LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFEiVEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3#
# The following code is autogenerated by the Explore command. Do not edit.
#
Explore:-Runtime:-Update( plots:-display(CURVES(Matrix(200, 2, [[.5,.5],[.499727205151898,.51651424933422],[.499046183768338,.530869183116155],[.497787821050396,.546981754053028],[.495995852743071,.563151516701292],[.493691029780341,.579178072181801],[.491091608481462,.59396292928113],[.487933611605656,.60918237341925],[.48417163259095,.624811178162111],[.479923655568534,.640261487321843],[.475039876506507,.656003576012482],[.470310477069373,.669729358564684],[.464512074931575,.685009546355756],[.45819051702442,.700153566315704],[.451631433260378,.714543814851831],[.445283537589727,.727425968507509],[.437257608805446,.742499038228492],[.43005915850958,.755047290089568],[.421233127667921,.769374557364092],[.413022178230764,.78180255550565],[.403592038086415,.795149905629756],[.394211395246947,.807566864043341],[.38401672989873,.820204858111],[.374297867327618,.831513357972189],[.363441844894844,.843380292648591],[.351767466804322,.855330338274218],[.341284700084083,.865410390504317],[.329638992091377,.875949644092106],[.317266359016559,.886448000947573],[.304838711743274,.896324816057109],[.292521711962305,.905500984006996],[.278522895076549,.915240890229007],[.265667485517899,.923580909790086],[.251666446541925,.932046293451247],[.23874628658101,.939317892469426],[.224381357345785,.946825476529331],[.210649494047315,.953460903119123],[.196086451826,.959945761378763],[.181655601525334,.965833921515457],[.166358951080717,.971513201719023],[.151458781563354,.976508381340081],[.136066580250622,.981129801341489],[.120662959991003,.985222062654007],[.106400167202223,.988547852742532],[.0899418659735091,.991843939421036],[.0751349374119452,.994322507256248],[.0592733750243558,.996474236001449],[.0440347205923133,.998057168789243],[.0270663310584235,.999266876252607],[.0123469307793924,.999847530053244],[-.00437200811017653,.999980885179708],[-.0196057287761022,.99961546753394],[-.0362476654134973,.99868437588526],[-.0505751484303313,.997435578101577],[-.0668408486454289,.995512160246708],[-.0824847950819056,.993149326857795],[-.0980360169441294,.990294747454764],[-.113432556090224,.986963094308838],[-.128117955757135,.983307137762948],[-.143860835196788,.978857035133121],[-.158721843383843,.974138562482356],[-.174195377744943,.96867469568166],[-.188041297757641,.963293071756552],[-.203150049202307,.956869847450125],[-.217416451680199,.950255579131222],[-.231433875230277,.94321367464903],[-.245831678068451,.935392680299117],[-.258859392031087,.927775426079264],[-.271957034535372,.9195704605507],[-.286116050887836,.910045857709049],[-.298648848543396,.901009806942055],[-.311170936194289,.891372774305993],[-.32358678174721,.881171345563231],[-.334704849354008,.87144671733495],[-.346234476158385,.860723838302002],[-.357336933799694,.849728917509881],[-.368826533425943,.837589970587104],[-.378623686469336,.826564088724357],[-.38951764885725,.813490033699191],[-.398961616738892,.801379542054682],[-.407935511422106,.789116963391608],[-.417164697317229,.775633117226912],[-.425982978803134,.761798590084073],[-.43360807644233,.748965933500915],[-.441282321710617,.73509553918713],[-.448344051081444,.721331452486716],[-.455514050033756,.706172137355766],[-.461304877573587,.692867337636048],[-.467476209311791,.677386565803272],[-.472804123183884,.662653807518664],[-.477623112567755,.647905923955365],[-.481985486469732,.633003724882184],[-.485955973772251,.61767239079352],[-.489632262262819,.601292881049923],[-.492518118002071,.586173681827446],[-.494909064970403,.571168935710194],[-.496959505721767,.555056785892179],[-.498482792275122,.538921790883737],[-.499404072218179,.524404357231852],[-.499948421062582,.507181662553419],[-.499947624535251,.492763100003501],[-.499419212550431,.475907467225156],[-.498385023073832,.459845687956385],[-.49702515983297,.445539092065862],[-.495005390836273,.429504162938308],[-.492449906959786,.413437368713178],[-.489388923351505,.397537901149477],[-.486092457457082,.38289268680678],[-.482217467671015,.367839817369399],[-.477719792640897,.352406640667209],[-.472745115071217,.337174767998623],[-.467121534676336,.321681543743714],[-.461747618761362,.308195055933868],[-.455232308634762,.29320651563682],[-.448200875949892,.278378758243372],[-.440967869541059,.264315172248149],[-.434017212140139,.251747991818593],[-.425286678699385,.23707179512109],[-.417502228316438,.224878409878816],[-.408007798497225,.210985058577506],[-.399217677467098,.198959726950403],[-.389166207946042,.186073794351447],[-.37920822775395,.174114866856881],[-.368426673334243,.161973689817104],[-.3581833304016,.151137990284958],[-.346777664739698,.139798318663444],[-.334550626327951,.128414372689718],[-.323602394202433,.118841908827251],[-.311470784854377,.10886581563075],[-.298615001704751,.0989649881159108],[-.285733691689422,.0896876099439182],[-.272996075828234,.0811048549071196],[-.258551839641691,.072038615973477],[-.245316003491164,.0643165616745078],[-.2309298874154,.0565235213697228],[-.217679955691247,.0498717550627859],[-.202975702486618,.0430526680239134],[-.188945095899546,.0370747892634153],[-.174091370752242,.0312866604953457],[-.159397938771223,.0260883024069934],[-.143849566475576,.0211395796009147],[-.128729617897237,.0168554196971351],[-.113135884140814,.0129678945698998],[-.0975557961963832,.00960947538875739],[-.0831515435435712,.00696265779728045],[-.0665556505403349,.00444945224411991],[-.051647982293547,.00267466792349552],[-.0357023363300146,.00127628572467248],[-.0204058283518766,.000416571362426232],[-.00339919444070735,1.15546563558278e-005],[.0113311907438187,.000128412373410436],[.0280376959833401,.000786731342259095],[.0432370349917847,.00187294913333685],[.0598162302289324,.00359087578772349],[.0740685259024661,.00551657917516013],[.0902249269041988,.00820790717506248],[.105739468059414,.0113087223056748],[.121138107886368,.0148963421930209],[.136359652346541,.0189531777342973],[.150855501241195,.0233002855620031],[.166370046897607,.0284907132459763],[.18099100471142,.0339074596031915],[.196188512680684,.0400977631146581],[.209764124982261,.0461288598396825],[.224551840215932,.0532601752074945],[.23848910642065,.0605424410495594],[.252157427304858,.0682400761360598],[.26616881651069,.0767339357835327],[.278821298799669,.0849594196519264],[.291515807208022,.0937752664498933],[.305208023760902,.103959519452916],[.317298974625144,.113578778142514],[.329350775215083,.123797837771732],[.341269732637713,.134575630827138],[.351914940485234,.144815717319482],[.362923985867448,.156072419712981],[.373493461804421,.16758063535746],[.384395484002807,.180249922786174],[.393659653200729,.19172726776135],[.4039224352397,.205302415500176],[.412782466634757,.217846433233734],[.421165751673984,.230520112778545],[.429746223209442,.244425776657308],[.437899648799921,.258662274849319],[.444908660018602,.271841536995772],[.451917634149337,.286059700044929],[.458319812121927,.300142676350048],[.464764083986276,.315624442410611],[.469918529219418,.329188478444048],[.475350034961685,.344944060862137],[.479974481767293,.359912538561731],[.484089858669872,.374872030572765],[.487741826339467,.389964047516089],[.490982030300283,.405466165198845],[.493878741566683,.422001354957223],[.49604557326925,.437240225940649],[.497723469698339,.452341341694852],[.499008813920056,.468532498826591],[.499766513486373,.484721518476134],[.5,.500000000410207]], datatype = float[8]),COLOUR(RGB,0.,0.,1.00000000)),plot(x^2,x = Student:-VectorCalculus:-`-`(T) .. T,scaling = constrained,color = red),view = [Student:-VectorCalculus:-`-`(T) .. T, Student:-VectorCalculus:-`-`(T) .. T]), ':-Plot2', [':-T'], ':-sliders'=["SliderT2"], ':-labels'=["LabelT2"],':-isplot'=true, ':-handlers'=[T=handlertab[T]]);
#
# End of autogenerated code.
#
Lower the slider towards 0. There is a point where the pixels of the two curves merge, beyond which both curves flatten out to the tangent line as the quadratic approximation morphs down to the linear approximation to the curve.
detailsOne calculates by hand the radius of the osculating circle at the origin of the graph 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, namely of the parametrized curve 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 at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYmUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjcvRjtRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYlUSIwRidGN0ZBRkFGK0ZB. One finds that the radius of curvature at the origin ( LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYmUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSsyRH5Db21tZW50RicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjcvRjtRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYlUSIwRidGN0ZBRkFGK0ZB) is 1/2, and the normal vector there is 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, so the center is along the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GNlEnbm9ybWFsRic= axis a distance 1/2 from the origin, i.e. at 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, so one can plot the circle with the graph in the following way just using 2D plotting.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Edit this line to change the osculating circle data: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Edit this line to change the function: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One sees the nice match at the origin which is the point of tangency of the osculating circle and the original curve. However, it is easier to use the Explore slider version.The Explore command makes it easy to control the zoom. At about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEpJmFwcHJveDtGJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk5GMkY80.23 or so you can barely distinguish the pixels of the red and blue curves
Edit the function in the plot command here too: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
SSJURzYi#
# The following code is autogenerated by the Explore command. Do not edit.
#
Explore:-Runtime:-Update( plots:-display(CURVES(Matrix(200, 2, [[.5,.5],[.499727205151898,.51651424933422],[.499046183768338,.530869183116155],[.497787821050396,.546981754053028],[.495995852743071,.563151516701292],[.493691029780341,.579178072181801],[.491091608481462,.59396292928113],[.487933611605656,.60918237341925],[.48417163259095,.624811178162111],[.479923655568534,.640261487321843],[.475039876506507,.656003576012482],[.470310477069373,.669729358564684],[.464512074931575,.685009546355756],[.45819051702442,.700153566315704],[.451631433260378,.714543814851831],[.445283537589727,.727425968507509],[.437257608805446,.742499038228492],[.43005915850958,.755047290089568],[.421233127667921,.769374557364092],[.413022178230764,.78180255550565],[.403592038086415,.795149905629756],[.394211395246947,.807566864043341],[.38401672989873,.820204858111],[.374297867327618,.831513357972189],[.363441844894844,.843380292648591],[.351767466804322,.855330338274218],[.341284700084083,.865410390504317],[.329638992091377,.875949644092106],[.317266359016559,.886448000947573],[.304838711743274,.896324816057109],[.292521711962305,.905500984006996],[.278522895076549,.915240890229007],[.265667485517899,.923580909790086],[.251666446541925,.932046293451247],[.23874628658101,.939317892469426],[.224381357345785,.946825476529331],[.210649494047315,.953460903119123],[.196086451826,.959945761378763],[.181655601525334,.965833921515457],[.166358951080717,.971513201719023],[.151458781563354,.976508381340081],[.136066580250622,.981129801341489],[.120662959991003,.985222062654007],[.106400167202223,.988547852742532],[.0899418659735091,.991843939421036],[.0751349374119452,.994322507256248],[.0592733750243558,.996474236001449],[.0440347205923133,.998057168789243],[.0270663310584235,.999266876252607],[.0123469307793924,.999847530053244],[-.00437200811017653,.999980885179708],[-.0196057287761022,.99961546753394],[-.0362476654134973,.99868437588526],[-.0505751484303313,.997435578101577],[-.0668408486454289,.995512160246708],[-.0824847950819056,.993149326857795],[-.0980360169441294,.990294747454764],[-.113432556090224,.986963094308838],[-.128117955757135,.983307137762948],[-.143860835196788,.978857035133121],[-.158721843383843,.974138562482356],[-.174195377744943,.96867469568166],[-.188041297757641,.963293071756552],[-.203150049202307,.956869847450125],[-.217416451680199,.950255579131222],[-.231433875230277,.94321367464903],[-.245831678068451,.935392680299117],[-.258859392031087,.927775426079264],[-.271957034535372,.9195704605507],[-.286116050887836,.910045857709049],[-.298648848543396,.901009806942055],[-.311170936194289,.891372774305993],[-.32358678174721,.881171345563231],[-.334704849354008,.87144671733495],[-.346234476158385,.860723838302002],[-.357336933799694,.849728917509881],[-.368826533425943,.837589970587104],[-.378623686469336,.826564088724357],[-.38951764885725,.813490033699191],[-.398961616738892,.801379542054682],[-.407935511422106,.789116963391608],[-.417164697317229,.775633117226912],[-.425982978803134,.761798590084073],[-.43360807644233,.748965933500915],[-.441282321710617,.73509553918713],[-.448344051081444,.721331452486716],[-.455514050033756,.706172137355766],[-.461304877573587,.692867337636048],[-.467476209311791,.677386565803272],[-.472804123183884,.662653807518664],[-.477623112567755,.647905923955365],[-.481985486469732,.633003724882184],[-.485955973772251,.61767239079352],[-.489632262262819,.601292881049923],[-.492518118002071,.586173681827446],[-.494909064970403,.571168935710194],[-.496959505721767,.555056785892179],[-.498482792275122,.538921790883737],[-.499404072218179,.524404357231852],[-.499948421062582,.507181662553419],[-.499947624535251,.492763100003501],[-.499419212550431,.475907467225156],[-.498385023073832,.459845687956385],[-.49702515983297,.445539092065862],[-.495005390836273,.429504162938308],[-.492449906959786,.413437368713178],[-.489388923351505,.397537901149477],[-.486092457457082,.38289268680678],[-.482217467671015,.367839817369399],[-.477719792640897,.352406640667209],[-.472745115071217,.337174767998623],[-.467121534676336,.321681543743714],[-.461747618761362,.308195055933868],[-.455232308634762,.29320651563682],[-.448200875949892,.278378758243372],[-.440967869541059,.264315172248149],[-.434017212140139,.251747991818593],[-.425286678699385,.23707179512109],[-.417502228316438,.224878409878816],[-.408007798497225,.210985058577506],[-.399217677467098,.198959726950403],[-.389166207946042,.186073794351447],[-.37920822775395,.174114866856881],[-.368426673334243,.161973689817104],[-.3581833304016,.151137990284958],[-.346777664739698,.139798318663444],[-.334550626327951,.128414372689718],[-.323602394202433,.118841908827251],[-.311470784854377,.10886581563075],[-.298615001704751,.0989649881159108],[-.285733691689422,.0896876099439182],[-.272996075828234,.0811048549071196],[-.258551839641691,.072038615973477],[-.245316003491164,.0643165616745078],[-.2309298874154,.0565235213697228],[-.217679955691247,.0498717550627859],[-.202975702486618,.0430526680239134],[-.188945095899546,.0370747892634153],[-.174091370752242,.0312866604953457],[-.159397938771223,.0260883024069934],[-.143849566475576,.0211395796009147],[-.128729617897237,.0168554196971351],[-.113135884140814,.0129678945698998],[-.0975557961963832,.00960947538875739],[-.0831515435435712,.00696265779728045],[-.0665556505403349,.00444945224411991],[-.051647982293547,.00267466792349552],[-.0357023363300146,.00127628572467248],[-.0204058283518766,.000416571362426232],[-.00339919444070735,1.15546563558278e-005],[.0113311907438187,.000128412373410436],[.0280376959833401,.000786731342259095],[.0432370349917847,.00187294913333685],[.0598162302289324,.00359087578772349],[.0740685259024661,.00551657917516013],[.0902249269041988,.00820790717506248],[.105739468059414,.0113087223056748],[.121138107886368,.0148963421930209],[.136359652346541,.0189531777342973],[.150855501241195,.0233002855620031],[.166370046897607,.0284907132459763],[.18099100471142,.0339074596031915],[.196188512680684,.0400977631146581],[.209764124982261,.0461288598396825],[.224551840215932,.0532601752074945],[.23848910642065,.0605424410495594],[.252157427304858,.0682400761360598],[.26616881651069,.0767339357835327],[.278821298799669,.0849594196519264],[.291515807208022,.0937752664498933],[.305208023760902,.103959519452916],[.317298974625144,.113578778142514],[.329350775215083,.123797837771732],[.341269732637713,.134575630827138],[.351914940485234,.144815717319482],[.362923985867448,.156072419712981],[.373493461804421,.16758063535746],[.384395484002807,.180249922786174],[.393659653200729,.19172726776135],[.4039224352397,.205302415500176],[.412782466634757,.217846433233734],[.421165751673984,.230520112778545],[.429746223209442,.244425776657308],[.437899648799921,.258662274849319],[.444908660018602,.271841536995772],[.451917634149337,.286059700044929],[.458319812121927,.300142676350048],[.464764083986276,.315624442410611],[.469918529219418,.329188478444048],[.475350034961685,.344944060862137],[.479974481767293,.359912538561731],[.484089858669872,.374872030572765],[.487741826339467,.389964047516089],[.490982030300283,.405466165198845],[.493878741566683,.422001354957223],[.49604557326925,.437240225940649],[.497723469698339,.452341341694852],[.499008813920056,.468532498826591],[.499766513486373,.484721518476134],[.5,.500000000410207]], datatype = float[8]),COLOUR(RGB,0.,0.,1.00000000)),plot(x^2,x = Student:-VectorCalculus:-`-`(T) .. T,scaling = constrained,color = red),view = [Student:-VectorCalculus:-`-`(T) .. T, Student:-VectorCalculus:-`-`(T) .. T]), ':-Plot0', [':-T'], ':-sliders'=["SliderT"], ':-labels'=["LabelT"],':-isplot'=true, ':-handlers'=[T=handlertab[T]]);
#
# End of autogenerated code.
#
Now repeat for the osculating circle of the graph 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 at the origin LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVy8lK2V4ZWN1dGFibGVHRj1GOQ==and zoom in until you barely can distinguish the red and blue pixels. Make a comment here about what an approximate vlaue is fhe slider where you can still distinguish red and blue pixels: