball. Intuitively,isaninteriorpointofifwecansqueezeanentireopenball centeredatwithin. Apointisaboundarypointofifallopenballscenteredatcontainboth pointsinandpointsnotin. Theboundaryofisthesetthatconsistsofalloftheboundarypointsof . Supposethat. Whichofthefollowingareelementsof? Whichofthefollowingareinteriorpointsof? Whichofthefollowingareboundarypointsof? Theboundaryis Wecannowgeneralizethenotionofopenandclosedintervalsfromtoopenand closedsetsin. (OpenandClosedSets) Asetisopenifeverypointinisaninteriorpoint. Asetisclosedifitcontainsallofitsboundarypoints. Determineifthefollowingsetsareopen,closed,orneither. Thesetisopenclosedneitheropennorclosed . Thesetisopenclosedneitheropennorclosed . Thesetisopenclosedneitheropennorclosed . (BoundedandUnbounded) Asetisboundedifthereisanopenballsuchthat Intuitively,thismeansthatwecanencloseallofthesetwithinalarge enoughballcenteredattheorigin. Asetthatisnotboundediscalledunbounded. Whichofthefollowingsetsarebounded? Let’snowlookatafewexamples. Considerthefunction. Thedomainofthefunctionisthesetofallforwhich,whichwecan writeinset Thepointisaninteriorpointaboundarypointnotanelement of. Thepointisaninteriorpointaboundarypointnotanelement of. Thedomainisopenclosedneitheropennorclosed andboundednotbounded . We’vealreadyfoundthedomainofthisfunctiontobe Thisistheregionboundedbytheellipse.Sincetheregionincludestheboundary (indicatedbytheuseof“”),thesetcontainsdoesnotcontain allofitsboundarypointsandhenceisclosed.Theregionisboundedunbounded asadiskofradius,centeredattheorigin,contains. Determineifthedomainofisopen,closed,orneither,andifitisbounded.Aswe cannotdivideby,wefindthedomaintobeInotherwords,thedomainisthesetof allpointsnotontheline.Foryourviewingpleasure,wehaveincludedagraph: Notehowwecandrawanopendiskaroundanypointinthedomainthatliesentirely insidethedomain,andalsonotehowtheonlyboundarypointsofthedomainarethe pointsontheline.Weconcludethedomainisanopensetaclosedsetneither opennorclosedset .Moreover,thesetisboundedunbounded . ←PreviousNext→