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[{"key": "dc.contributor.advisor", "value": "Laitinen, Anni", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Piironen, Aleksi", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2025-05-27T18:25:23Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2025-05-27T18:25:23Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2025", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/102804", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 pro gradu -tutkielmassa tutkitaan k\u00e4\u00e4nteiskuvauslausetta ja k\u00e4\u00e4nteisfunktiota. K\u00e4\u00e4nteiskuvauslause antaa ehdot k\u00e4\u00e4nteiskuvauksen ja sen derivaatan olemassaololle. Lukion pitk\u00e4ss\u00e4 matematiikassa k\u00e4sitelt\u00e4v\u00e4 k\u00e4\u00e4nteisfunktio on k\u00e4\u00e4nteiskuvauksen yksiuloitteinen tapaus.\t\t\nK\u00e4\u00e4nteiskuvauslauseen mukaan avoimessa joukossa $A \\subset \\mathbb{R}^n$ m\u00e4\u00e4ritellyll\u00e4 jatkuvasti derivoituvalla kuvauksella $f:A \\to \\mathbb{R}^n$ on k\u00e4\u00e4nteiskuvaus pisteen $x_0 \\in A$ l\u00e4hiymp\u00e4rist\u00f6ss\u00e4, jos derivaattakuvaus $f'(x_0)$ on k\u00e4\u00e4ntyv\u00e4. Lis\u00e4ksi lauseesta saadaan kyseisen k\u00e4\u00e4nteiskuvaksen derivoituvuus pisteess\u00e4 $f(x_0)$ sek\u00e4 kaava k\u00e4\u00e4nteiskuvauksen derivaatalle pisteess\u00e4 $f(x_0)$.\nTy\u00f6n alussa esitell\u00e4\u00e4n tarvittavat esitiedot, joista suurin osa liittyy vektorianalyysiin. Esitietojen j\u00e4lkeen k\u00e4\u00e4nteiskuvauslause todistetaan kahdesta eri l\u00e4hestymiskulmasta. Ensimm\u00e4isess\u00e4 l\u00e4hestymiskulmassa todistus pohjautuu implisiittifunktiolauseeseen, jonka todistusta alustetaan Cousinin lemmalla sek\u00e4 moniuloitteisella differentiaalilaskennan v\u00e4liarvolauseella. Toisessa l\u00e4hestymiskulmassa todistuksessa ei k\u00e4ytet\u00e4 implisiittifunktiolausetta, vaan sen todistus pohjautuu kiintopistelauseeseen.\nTy\u00f6n lopussa tutkitaan, miten ja mill\u00e4 kursseilla k\u00e4\u00e4nteisfunktiota on k\u00e4sitelty lukion pitk\u00e4ss\u00e4 matematiikassa 2000-luvun aikana. Tutkintaan k\u00e4ytet\u00e4\u00e4n 2000-luvun lukion opetussuunnitelmia ja niihin perustuvia kirjasarjoja. Kirjasarjojen perusteella huomataan, ett\u00e4 k\u00e4\u00e4nteisfunktion k\u00e4sittely lukiomatematiikassa on pysynyt koko 2000-luvun l\u00e4hes identtisen\u00e4. Ainoa suurempi muutos tapahtui vuoden 2015 lukion opetussuunnitelmassa, kun k\u00e4\u00e4nteisfunktio siirrettiin pakollisilta kursseilta valinnaisille kursseille.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2025-05-27T18:25:23Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-05-27T18:25:23Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "37", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "k\u00e4\u00e4nteiskuvauslause", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "k\u00e4\u00e4nteisfunktio", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "K\u00e4\u00e4nteiskuvauslause ja k\u00e4\u00e4nteisfunktio lukiomatematiikassa", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202505274625", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": "fi", "element": "rights", "qualifier": "copyright", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "lukio", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "opetussuunnitelmat", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "funktiot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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