https://peda.net/id/48dbd2601d9 2024-05-29T12:03:15+03:00 https://peda.net/:static/548/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license noopener" href="https://peda.net/info" target="_blank">Peda.net-yleislisenssi</a></div> https://peda.net/id/59e38cd81d9 2026-02-27T07:11:18+02:00 <p><b><span><!--filtered attribute: data-contrast="none"-->MAB10 Lyhyen matematiikan kertaus (2 op)</span></b><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Yleiset tavoitteet</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="9"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="1"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->syventää ymmärrystään lukion pitkän matematiikan sisällöstä</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="8"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="2"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->perehtyä aiempien vuosien YO-koetehtäviin</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="7"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="3"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->kertaa ja syventää laskinohjelmistojen hallintaa</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="6"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="4"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->valmistaa YO-kokeisiin sekä jatko-opintoihin</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="5"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="5"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->harjoitteluttaa YO-koetilannetta osallistumalla preliminäärikokeeseen</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-contrast="auto"-->Keskeiset sisällöt: koko lukion lyhyen matematiikan oppimäärä</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Arviointi: Arviointi hyväksytty/hylätty.</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Laaja-alainen osaaminen </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Monitieteinen ja luova osaaminen </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Opintojakson aikana syvennetään opiskelijan ymmärrystä matematiikan käsitteiden ja merkintöjen oikeanlaisesta käytöstä sekä opetellaan yhdistelemään opittuja matematiikan eri osa-alueita. Matematiikan eri osa-alueiden kertaamisessa hyödynnetään matemaattisia ohjelmistoja.</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Opintojakson arviointi</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Mahdollinen preliminääri (ylioppilaskirjoituksiin valmentava harjoitus) voidaan sisällyttää opintojakson hyväksyttyyn suoritusmerkintään summatiivisena arviointina. Opintojakson alussa käydään läpi opintojakson tavoitteet, keskeiset sisällöt sekä arviointiperusteet. Opintojakson hyväksytty suorittaminen edellyttää opintojakson alussa sovittujen tehtävien tekemistä ohjeistuksen mukaan opintojakson aikana annettuun määräaikaan mennessä. </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Opintojakso on tarkoitettu ylioppilaskokeeseen ilmoittautuneille. Olennaisena osana aihealueiden ja ohjelmistojen kertaamisessa on lähiopetukseen osallistuminen, eikä opintojakson itsenäinen suorittaminen ole pääsääntöisesti mahdollista</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><b><span><!--filtered attribute: data-contrast="none"-->MAB15 Funktiot ja yhtälöt 1 (MAA2 1op) paikallinen opintojakso</span></b><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Keskeiset sisällöt: </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="39"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="1"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->polynomien tekijät </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="38"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="2"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->potenssifunktio ja potenssiyhtälö (eksponenttina positiivinen kokonaisluku) </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="37"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="3"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->rationaalifunktiot ja -yhtälöt </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="36"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="4"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->juurifunktiot ja -yhtälöt </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-contrast="auto"--> Yleiset tavoitteet: </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="35"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="5"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->tutustuu ilmiöiden matemaattiseen mallintamiseen polynomi-, rationaali- ja juurifunktioiden avulla, tuntee polynomi-, rationaali- ja juurifunktioiden ominaisuudet ja osaa ratkaista niihin liittyviä yhtälöitä sekä tietää polynomifunktion nollakohtien ja polynomin tekijöiden välisen yhteyden </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="34"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="6"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa ratkaista yksinkertaisia polynomiepäyhtälöitä </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="33"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="7"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa käyttää ohjelmistoja matemaattisessa mallintamisessa, polynomi-, rationaali- ja juurifunktioiden tutkimisessa sekä polynomi-, rationaali- ja juuriyhtälöiden ja polynomiepäyhtälöiden ratkaisemisessa sovellusten yhteydessä. </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-contrast="auto"--> Arviointi: Arviointi hyväksytty/hylätty. </span></p> <p><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><b><span><!--filtered attribute: data-contrast="none"-->MAB11 Analyyttinen geometria ja vektorit (MAA4 3op vastaavuus) paikallinen opintojakso</span></b><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Yleiset tavoitteet</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Moduulin tavoitteena on, että opiskelija</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559739":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="46"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="8"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->ymmärtää, kuinka analyyttinen geometria luo yhteyksiä geometristen ja algebrallisten käsitteiden välille</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="46"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="9"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->ymmärtää yhtälön geometrisen merkityksen</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="46"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="10"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa ratkaista muotoa | f(x) | = a tai | f(x) | = | g(x) | olevia itseisarvoyhtälöitä</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> </ul> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="46"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="1"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->ymmärtää vektorikäsitteen ja perehtyy vektorilaskennan perusteisiin</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="46"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="2"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa tutkia kaksiulotteisen koordinaatiston pisteitä, etäisyyksiä ja kulmia vektoreiden avulla</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="46"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="3"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa ratkaista tasogeometrian ongelmia vektoreiden avulla</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="46"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="4"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa käyttää ohjelmistoja käyrien ja vektoreiden tutkimisessa sekä niihin liittyvissä sovelluksissa.</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-contrast="auto"-->Keskeiset sisällöt</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="47"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="5"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->käyrän yhtälö</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="47"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="6"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->suoran, ympyrän ja paraabelin yhtälö</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="47"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="7"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->yhtälöryhmä</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="47"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="8"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->suorien yhdensuuntaisuus ja kohtisuoruus</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="47"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="9"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->itseisarvoyhtälö</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="47"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="10"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->pisteen etäisyys suorasta</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="47"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="11"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->vektoreiden perusominaisuudet</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="47"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="12"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->tason vektoreiden yhteen- ja vähennyslasku sekä tason vektorin kertominen luvulla</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="47"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="13"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->tason vektoreiden pistetulo, tason vektoreiden välinen kulma</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559739":0,"335559740":240}"--> </span><span><!--filtered attribute: data-contrast="auto"-->Arviointi</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Tällä opintojaksolla voisi esimerkiksi käyttää arvioinnin monipuolistamisessa testejä, jotka opiskelija korjaa itse pisteytysohjeen perusteella.</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Arviointi hyväksytty/hylätty. </span></p> <p><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><b><span><!--filtered attribute: data-contrast="none"-->MAB16 Funktiot ja yhtälöt 2 (MAA5 2op vastaavuus) paikallinen opintojakso</span></b><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Yleiset tavoitteet</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Moduulin tavoitteena on, että opiskelija</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="32"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="1"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->tutustuu ilmiöiden matemaattiseen mallintamiseen sini- ja kosinifunktioiden sekä eksponentti- ja logaritmifunktioiden avulla</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="31"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="2"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->tutkii sini- ja kosinifunktioita yksikköympyrän symmetrioiden avulla</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="30"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="3"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa ratkaista sellaisia trigonometrisia yhtälöitä, jotka ovat tyyppiä sin f(x) = a tai sin f(x) = sin g(x)</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="29"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="4"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa soveltaa sini- ja kosinifunktioiden yhteyttä </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="28"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="5"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->tuntee eksponentti- ja logaritmifunktioiden ominaisuudet ja osaa ratkaista niihin liittyviä yhtälöitä</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="27"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="6"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa käyttää ohjelmistoja funktioiden tutkimisessa, yhtälöiden ratkaisemisessa ja sovellusten yhteydessä.</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-contrast="auto"-->Keskeiset sisällöt</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="26"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="7"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->suunnattu kulma ja radiaani</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="25"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="8"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->yksikköympyrä</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="24"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="9"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->sini- ja kosinifunktiot symmetria- ja jaksollisuusominaisuuksineen</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="23"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="10"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->sini- ja kosiniyhtälöiden ratkaiseminen</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="22"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="11"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->murtopotenssi ja sen yhteys juureen</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="21"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="12"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->eksponenttifunktiot ja -yhtälöt</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="20"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="13"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->logaritmi ja logaritmin laskusäännöt</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="19"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="14"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->logaritmifunktiot ja -yhtälöt</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-contrast="auto"-->Arviointi hyväksytty/hylätty. </span></p> <p><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><b><span><!--filtered attribute: data-contrast="none"-->MAB13 Derivaatta (MAA6 </span></b><b><span><!--filtered attribute: data-contrast="none"-->1op</span></b><b><span><!--filtered attribute: data-contrast="none"--> vastaavuus) paikallinen opintojakso </span></b><b><span><!--filtered attribute: data-contrast="none"-->PUUSSA 2OP – KORJATTAVA!</span></b><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Yleiset tavoitteet</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Moduulin tavoitteena on, että opiskelija</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="18"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="1"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa derivoida yhdistettyjä funktioita</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="17"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="2"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->hallitsee funktioiden kulun tutkimisen derivaatan avulla ja osaa määrittää niiden ääriarvot suljetulla välillä</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="16"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="3"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa käyttää ohjelmistoja raja-arvon, jatkuvuuden ja derivaatan tutkimisessa sovellusten yhteydessä.</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-contrast="auto"-->Keskeiset sisällöt: </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="15"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="4"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->sini- ja kosinifunktioiden sekä eksponentti- ja logaritmifunktioiden derivaatat </span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="14"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="5"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->funktioiden tulon ja osamäärän derivaatta</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="13"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="6"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->yhdistetty funktio ja sen derivointi</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="12"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="7"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->funktion kulun tutkiminen ja ääriarvojen määrittäminen</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-contrast="auto"-->Arviointi hyväksytty/hylätty. </span></p> <p><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><b><span><!--filtered attribute: data-contrast="none"-->MAB14 </span></b><b><span><!--filtered attribute: data-contrast="none"-->Integraalilaskenta</span></b><b><span><!--filtered attribute: data-contrast="none"--> (MAA07 2 op vastaavuus) paikallinen opintojakso </span></b><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Tavoitteet</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Moduulin tavoitteena on, että opiskelija</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="12"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="8"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->ymmärtää integraalifunktion käsitteen ja oppii määrittämään yksinkertaisten funktioiden integraalifunktioita</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="12"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="9"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->ymmärtää määrätyn integraalin käsitteen ja sen yhteyden pinta-alaan sekä tutustuu numeeriseen menetelmään määrätyn integraalin määrittämisessä</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="12"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="10"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa määrittää pinta-aloja ja tilavuuksia määrätyn integraalin avulla</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="12"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="11"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->perehtyy integraalilaskennan sovelluksiin</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="12"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="12"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa käyttää ohjelmistoja funktion ominaisuuksien tutkimisessa, integraalifunktion määrittämisessä, määrätyn integraalin laskemisessa sovellusten yhteydessä sekä numeerisessa integroinnissa.</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-contrast="auto"-->Keskeiset sisällöt</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="12"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="1"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->integraalifunktio ja tärkeimpien alkeisfunktioiden integrointi</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="12"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="2"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->määrätty integraali</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="12"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="3"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->suorakaidesääntö</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext="-"--><!--filtered attribute: data-font="Calibri"--><!--filtered attribute: data-listid="12"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Calibri","469769242":[8226],"469777803":"left","469777804":"-","469777815":"hybridMultilevel"}"--><!--filtered attribute: data-aria-posinset="4"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->pinta-alan ja tilavuuden laskeminen</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559685":720,"335559740":240}"--> Arviointi hyväksytty/hylätty. </span></p> <p><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><b><span><!--filtered attribute: data-contrast="none"-->MAB17 3D-Geometria (MAA10 2op vastaavuus) paikallinen opintojakso</span></b><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Yleiset tavoitteet</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Moduulin tavoitteena on, että opiskelija</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559739":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="42"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="5"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->syventää vektorilaskennan tuntemustaan ja oppii käyttämään vektoreita kolmiulotteisessa avaruudessa</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="42"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="6"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->oppii tutkimaan xyz-koordinaatiston pisteitä, suoria ja tasoja vektoreiden avulla</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="42"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="7"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->vahvistaa avaruusgeometrian osaamistaan ääriarvosovellusten yhteydessä</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="42"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="8"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->tutustuu kahden muuttujan funktioon</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="42"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="9"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa käyttää ohjelmistoja vektoreiden, suorien, tasojen ja pintojen havainnollistamisessa sekä vektorilaskennassa.</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span><span><!--filtered attribute: data-contrast="auto"-->Keskeiset sisällöt</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="43"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="10"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->vektoriesitys kolmiulotteisessa koordinaatistossa</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="43"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="11"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->piste- ja ristitulo</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> </ul> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="43"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="1"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->piste, suora ja taso avaruudessa</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="43"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="2"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->kulma avaruudessa</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="43"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="3"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->yhden muuttujan differentiaali- ja integraalilaskennan sovelluksia avaruusgeometriassa</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="43"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="4"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->kahden muuttujan funktio ja pinta avaruudessa</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span><span><!--filtered attribute: data-contrast="auto"-->Arviointi:</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Tällä opintojaksolla voisi esimerkiksi käyttää arvioinnin monipuolistamisessa joka luvun jälkeen pidettäviä testejä.</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559685":720,"335559740":240}"-->Arviointi hyväksytty/hylätty. </span></p> <p><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><b><span><!--filtered attribute: data-contrast="none"-->MAB18 Algoritmit ja lukuteoria (MAA11 2op vastaavuus) paikallinen opintojakso</span></b><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Yleiset tavoitteet</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Moduulin tavoitteena on, että opiskelija</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559739":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="40"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="5"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->tietää, mikä on algoritmi, sekä oppii tutkimaan, kuinka algoritmit toimivat</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="40"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="6"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->laatii yksinkertaisiin matemaattisiin ongelmiin liittyviä algoritmeja</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="40"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="7"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->oppii ohjelmoimaan yksinkertaisia algoritmeja</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="40"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="8"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->perehtyy logiikan käsitteisiin</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="40"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="9"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->hallitsee lukuteorian peruskäsitteet ja perehtyy alkulukujen ominaisuuksiin</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="40"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="10"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa tutkia kokonaislukujen jaollisuutta</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="40"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="11"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->käyttää ohjelmistoja ohjelmoinnissa ja lukujen tutkimisessa.</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span><span><!--filtered attribute: data-contrast="auto"-->Keskeiset sisällöt</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="41"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="1"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->Algoritmisen ajattelun peruskäsitteet: peräkkäisyys, valinta ja toisto</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="41"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="2"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->vuokaavio</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="41"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="3"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->yksinkertaisten algoritmien, lajittelualgoritmien tai yhtälön numeerisen ratkaisuun liittyvän algoritmin ohjelmointi</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="41"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="4"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->konnektiivit ja totuusarvot</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="41"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="5"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->kokonaislukujenjaollisuus, jakoyhtälö ja kongruenssi</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="41"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="6"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->Eukleideen algoritmi</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="41"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="7"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->aritmetiikan peruslause</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span><span><!--filtered attribute: data-contrast="auto"-->Arviointi:</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Tällä kurssilla voisi esimerkiksi käyttää arvioinnin monipuolistamisessa Abitti-ympäristössä tehtäviä palautettavia tuntitehtäviä.</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559685":720,"335559740":240}"-->Arviointi hyväksytty/hylätty. </span></p> <p><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><b><span><!--filtered attribute: data-contrast="none"-->MAB19 Analyysi ja jatkuva jakauma (MAA12 2op vastaavuus) paikallinen opintojakso</span></b><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Yleiset tavoitteet</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></p> <p><span><!--filtered attribute: data-contrast="auto"-->Moduulin tavoitteena on, että opiskelija</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559739":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="44"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="8"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->syventää ymmärrystään analyysin peruskäsitteistä</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="44"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="9"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa muodostaa ja tutkia aidosti monotonisten funktioiden käänteisfunktioita</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="44"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="10"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->täydentää integraalilaskennan taitojaan</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> </ul> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="44"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="1"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->perehtyy jatkuvan todennäköisyysjakauman käsitteeseen ja oppii soveltamaan normaalijakaumaa</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="44"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="2"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->osaa käyttää ohjelmistoja funktion ominaisuuksien tutkimisessa ja epäoleellisten integraalien laskemisessa sovellusten yhteydessä.</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> </ul> <p><span><!--filtered attribute: data-contrast="auto"-->Keskeiset sisällöt</span><span><!--filtered attribute: data-ccp-props="{"201341983":0,"335559739":0,"335559740":240}"--> </span></p> <ul> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="45"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="3"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->paloittain määritelty funktio</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> <li aria-setsize="-1"><!--filtered attribute: data-leveltext=""--><!--filtered attribute: data-font="Symbol"--><!--filtered attribute: data-listid="45"--><!--filtered attribute: data-list-defn-props="{"335552541":1,"335559685":720,"335559991":360,"469769226":"Symbol","469769242":[8226],"469777803":"left","469777804":"","469777815":"multilevel"}"--><!--filtered attribute: data-aria-posinset="4"--><!--filtered attribute: data-aria-level="1"--><span><!--filtered attribute: data-contrast="auto"-->funktion jatkuvuuden ja derivoituvuuden tutkiminen</span><span><!--filtered attribute: data-ccp-props="{"134233117":true,"134233118":true,"201341983":0,"335559740":240}"--> </span></li> </ul><p class="warning">Varoitus: sisältö oli liian pitkä, ja sen loppu on poistettu. Koko sisältö näkyy vain muokkausnäkymässä.</p> 2024-05-29T12:03:44+03:00