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[{"key": "dc.contributor.advisor", "value": "Juutinen, Petri", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Zykina, Aleksandra", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2020-01-21T05:17:30Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2020-01-21T05:17:30Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/67416", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa k\u00e4sitell\u00e4\u00e4n derivaattaa ja derivoituvuutta yhden reaalimuuttujan reaaliarvoisten funktioiden, usean reaalimuuttujan funktioiden ja yhden kompleksimuuttujan kompleksiarvoisten funktioiden tapauksissa. Lis\u00e4ksi k\u00e4yd\u00e4\u00e4n l\u00e4pi derivoituvien,\ndifferentioituvien ja analyyttisten funktioiden perusominaisuuksia (muun muassa derivointis\u00e4\u00e4nt\u00f6j\u00e4 kussakin tapauksessa) m\u00e4\u00e4ritelmien, lauseiden, todistusten ja esimerkkien avulla. T\u00e4m\u00e4 mahdollistaa kyseisten funktioluokkien yht\u00e4l\u00e4isyyksien\nja erojen tarkastelun. Tutkielmassa huomataan esimerkiksi, ett\u00e4 kompleksisesta derivoituvuudesta seuraa automaattisesti derivoituvuus \u00e4\u00e4rett\u00f6m\u00e4n monta kertaa, mutta reaalisen derivoituvuuden tapauksessat\u00e4m\u00e4 ei tietenk\u00e4\u00e4n p\u00e4de. Kolmannessa luvussa k\u00e4yd\u00e4\u00e4n lyhyesti l\u00e4pi Cauchyn ja Riemannin yht\u00e4l\u00f6t ja niiden yhteys analyyttisiin funktioihin. Lopuksi tarkastellaan Rollen lausetta ja v\u00e4liarvolausetta eri tilanteissa.\nTutkielmassa esitell\u00e4\u00e4n lis\u00e4ksi esimerkkej\u00e4 ja sovelluksia aiheeseen liittyen. Lukujen 1, 3 ja 4 alussa k\u00e4sitell\u00e4\u00e4n lyhyesti aiheeseen liittyv\u00e4\u00e4 historiaa.\n\nTutkielman j\u00e4lkeen lukija on esimerkiksi tietoinen siit\u00e4, ett\u00e4 Rollen lause ja v\u00e4liarvolause eiv\u00e4t suoraan sovellu usean reaalimuuttujan funktioille, eiv\u00e4tk\u00e4 analyyttisille funktioille. Kuitenkin molemmista lauseista voidaan johtaa ja todistaa n\u00e4ille funktioille\nsoveltuvat versiot.\n\nTutkielman tarkoituksena on antaa lukijalle hyv\u00e4 ja tiivistetty kokonaiskuva derivaatasta ja derivoituvuudesta eri tilanteissa, tuoda esille eri tilanteiden v\u00e4lisi\u00e4 eroja niin teorian kuin esimerkkienkin osalta sek\u00e4 tarkastella, miten Rollen lause ja v\u00e4liarvolause muuttuvat tilanteesta riippuen.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2020-01-21T05:17:30Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2020-01-21T05:17:30Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "51", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Derivaatta ja v\u00e4liarvolause", "language": "", "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-202001211356", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "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": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "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": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "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.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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