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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Skrá inn</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Síður fyrir útskráða notendur <a href="/wiki/Hj%C3%A1lp:Introduction" aria-label="Læra meira um breytingar"><span>Læra meira</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Kerfiss%C3%AD%C3%B0a:Framl%C3%B6g_m%C3%ADn" title="Listi yfir breytingar frá þessu IP-staðfangi [y]" accesskey="y"><span>Framlög</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Kerfiss%C3%AD%C3%B0a:Spjallas%C3%AD%C3%B0a_m%C3%ADn" title="Spjall um breytingar frá þessu IP-staðfangi [n]" accesskey="n"><span>Spjall</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Vefsíða"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Efni" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Efni</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">færa í hliðarstiku</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">fela</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Byrjun</div> </a> </li> <li id="toc-Meðalgildisregla_Cauchy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Meðalgildisregla_Cauchy"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Meðalgildisregla Cauchy</span> </div> </a> <button aria-controls="toc-Meðalgildisregla_Cauchy-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Sýna/fela Meðalgildisregla Cauchy undirkafla</span> </button> <ul id="toc-Meðalgildisregla_Cauchy-sublist" class="vector-toc-list"> <li id="toc-Sönnun" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sönnun"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Sönnun</span> </div> </a> <ul id="toc-Sönnun-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tenglar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tenglar"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Tenglar</span> </div> </a> <ul id="toc-Tenglar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heimildir" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Heimildir"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Heimildir</span> </div> </a> <ul id="toc-Heimildir-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Efni" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Sýna/fela efnisyfirlit" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Sýna/fela efnisyfirlit</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Meðalgildissetningin</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Fara á grein á öðru tungumáli. Til staðar á 52 tungumálum" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-52" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">52 tungumál</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%9B%E1%8B%95%E1%8A%A8%E1%88%8B%E1%8B%8A_%E1%8B%8B%E1%8C%8B_%E1%8A%A5%E1%88%AD%E1%8C%8D%E1%8C%A5" title="ማዕከላዊ ዋጋ እርግጥ – amharíska" lang="am" hreflang="am" data-title="ማዕከላዊ ዋጋ እርግጥ" data-language-autonym="አማርኛ" data-language-local-name="amharíska" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%A7%D9%84%D9%82%D9%8A%D9%85%D8%A9_%D8%A7%D9%84%D9%85%D8%AA%D9%88%D8%B3%D8%B7%D8%A9" title="مبرهنة القيمة المتوسطة – arabíska" lang="ar" hreflang="ar" data-title="مبرهنة القيمة المتوسطة" data-language-autonym="العربية" data-language-local-name="arabíska" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Orta_qiym%C9%99t_teoremi" title="Orta qiymət teoremi – aserska" lang="az" hreflang="az" data-title="Orta qiymət teoremi" data-language-autonym="Azərbaycanca" data-language-local-name="aserska" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%A1%E0%A6%BC_%E0%A6%AE%E0%A6%BE%E0%A6%A8_%E0%A6%89%E0%A6%AA%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A7%8D%E0%A6%AF" title="গড় মান উপপাদ্য – bengalska" lang="bn" hreflang="bn" data-title="গড় মান উপপাদ্য" data-language-autonym="বাংলা" data-language-local-name="bengalska" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_del_valor_mitj%C3%A0" title="Teorema del valor mitjà – katalónska" lang="ca" hreflang="ca" data-title="Teorema del valor mitjà" data-language-autonym="Català" data-language-local-name="katalónska" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%D9%85%DB%8C_%D8%A8%DB%95%DA%BE%D8%A7%DB%8C_%D9%86%D8%A7%D9%88%DB%95%D9%86%D8%AF" title="تیۆرمی بەھای ناوەند – miðkúrdíska" lang="ckb" hreflang="ckb" data-title="تیۆرمی بەھای ناوەند" data-language-autonym="کوردی" data-language-local-name="miðkúrdíska" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/V%C4%9Bta_o_st%C5%99edn%C3%AD_hodnot%C4%9B_diferenci%C3%A1ln%C3%ADho_po%C4%8Dtu" title="Věta o střední hodnotě diferenciálního počtu – tékkneska" lang="cs" hreflang="cs" data-title="Věta o střední hodnotě diferenciálního počtu" data-language-autonym="Čeština" data-language-local-name="tékkneska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Theorem_gwerth-cymedrig" title="Theorem gwerth-cymedrig – velska" lang="cy" hreflang="cy" data-title="Theorem gwerth-cymedrig" data-language-autonym="Cymraeg" data-language-local-name="velska" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Middelv%C3%A6rdis%C3%A6tningen" title="Middelværdisætningen – danska" lang="da" hreflang="da" data-title="Middelværdisætningen" data-language-autonym="Dansk" data-language-local-name="danska" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mittelwertsatz_der_Differentialrechnung" title="Mittelwertsatz der Differentialrechnung – þýska" lang="de" hreflang="de" data-title="Mittelwertsatz der Differentialrechnung" data-language-autonym="Deutsch" data-language-local-name="þýska" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%8E%CF%81%CE%B7%CE%BC%CE%B1_%CE%BC%CE%AD%CF%83%CE%B7%CF%82_%CF%84%CE%B9%CE%BC%CE%AE%CF%82" title="Θεώρημα μέσης τιμής – gríska" lang="el" hreflang="el" data-title="Θεώρημα μέσης τιμής" data-language-autonym="Ελληνικά" data-language-local-name="gríska" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Mean_value_theorem" title="Mean value theorem – enska" lang="en" hreflang="en" data-title="Mean value theorem" data-language-autonym="English" data-language-local-name="enska" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Meznombra_valora_teoremo" title="Meznombra valora teoremo – esperantó" lang="eo" hreflang="eo" data-title="Meznombra valora teoremo" data-language-autonym="Esperanto" data-language-local-name="esperantó" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teorema_del_valor_medio" title="Teorema del valor medio – spænska" lang="es" hreflang="es" data-title="Teorema del valor medio" data-language-autonym="Español" data-language-local-name="spænska" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lagrange%27i_keskv%C3%A4%C3%A4rtusteoreem" title="Lagrange'i keskväärtusteoreem – eistneska" lang="et" hreflang="et" data-title="Lagrange'i keskväärtusteoreem" data-language-autonym="Eesti" data-language-local-name="eistneska" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Batez_besteko_balioaren_teorema" title="Batez besteko balioaren teorema – baskneska" lang="eu" hreflang="eu" data-title="Batez besteko balioaren teorema" data-language-autonym="Euskara" data-language-local-name="baskneska" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D9%85%D9%82%D8%AF%D8%A7%D8%B1_%D9%85%DB%8C%D8%A7%D9%86%DA%AF%DB%8C%D9%86" title="قضیه مقدار میانگین – persneska" lang="fa" hreflang="fa" data-title="قضیه مقدار میانگین" data-language-autonym="فارسی" data-language-local-name="persneska" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Differentiaalilaskennan_v%C3%A4liarvolause" title="Differentiaalilaskennan väliarvolause – finnska" lang="fi" hreflang="fi" data-title="Differentiaalilaskennan väliarvolause" data-language-autonym="Suomi" data-language-local-name="finnska" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_des_accroissements_finis" title="Théorème des accroissements finis – franska" lang="fr" hreflang="fr" data-title="Théorème des accroissements finis" data-language-autonym="Français" data-language-local-name="franska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_do_valor_medio" title="Teorema do valor medio – galisíska" lang="gl" hreflang="gl" data-title="Teorema do valor medio" data-language-autonym="Galego" data-language-local-name="galisíska" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%A2%D7%A8%D7%9A_%D7%94%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%A9%D7%9C_%D7%9C%D7%92%D7%A8%D7%90%D7%A0%D7%96%27" title="משפט הערך הממוצע של לגראנז' – hebreska" lang="he" hreflang="he" data-title="משפט הערך הממוצע של לגראנז'" data-language-autonym="עברית" data-language-local-name="hebreska" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AE%E0%A4%BE%E0%A4%A7%E0%A5%8D%E0%A4%AF%E0%A4%AE%E0%A4%BE%E0%A4%A8_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="माध्यमान प्रमेय – hindí" lang="hi" hreflang="hi" data-title="माध्यमान प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="hindí" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Riemannov_teorem" title="Riemannov teorem – króatíska" lang="hr" hreflang="hr" data-title="Riemannov teorem" data-language-autonym="Hrvatski" data-language-local-name="króatíska" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Lagrange-f%C3%A9le_k%C3%B6z%C3%A9p%C3%A9rt%C3%A9kt%C3%A9tel" title="Lagrange-féle középértéktétel – ungverska" lang="hu" hreflang="hu" data-title="Lagrange-féle középértéktétel" data-language-autonym="Magyar" data-language-local-name="ungverska" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teorema_nilai_purata" title="Teorema nilai purata – indónesíska" lang="id" hreflang="id" data-title="Teorema nilai purata" data-language-autonym="Bahasa Indonesia" data-language-local-name="indónesíska" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_di_Lagrange" title="Teorema di Lagrange – ítalska" lang="it" hreflang="it" data-title="Teorema di Lagrange" data-language-autonym="Italiano" data-language-local-name="ítalska" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%B3%E5%9D%87%E5%80%A4%E3%81%AE%E5%AE%9A%E7%90%86" title="平均値の定理 – japanska" lang="ja" hreflang="ja" data-title="平均値の定理" data-language-autonym="日本語" data-language-local-name="japanska" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8F%89%EA%B7%A0%EA%B0%92_%EC%A0%95%EB%A6%AC" title="평균값 정리 – kóreska" lang="ko" hreflang="ko" data-title="평균값 정리" data-language-autonym="한국어" data-language-local-name="kóreska" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Teorema_de_Lagrange" title="Teorema de Lagrange – Lombard" lang="lmo" hreflang="lmo" data-title="Teorema de Lagrange" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Lagran%C5%BEo_vidutin%C4%97s_reik%C5%A1m%C4%97s_teorema" title="Lagranžo vidutinės reikšmės teorema – litháíska" lang="lt" hreflang="lt" data-title="Lagranžo vidutinės reikšmės teorema" data-language-autonym="Lietuvių" data-language-local-name="litháíska" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Lagran%C5%BEa_teor%C4%93ma" title="Lagranža teorēma – lettneska" lang="lv" hreflang="lv" data-title="Lagranža teorēma" data-language-autonym="Latviešu" data-language-local-name="lettneska" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B8_%D0%B7%D0%B0_%D1%81%D1%80%D0%B5%D0%B4%D0%BD%D0%B0_%D0%B2%D1%80%D0%B5%D0%B4%D0%BD%D0%BE%D1%81%D1%82" title="Теореми за средна вредност – makedónska" lang="mk" hreflang="mk" data-title="Теореми за средна вредност" data-language-autonym="Македонски" data-language-local-name="makedónska" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Middelwaardestelling" title="Middelwaardestelling – hollenska" lang="nl" hreflang="nl" data-title="Middelwaardestelling" data-language-autonym="Nederlands" data-language-local-name="hollenska" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Middelverdisetningen" title="Middelverdisetningen – norskt bókmál" lang="nb" hreflang="nb" data-title="Middelverdisetningen" data-language-autonym="Norsk bokmål" data-language-local-name="norskt bókmál" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Lagrange%E2%80%99a_(rachunek_r%C3%B3%C5%BCniczkowy)" title="Twierdzenie Lagrange’a (rachunek różniczkowy) – pólska" lang="pl" hreflang="pl" data-title="Twierdzenie Lagrange’a (rachunek różniczkowy)" data-language-autonym="Polski" data-language-local-name="pólska" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teorema_%C3%ABd_Cauchy_dle_ch%C3%ABrs%C3%B9e_fin%C3%ACe" title="Teorema ëd Cauchy dle chërsùe finìe – Piedmontese" lang="pms" hreflang="pms" data-title="Teorema ëd Cauchy dle chërsùe finìe" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teorema_do_valor_m%C3%A9dio" title="Teorema do valor médio – portúgalska" lang="pt" hreflang="pt" data-title="Teorema do valor médio" data-language-autonym="Português" data-language-local-name="portúgalska" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teorema_cre%C8%99terilor_finite" title="Teorema creșterilor finite – rúmenska" lang="ro" hreflang="ro" data-title="Teorema creșterilor finite" data-language-autonym="Română" data-language-local-name="rúmenska" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%BA%D0%BE%D0%BD%D0%B5%D1%87%D0%BD%D1%8B%D1%85_%D0%BF%D1%80%D0%B8%D1%80%D0%B0%D1%89%D0%B5%D0%BD%D0%B8%D0%B9" title="Формула конечных приращений – rússneska" lang="ru" hreflang="ru" data-title="Формула конечных приращений" data-language-autonym="Русский" data-language-local-name="rússneska" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Lagrangeova_veta_o_strednej_hodnote" title="Lagrangeova veta o strednej hodnote – slóvakíska" lang="sk" hreflang="sk" data-title="Lagrangeova veta o strednej hodnote" data-language-autonym="Slovenčina" data-language-local-name="slóvakíska" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Izrek_o_povpre%C4%8Dni_vrednosti" title="Izrek o povprečni vrednosti – slóvenska" lang="sl" hreflang="sl" data-title="Izrek o povprečni vrednosti" data-language-autonym="Slovenščina" data-language-local-name="slóvenska" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teorema_e_vler%C3%ABs_mesatare" title="Teorema e vlerës mesatare – albanska" lang="sq" hreflang="sq" data-title="Teorema e vlerës mesatare" data-language-autonym="Shqip" data-language-local-name="albanska" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Лагранжова теорема – serbneska" lang="sr" hreflang="sr" data-title="Лагранжова теорема" data-language-autonym="Српски / srpski" data-language-local-name="serbneska" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Medelv%C3%A4rdessatsen" title="Medelvärdessatsen – sænska" lang="sv" hreflang="sv" data-title="Medelvärdessatsen" data-language-autonym="Svenska" data-language-local-name="sænska" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%9F%E0%AF%88%E0%AE%AE%E0%AE%A4%E0%AE%BF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%A4%E0%AF%8D_%E0%AE%A4%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="இடைமதிப்புத் தேற்றம் – tamílska" lang="ta" hreflang="ta" data-title="இடைமதிப்புத் தேற்றம்" data-language-autonym="தமிழ்" data-language-local-name="tamílska" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%84%E0%B9%88%E0%B8%B2%E0%B8%A1%E0%B8%B1%E0%B8%8A%E0%B8%8C%E0%B8%B4%E0%B8%A1" title="ทฤษฎีบทค่ามัชฌิม – taílenska" lang="th" hreflang="th" data-title="ทฤษฎีบทค่ามัชฌิม" data-language-autonym="ไทย" data-language-local-name="taílenska" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Ortalama_de%C4%9Fer_teoremi" title="Ortalama değer teoremi – tyrkneska" lang="tr" hreflang="tr" data-title="Ortalama değer teoremi" data-language-autonym="Türkçe" data-language-local-name="tyrkneska" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BF%D1%80%D0%BE_%D1%81%D0%B5%D1%80%D0%B5%D0%B4%D0%BD%D1%94_%D0%B7%D0%BD%D0%B0%D1%87%D0%B5%D0%BD%D0%BD%D1%8F" title="Теорема про середнє значення – úkraínska" lang="uk" hreflang="uk" data-title="Теорема про середнє значення" data-language-autonym="Українська" data-language-local-name="úkraínska" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%DB%81_%D9%82%D8%AF%D8%B1_%D9%88%D8%B3%D8%B7%DB%8C" title="قضیہ قدر وسطی – úrdú" lang="ur" hreflang="ur" data-title="قضیہ قدر وسطی" data-language-autonym="اردو" data-language-local-name="úrdú" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Lagranj_formulasi" title="Lagranj formulasi – úsbekska" lang="uz" hreflang="uz" 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class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Mean_value_theorem" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q189136" title="Tengill á tengdan hlut í gagnagrunni [g]" accesskey="g"><span>Wikidata hlutur</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Síðuverkfæri"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Útlit"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Útlit</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">færa í hliðarstiku</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">fela</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Úr Wikipediu, frjálsa alfræðiritinu</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="is" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Mynd:Mvt2_is.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Mvt2_is.svg/250px-Mvt2_is.svg.png" decoding="async" width="250" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Mvt2_is.svg/375px-Mvt2_is.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Mvt2_is.svg/500px-Mvt2_is.svg.png 2x" data-file-width="744" data-file-height="657" /></a><figcaption>Fyrir sérhvert fall, sem samfellt er á bilinu [<i>a</i>, <i>b</i>] og diffranlegt á bilinu (<i>a</i>, <i>b</i>) er til minnst eitt gildi, <i>c</i> (eða <i>t</i>) á bilinu (<i>a</i>, <i>b</i>) þannig að <a href="/wiki/Sni%C3%B0ill" title="Sniðill">sniðillinn</a> sem tengir saman endapunkta bilsins [<i>a</i>, <i>b</i>] sé samsíða <a href="/wiki/Snertill" title="Snertill">snertilínu</a> við <i>f(x)</i> í x = c = t.</figcaption></figure> <p><b>Meðalgildissetningin</b> er mikilvæg <a href="/wiki/Setning_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Setning (stærðfræði)">setning</a> í <a href="/wiki/%C3%96rsm%C3%A6%C3%B0areikningur" title="Örsmæðareikningur">örsmæðareikningi</a> sem segir í stuttu máli að snertill <a href="/wiki/%C3%9Ej%C3%A1lt_fall" title="Þjált fall">þjáls ferils</a> á gefnu bili er í einhverjum punkti samsíða <a href="/wiki/Sni%C3%B0ill" title="Sniðill">sniðli</a> <a href="/wiki/Fall_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Fall (stærðfræði)">fallsins</a>. <a href="/wiki/Lagrange" class="mw-redirect" title="Lagrange">Lagrange</a> setti regluna fram á <a href="/wiki/18._%C3%B6ld" class="mw-redirect" title="18. öld">18. öld</a>, en <a href="/wiki/Cauchy" class="mw-redirect" title="Cauchy">Cauchy</a> setti hana stuttu síðar fram í almennara formi. </p><p><br /> Hægt er að túlka regluna á eftirfarandi hátt: Ef bíl er ekið 100 kílómetra vegalengd á klukkustund þá hefur bílinn farið yfir á 100km/klst að meðaltali. Samkvæmt því ætti hann á einhverjum tímapunkti að hafa ekið á nákvæmlega hraðanum 100km/klst. Þ.e. ef hann hefur ekki haldið nákvæmlega hraðanum 100km/klst alla leið hlýtur hann að hafa keyrt hægar en 100km/klst stundum og stundum hraðar. </p><p>Almennt séð segir setningin að þegar maður hefur fall <i>f</i> : [<i>a</i>, <i>b</i>] → <b>R</b> sem er samfellt á lokaða bilinu [<i>a</i>, <i>b</i>] og deildanlegt á opna bilinu (<i>a</i>, <i>b</i>), þá er til eitthvert <i>c</i> á milli <i>a</i> og <i>b</i> þ.a. </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e68eccc69d29a2a10d669fdd0a7f038417277b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.646ex; height:6.009ex;" alt="{\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}.}"></span></dd></dl></dd></dl> <p>Hérna táknar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/151d6d9bd64c4cd7e2010fb08ef6bc4e7c56e97b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.821ex; height:3.009ex;" alt="{\displaystyle f'(c)}"></span> afleiðu fallsins <i>f</i> í punktinum <i>c</i> og <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f(b)-f(a)}{b-a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(b)-f(a)}{b-a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fed83c0443dce2a8d4c231747a1d33f72b239fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.08ex; height:6.009ex;" alt="{\displaystyle {\frac {f(b)-f(a)}{b-a}}}"></span> táknar meðal breytingu á fallinu yfir bilið [<i>a</i>, <i>b</i>] eins og beina línan sýnir á sýnimyndinni til hægri. </p><p>Til eru mismunandi útgáfur af þessari setningu og eru þær listaðar hér fyrir neðan. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Meðalgildisregla_Cauchy"><span id="Me.C3.B0algildisregla_Cauchy"></span>Meðalgildisregla Cauchy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Me%C3%B0algildissetningin&veaction=edit&section=1" title="Breyti hluta: Meðalgildisregla Cauchy" class="mw-editsection-visualeditor"><span>breyta</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Me%C3%B0algildissetningin&action=edit&section=1" title="Edit section's source code: Meðalgildisregla Cauchy"><span>breyta frumkóða</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1754226">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox{border:1px solid #a2a9b1;border-spacing:3px;background-color:#f8f9fa;color:black;margin:0.5em 0 0.5em 1em;padding:0.2em;clear:right;font-size:88%;line-height:1.5em;width:22em}@media screen and (min-width:720px){.mw-parser-output .infobox{float:right}}@media screen and (max-width:720px){.mw-parser-output .infobox{float:none}}.mw-parser-output .infobox-header,.mw-parser-output .infobox-label,.mw-parser-output .infobox-above,.mw-parser-output .infobox-full-data,.mw-parser-output .infobox-data,.mw-parser-output .infobox-below,.mw-parser-output .infobox-subheader,.mw-parser-output .infobox-image,.mw-parser-output .infobox-navbar,.mw-parser-output .infobox th,.mw-parser-output .infobox td{vertical-align:top}.mw-parser-output .infobox-label,.mw-parser-output .infobox-data{text-align:left}.mw-parser-output .infobox .infobox-above,.mw-parser-output .infobox .infobox-title,.mw-parser-output .infobox caption{font-size:125%;font-weight:bold;text-align:center}.mw-parser-output .infobox-title,.mw-parser-output .infobox caption{padding:0.2em}.mw-parser-output .infobox .infobox-header,.mw-parser-output .infobox .infobox-above,.mw-parser-output .infobox .infobox-subheader,.mw-parser-output .infobox .infobox-image,.mw-parser-output .infobox .infobox-full-data,.mw-parser-output .infobox .infobox-below{text-align:center}.mw-parser-output .infobox .infobox-navbar{text-align:right}</style> <table align="right" class="infobox" style="width: 180px; margin: 0 0 1em 1em"> <tbody><tr style="background:#ccccff" align="center"> <td style="border-bottom: 2px solid #303060"><b><a href="/wiki/%C3%96rsm%C3%A6%C3%B0areikningur" title="Örsmæðareikningur">Örsmæðareikningur</a></b> </td></tr> <tr> <td align="center"> <p><a href="/wiki/Undirst%C3%B6%C3%B0usetning_%C3%B6rsm%C3%A6%C3%B0areiknings" title="Undirstöðusetning örsmæðareiknings">Undirstöðusetning</a> <br /> <a href="/wiki/Markgildi" title="Markgildi">Markgildi</a> <br /> <a href="/wiki/Samfelldni" title="Samfelldni">Samfelldni</a> <br /> <a href="/wiki/Vigurgreining" title="Vigurgreining">Vigurgreining</a><br /><a href="/w/index.php?title=%C3%9Einreikningur&action=edit&redlink=1" class="new" title="Þinreikningur (síða er ekki enn til)">Þinreikningur</a><br /> <a class="mw-selflink selflink">Meðalgildissetningin</a> </p> </td></tr> <tr style="background:#ccccff" align="center"> <td><b><a href="/wiki/Deildun" class="mw-redirect" title="Deildun">Deildun</a></b> (<i>diffrun</i>) </td></tr> <tr> <td align="center"> <p><a href="/wiki/Margfeldisreglan" class="mw-redirect" title="Margfeldisreglan">Margfeldisreglan</a> <br /> <a href="/wiki/Brotareglan" class="mw-redirect" title="Brotareglan">Brotareglan</a> <br /> <a href="/wiki/Ke%C3%B0jureglan" class="mw-redirect" title="Keðjureglan">Keðjureglan</a><br /> <a href="/wiki/F%C3%B3lgi%C3%B0_fall" title="Fólgið fall">Fólgið fall</a> <br /> <a href="/wiki/Setning_Taylors" title="Setning Taylors">Setning Taylors</a> <br /> <a href="/w/index.php?title=Listi_yfir_aflei%C3%B0ur&action=edit&redlink=1" class="new" title="Listi yfir afleiður (síða er ekki enn til)">Listi yfir afleiður</a> </p> </td></tr> <tr style="background:#ccccff" align="center"> <td><b><a href="/wiki/Heildun" title="Heildun">Heildun</a></b> (<i>tegrun</i>) </td></tr> <tr> <td align="center"> <p><a href="/w/index.php?title=Listi_yfir_heildi&action=edit&redlink=1" class="new" title="Listi yfir heildi (síða er ekki enn til)">Listi yfir heildi</a> <br /> <a href="/wiki/%C3%93eiginlegt_heildi" title="Óeiginlegt heildi">Óeiginlegt heildi</a> <br /> <a href="/wiki/Hlutheildun" title="Hlutheildun">Hlutheildun</a><br /> <a href="/w/index.php?title=Hringheildun&action=edit&redlink=1" class="new" title="Hringheildun (síða er ekki enn til)">Hringheildun</a><br /> <a href="/w/index.php?title=Heildun_sn%C3%BA%C3%B0a&action=edit&redlink=1" class="new" title="Heildun snúða (síða er ekki enn til)">Heildun snúða</a><br /> <a href="/w/index.php?title=Innsetningara%C3%B0fer%C3%B0in&action=edit&redlink=1" class="new" title="Innsetningaraðferðin (síða er ekki enn til)">Innsetningaraðferðin</a><br /> <a href="/w/index.php?title=Innsetning_hornafalla&action=edit&redlink=1" class="new" title="Innsetning hornafalla (síða er ekki enn til)">Innsetning hornafalla</a><br /><a href="/w/index.php?title=Heildun_r%C3%A6%C3%B0ra_falla&action=edit&redlink=1" class="new" title="Heildun ræðra falla (síða er ekki enn til)">Heildun ræðra falla</a> </p> </td></tr></tbody></table> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Mynd:Cauchy.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy.svg/220px-Cauchy.svg.png" decoding="async" width="220" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy.svg/330px-Cauchy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy.svg/440px-Cauchy.svg.png 2x" data-file-width="528" data-file-height="468" /></a><figcaption>Myndræn túlkun á reglunni.</figcaption></figure> <p>Gerum ráð fyrir að <i>f</i> og <i>g</i> séu deildanleg föll á bilinu [<i>a</i>,<i>b</i>] og að <i>g'(x)</i> sé aldrei núll. Þá er til <i>t</i> ∈ ]<i>a</i>,<i>b</i>[ þannig að: </p> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f'(t)}{g'(t)}}={\frac {f(b)-f(a)}{g(b)-g(a)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f'(t)}{g'(t)}}={\frac {f(b)-f(a)}{g(b)-g(a)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f63ec8d50336510c4b2cfe4a1a1f204b44837b0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.668ex; height:6.509ex;" alt="{\displaystyle {\frac {f'(t)}{g'(t)}}={\frac {f(b)-f(a)}{g(b)-g(a)}}}"></span></center> <div class="mw-heading mw-heading3"><h3 id="Sönnun"><span id="S.C3.B6nnun"></span>Sönnun</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Me%C3%B0algildissetningin&veaction=edit&section=2" title="Breyti hluta: Sönnun" class="mw-editsection-visualeditor"><span>breyta</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Me%C3%B0algildissetningin&action=edit&section=2" title="Edit section's source code: Sönnun"><span>breyta frumkóða</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><i>Sönnunin byggir á skilning á <a href="/wiki/Deildun" class="mw-redirect" title="Deildun">deildun</a> og <a href="/w/index.php?title=Regla_rolles&action=edit&redlink=1" class="new" title="Regla rolles (síða er ekki enn til)">reglu Rolles</a>.</i></dd></dl> <p>Skilgreinum nýtt fall <i>G(x)</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(x)=(g(b)-g(a))(f(x)-(fa))-(g(x)-g(a))(f(b)-f(a))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(x)=(g(b)-g(a))(f(x)-(fa))-(g(x)-g(a))(f(b)-f(a))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6417b20f77db9e8e1cf571de1f9cb2bd7c29f8e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.13ex; height:2.843ex;" alt="{\displaystyle G(x)=(g(b)-g(a))(f(x)-(fa))-(g(x)-g(a))(f(b)-f(a))}"></span></dd></dl> <p>Þar sem <i>G(a)=0</i> og <i>G(b)=0</i> er til tala <i>t</i> ∈ ]<i>a</i>,<i>b</i>[ svoleiðis að <i>G'(t)=0</i>. Nú er G'(x): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g(b)-g(a))f'(t)-g'(t)(f(b)-f(a))=0\Leftrightarrow (g(b)-g(a))f'(t)=g'(t)(f(b)-f(a))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇔</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g(b)-g(a))f'(t)-g'(t)(f(b)-f(a))=0\Leftrightarrow (g(b)-g(a))f'(t)=g'(t)(f(b)-f(a))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94e31f909eaf1ae01e61c101ba076f309a7fb5aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:83.587ex; height:3.009ex;" alt="{\displaystyle (g(b)-g(a))f'(t)-g'(t)(f(b)-f(a))=0\Leftrightarrow (g(b)-g(a))f'(t)=g'(t)(f(b)-f(a))}"></span></dd></dl> <p>Látum <i>x=t</i> og við fáum: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'(t)=(g(b)-g(a))f'(t)-g'(t)(f(b)-f(a))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'(t)=(g(b)-g(a))f'(t)-g'(t)(f(b)-f(a))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc645bb7fdea8bb9c78e24f83c58ab30d9df92e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.986ex; height:3.009ex;" alt="{\displaystyle G'(t)=(g(b)-g(a))f'(t)-g'(t)(f(b)-f(a))}"></span></dd></dl> <p>Þar sem <i>g'</i> er aldrei núll á bilinu ]<i>a</i>,<i>b</i>[ og <i>g(a) ≠ g(b)</i> getum við deilt og fengið: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {f'(t)}{g'(t)}}={\frac {f(b)-f(a)}{g(b)-g(a)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f'(t)}{g'(t)}}={\frac {f(b)-f(a)}{g(b)-g(a)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f63ec8d50336510c4b2cfe4a1a1f204b44837b0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.668ex; height:6.509ex;" alt="{\displaystyle {\frac {f'(t)}{g'(t)}}={\frac {f(b)-f(a)}{g(b)-g(a)}}}"></span></dd></dl> <p><b>QED</b> </p><p>Meðalgildisregla Cauchy's er einnig stundum kölluð útvíkkaða meðalgildissetning<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Tenglar">Tenglar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Me%C3%B0algildissetningin&veaction=edit&section=3" title="Breyti hluta: Tenglar" class="mw-editsection-visualeditor"><span>breyta</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Me%C3%B0algildissetningin&action=edit&section=3" title="Edit section's source code: Tenglar"><span>breyta frumkóða</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1870430">.mw-parser-output .citation a.new{color:black}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .citation a.new{color:white}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .citation a.new{color:white}}.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style><style data-mw-deduplicate="TemplateStyles:r1873994">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#2C882D;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911F}html.skin-theme-clientpref-night .mw-parser-output .cs1-visible-error,html.skin-theme-clientpref-night .mw-parser-output .cs1-hidden-error{color:#f8a397}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-visible-error,html.skin-theme-clientpref-os .mw-parser-output .cs1-hidden-error{color:#f8a397}html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911F}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.visindavefur.is/svar.php?id=47964">„Getið þið útskýrt reglu Rolles og meðalgildissetninguna?“</a>. <i><a href="/wiki/V%C3%ADsindavefurinn" title="Vísindavefurinn">Vísindavefurinn</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=V%C3%ADsindavefurinn&rft.atitle=Geti%C3%B0+%C3%BEi%C3%B0+%C3%BAtsk%C3%BDrt+reglu+Rolles+og+me%C3%B0algildissetninguna%3F&rft_id=https%3A%2F%2Fwww.visindavefur.is%2Fsvar.php%3Fid%3D47964&rfr_id=info%3Asid%2Fis.wikipedia.org%3AMe%C3%B0algildissetningin" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Heimildir">Heimildir</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Me%C3%B0algildissetningin&veaction=edit&section=4" title="Breyti hluta: Heimildir" class="mw-editsection-visualeditor"><span>breyta</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Me%C3%B0algildissetningin&action=edit&section=4" title="Edit section's source code: Heimildir"><span>breyta frumkóða</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1744314">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://books.google.com/books?id=TKNhCXjTfDcC&pg=PA132&lpg=PA132&dq=%22extended+mean+value+theorem%22&source=bl&ots=_UR8flmRyx&sig=TDPMlifs9l5lNJZEulyRRt1woGs&hl=en&ei=d_-SSq-bFOOMjAe9zZX8DQ&sa=X&oi=book_result&ct=result&resnum=2#v=onepage&q=%22extended%20mean%20value%20theorem%22&f=false">Advanced calculus: an introduction to modern analysis By 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