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href="#Istoric"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Istoric</span> </div> </a> <ul id="toc-Istoric-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tangenta_la_o_curbă" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tangenta_la_o_curbă"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Tangenta la o curbă</span> </div> </a> <ul id="toc-Tangenta_la_o_curbă-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abordarea_analitică" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Abordarea_analitică"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Abordarea analitică</span> </div> </a> <button aria-controls="toc-Abordarea_analitică-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon 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class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografie"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bibliografie</span> </div> </a> <ul id="toc-Bibliografie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Legături_externe" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Legături_externe"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Legături externe</span> </div> </a> <ul id="toc-Legături_externe-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="Cuprins" 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="Comută cuprinsul" > <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">Comută cuprinsul</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">Tangentă (geometrie)</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="Mergeți la un articol în altă limbă. Disponibil în 65 limbi" > <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-65" 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">65 limbi</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%85%D8%A7%D8%B3" title="مماس – arabă" lang="ar" hreflang="ar" data-title="مماس" data-language-autonym="العربية" data-language-local-name="arabă" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Tanxente" title="Tanxente – asturiană" lang="ast" hreflang="ast" data-title="Tanxente" data-language-autonym="Asturianu" data-language-local-name="asturiană" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%BE%D0%BF%D0%B8%D1%80%D0%B0%D1%82%D0%B5%D0%BB%D0%BD%D0%B0" title="Допирателна – bulgară" lang="bg" hreflang="bg" data-title="Допирателна" data-language-autonym="Български" data-language-local-name="bulgară" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%8D%E0%A6%AA%E0%A6%B0%E0%A7%8D%E0%A6%B6%E0%A6%95" title="স্পর্শক – bengaleză" lang="bn" hreflang="bn" data-title="স্পর্শক" data-language-autonym="বাংলা" data-language-local-name="bengaleză" 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/Tangent" title="Tangent – catalană" lang="ca" hreflang="ca" data-title="Tangent" data-language-autonym="Català" data-language-local-name="catalană" 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/%D9%84%DB%8E%DA%A9%DB%95%D9%88%D8%AA" title="لێکەوت – kurdă centrală" lang="ckb" hreflang="ckb" data-title="لێکەوت" data-language-autonym="کوردی" data-language-local-name="kurdă centrală" 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/Te%C4%8Dna" title="Tečna – cehă" lang="cs" hreflang="cs" data-title="Tečna" data-language-autonym="Čeština" data-language-local-name="cehă" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A1%C4%95%D1%80%D1%82%C4%95%D0%BD%D0%B5%D0%B2%C4%95%D1%88" title="Сĕртĕневĕш – ciuvașă" lang="cv" hreflang="cv" data-title="Сĕртĕневĕш" data-language-autonym="Чӑвашла" data-language-local-name="ciuvașă" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Tangiad" title="Tangiad – galeză" lang="cy" hreflang="cy" data-title="Tangiad" data-language-autonym="Cymraeg" data-language-local-name="galeză" 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/Tangent_(geometri)" title="Tangent (geometri) – daneză" lang="da" hreflang="da" data-title="Tangent (geometri)" data-language-autonym="Dansk" data-language-local-name="daneză" 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/Tangente" title="Tangente – germană" lang="de" hreflang="de" data-title="Tangente" data-language-autonym="Deutsch" data-language-local-name="germană" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Tangent" title="Tangent – engleză" lang="en" hreflang="en" data-title="Tangent" data-language-autonym="English" data-language-local-name="engleză" 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/Tan%C4%9Danto" title="Tanĝanto – esperanto" lang="eo" hreflang="eo" data-title="Tanĝanto" data-language-autonym="Esperanto" data-language-local-name="esperanto" 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/Tangente_(geometr%C3%ADa)" title="Tangente (geometría) – spaniolă" lang="es" hreflang="es" data-title="Tangente (geometría)" data-language-autonym="Español" data-language-local-name="spaniolă" 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/Puutuja" title="Puutuja – estonă" lang="et" hreflang="et" data-title="Puutuja" data-language-autonym="Eesti" data-language-local-name="estonă" 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/Zuzen_ukitzaile" title="Zuzen ukitzaile – bască" lang="eu" hreflang="eu" data-title="Zuzen ukitzaile" data-language-autonym="Euskara" data-language-local-name="bască" 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%85%D9%85%D8%A7%D8%B3" title="مماس – persană" lang="fa" hreflang="fa" data-title="مماس" data-language-autonym="فارسی" data-language-local-name="persană" 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/Tangentti_(geometria)" title="Tangentti (geometria) – finlandeză" lang="fi" hreflang="fi" data-title="Tangentti (geometria)" data-language-autonym="Suomi" data-language-local-name="finlandeză" 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/Tangente_(g%C3%A9om%C3%A9trie)" title="Tangente (géométrie) – franceză" lang="fr" hreflang="fr" data-title="Tangente (géométrie)" data-language-autonym="Français" data-language-local-name="franceză" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Tangent" title="Tangent – frizonă nordică" lang="frr" hreflang="frr" data-title="Tangent" data-language-autonym="Nordfriisk" data-language-local-name="frizonă nordică" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Tadhla%C3%AD" title="Tadhlaí – irlandeză" lang="ga" hreflang="ga" data-title="Tadhlaí" data-language-autonym="Gaeilge" data-language-local-name="irlandeză" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Tanxente" title="Tanxente – galiciană" lang="gl" hreflang="gl" data-title="Tanxente" data-language-autonym="Galego" data-language-local-name="galiciană" 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%99%D7%A7" title="משיק – ebraică" lang="he" hreflang="he" data-title="משיק" data-language-autonym="עברית" data-language-local-name="ebraică" 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%B8%E0%A5%8D%E0%A4%AA%E0%A4%B0%E0%A5%8D%E0%A4%B6%E0%A4%B0%E0%A5%87%E0%A4%96%E0%A4%BE" title="स्पर्शरेखा – hindi" lang="hi" hreflang="hi" data-title="स्पर्शरेखा" data-language-autonym="हिन्दी" data-language-local-name="hindi" 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/Tangenta" title="Tangenta – croată" lang="hr" hreflang="hr" data-title="Tangenta" data-language-autonym="Hrvatski" data-language-local-name="croată" 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/%C3%89rint%C5%91_(k%C3%B6r)" title="Érintő (kör) – maghiară" lang="hu" hreflang="hu" data-title="Érintő (kör)" data-language-autonym="Magyar" data-language-local-name="maghiară" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%87%D5%B8%D5%B7%D5%A1%D6%83%D5%B8%D5%B2" title="Շոշափող – armeană" lang="hy" hreflang="hy" data-title="Շոշափող" data-language-autonym="Հայերեն" data-language-local-name="armeană" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Garis_singgung" title="Garis singgung – indoneziană" lang="id" hreflang="id" data-title="Garis singgung" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneziană" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Snertill" title="Snertill – islandeză" lang="is" hreflang="is" data-title="Snertill" data-language-autonym="Íslenska" data-language-local-name="islandeză" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tangente_(geometria)" title="Tangente (geometria) – italiană" lang="it" hreflang="it" data-title="Tangente (geometria)" data-language-autonym="Italiano" data-language-local-name="italiană" 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/%E6%8E%A5%E7%B7%9A" title="接線 – japoneză" lang="ja" hreflang="ja" data-title="接線" data-language-autonym="日本語" data-language-local-name="japoneză" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%96%D0%B0%D0%BD%D0%B0%D0%BC%D0%B0" title="Жанама – kazahă" lang="kk" hreflang="kk" data-title="Жанама" data-language-autonym="Қазақша" data-language-local-name="kazahă" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%91%EC%84%A0" title="접선 – coreeană" lang="ko" hreflang="ko" data-title="접선" data-language-autonym="한국어" data-language-local-name="coreeană" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%B9%D0%BB%D0%B0%D0%BD%D0%B0%D0%B3%D0%B0_%D0%B6%D0%B0%D0%BD%D1%8B%D0%BC%D0%B0" title="Айланага жаныма – kârgâză" lang="ky" hreflang="ky" data-title="Айланага жаныма" data-language-autonym="Кыргызча" data-language-local-name="kârgâză" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Tangent" title="Tangent – Lombard" lang="lmo" hreflang="lmo" data-title="Tangent" 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/Liestin%C4%97" title="Liestinė – lituaniană" lang="lt" hreflang="lt" data-title="Liestinė" data-language-autonym="Lietuvių" data-language-local-name="lituaniană" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%BE%D0%BF%D0%B8%D1%80%D0%BA%D0%B0" title="Допирка – macedoneană" lang="mk" hreflang="mk" data-title="Допирка" data-language-autonym="Македонски" data-language-local-name="macedoneană" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%8A%E0%B4%9F%E0%B5%81%E0%B4%B5%E0%B4%B0" title="തൊടുവര – malayalam" lang="ml" hreflang="ml" data-title="തൊടുവര" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Tangen" title="Tangen – malaeză" lang="ms" hreflang="ms" data-title="Tangen" data-language-autonym="Bahasa Melayu" data-language-local-name="malaeză" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B8%E0%A5%8D%E0%A4%AA%E0%A4%B0%E0%A5%8D%E0%A4%B6_%E0%A4%B0%E0%A5%87%E0%A4%96%E0%A4%BE" title="स्पर्श रेखा – nepaleză" lang="ne" hreflang="ne" data-title="स्पर्श रेखा" data-language-autonym="नेपाली" data-language-local-name="nepaleză" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Raaklijn" title="Raaklijn – neerlandeză" lang="nl" hreflang="nl" data-title="Raaklijn" data-language-autonym="Nederlands" data-language-local-name="neerlandeză" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Tangent" title="Tangent – norvegiană nynorsk" lang="nn" hreflang="nn" data-title="Tangent" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegiană nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tangent_(matematikk)" title="Tangent (matematikk) – norvegiană bokmål" lang="nb" hreflang="nb" data-title="Tangent (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norvegiană bokmål" 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/Styczna" title="Styczna – poloneză" lang="pl" hreflang="pl" data-title="Styczna" data-language-autonym="Polski" data-language-local-name="poloneză" 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/Tangenta" title="Tangenta – Piedmontese" lang="pms" hreflang="pms" data-title="Tangenta" 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/Tangente_(geometria)" title="Tangente (geometria) – portugheză" lang="pt" hreflang="pt" data-title="Tangente (geometria)" data-language-autonym="Português" data-language-local-name="portugheză" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Patan_siq%27i" title="Patan siq'i – quechua" lang="qu" hreflang="qu" data-title="Patan siq'i" data-language-autonym="Runa Simi" data-language-local-name="quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%B0%D1%81%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D1%8F%D0%BC%D0%B0%D1%8F" title="Касательная прямая – rusă" lang="ru" hreflang="ru" data-title="Касательная прямая" data-language-autonym="Русский" data-language-local-name="rusă" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Tangenta" title="Tangenta – sârbo-croată" lang="sh" hreflang="sh" data-title="Tangenta" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="sârbo-croată" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Tangent_(geometry)" title="Tangent (geometry) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Tangent (geometry)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Tangenta" title="Tangenta – slovenă" lang="sl" hreflang="sl" data-title="Tangenta" data-language-autonym="Slovenščina" data-language-local-name="slovenă" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Tangent" title="Tangent – somaleză" lang="so" hreflang="so" data-title="Tangent" data-language-autonym="Soomaaliga" data-language-local-name="somaleză" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Tangjentja" title="Tangjentja – albaneză" lang="sq" hreflang="sq" data-title="Tangjentja" data-language-autonym="Shqip" data-language-local-name="albaneză" 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%A2%D0%B0%D0%BD%D0%B3%D0%B5%D0%BD%D1%82%D0%B0" title="Тангента – sârbă" lang="sr" hreflang="sr" data-title="Тангента" data-language-autonym="Српски / srpski" data-language-local-name="sârbă" 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/Tangent_(matematik)" title="Tangent (matematik) – suedeză" lang="sv" hreflang="sv" data-title="Tangent (matematik)" data-language-autonym="Svenska" data-language-local-name="suedeză" 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%A4%E0%AF%8A%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%81" title="தொடுகோடு – tamilă" lang="ta" hreflang="ta" data-title="தொடுகோடு" data-language-autonym="தமிழ்" data-language-local-name="tamilă" 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%B9%80%E0%B8%AA%E0%B9%89%E0%B8%99%E0%B8%AA%E0%B8%B1%E0%B8%A1%E0%B8%9C%E0%B8%B1%E0%B8%AA" title="เส้นสัมผัส – thailandeză" lang="th" hreflang="th" data-title="เส้นสัมผัส" data-language-autonym="ไทย" data-language-local-name="thailandeză" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Tangent" title="Tangent – tagalog" lang="tl" hreflang="tl" data-title="Tangent" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Te%C4%9Fet" title="Teğet – turcă" lang="tr" hreflang="tr" data-title="Teğet" data-language-autonym="Türkçe" data-language-local-name="turcă" 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%94%D0%BE%D1%82%D0%B8%D1%87%D0%BD%D0%B0" title="Дотична – ucraineană" lang="uk" hreflang="uk" data-title="Дотична" data-language-autonym="Українська" data-language-local-name="ucraineană" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ti%E1%BA%BFp_tuy%E1%BA%BFn" title="Tiếp tuyến – vietnameză" lang="vi" hreflang="vi" data-title="Tiếp tuyến" data-language-autonym="Tiếng Việt" data-language-local-name="vietnameză" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%88%87%E7%BA%BF" title="切线 – chineză wu" lang="wuu" hreflang="wuu" data-title="切线" data-language-autonym="吴语" data-language-local-name="chineză wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%88%87%E7%BA%BF" title="切线 – chineză" lang="zh" hreflang="zh" data-title="切线" data-language-autonym="中文" data-language-local-name="chineză" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%9B%B8%E5%88%87" title="相切 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="相切" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%88%87%E7%B7%9A" title="切線 – cantoneză" lang="yue" hreflang="yue" data-title="切線" data-language-autonym="粵語" data-language-local-name="cantoneză" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q131251#sitelinks-wikipedia" title="Modifică legăturile interlinguale" class="wbc-editpage">Modifică legăturile</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Spații de nume"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Tangent%C4%83_(geometrie)" title="Vedeți conținutul paginii [a]" accesskey="a"><span>Articol</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discu%C8%9Bie:Tangent%C4%83_(geometrie)" rel="discussion" title="Discuții despre această pagină [t]" accesskey="t"><span>Discuție</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">română</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vizualizări"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Tangent%C4%83_(geometrie)"><span>Lectură</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&veaction=edit" title="Modificați această pagină cu EditorulVizual [v]" accesskey="v"><span>Modificare</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=edit" title="Modificați codul sursă al acestei pagini [e]" accesskey="e"><span>Modificare sursă</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=history" title="Versiunile anterioare ale paginii și autorii lor. [h]" accesskey="h"><span>Istoric</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Unelte" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Unelte</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Unelte</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ascunde</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Mai multe opțiuni" > <div class="vector-menu-heading"> Acțiuni </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Tangent%C4%83_(geometrie)"><span>Lectură</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&veaction=edit" title="Modificați această pagină cu EditorulVizual [v]" accesskey="v"><span>Modificare</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=edit" title="Modificați codul sursă al acestei pagini [e]" accesskey="e"><span>Modificare sursă</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=history"><span>Istoric</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:Ce_se_leag%C4%83_aici/Tangent%C4%83_(geometrie)" title="Lista tuturor paginilor wiki care conduc spre această pagină [j]" accesskey="j"><span>Ce trimite aici</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:Modific%C4%83ri_corelate/Tangent%C4%83_(geometrie)" rel="nofollow" title="Schimbări recente în legătură cu această pagină [k]" accesskey="k"><span>Schimbări corelate</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:Trimite_fi%C8%99ier" title="Încărcare fișiere [u]" accesskey="u"><span>Trimite fișier</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:Pagini_speciale" title="Lista tuturor paginilor speciale [q]" accesskey="q"><span>Pagini speciale</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&oldid=15808076" title="Legătură permanentă către această versiune a acestei pagini"><span>Legătură permanentă</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=info" title="Mai multe informații despre această pagină"><span>Informații despre pagină</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:Citeaz%C4%83&page=Tangent%C4%83_%28geometrie%29&id=15808076&wpFormIdentifier=titleform" title="Informații cu privire la modul de citare a acestei pagini"><span>Citează acest articol</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fro.wikipedia.org%2Fwiki%2FTangent%25C4%2583_%28geometrie%29"><span>Obține URL scurtat</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fro.wikipedia.org%2Fwiki%2FTangent%25C4%2583_%28geometrie%29"><span>Descărcați codul QR</span></a></li> </ul> </div> </div> <div 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<div id="siteSub" class="noprint">De la Wikipedia, enciclopedia liberă</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="ro" dir="ltr"><div role="note" class="dezambiguizare">Acest articol se referă la <a href="/wiki/Obiect_matematic" title="Obiect matematic">obiecte matematice</a> tangente.  Pentru funcția trigonometrică, vedeți <a href="/wiki/Tangent%C4%83" title="Tangentă">tangentă</a>.  </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Tangent_to_a_curve.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Tangent_to_a_curve.svg/220px-Tangent_to_a_curve.svg.png" decoding="async" width="220" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Tangent_to_a_curve.svg/330px-Tangent_to_a_curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Tangent_to_a_curve.svg/440px-Tangent_to_a_curve.svg.png 2x" data-file-width="400" data-file-height="280" /></a><figcaption>Tangenta la o curbă; dreapta roșie este tangentă la curbă în punctul marcat</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Image_Tangent-plane.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Image_Tangent-plane.svg/220px-Image_Tangent-plane.svg.png" decoding="async" width="220" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Image_Tangent-plane.svg/330px-Image_Tangent-plane.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Image_Tangent-plane.svg/440px-Image_Tangent-plane.svg.png 2x" data-file-width="440" data-file-height="283" /></a><figcaption>Plan tangent la o sferă</figcaption></figure> <p>În <a href="/wiki/Geometrie" title="Geometrie">geometrie</a> o <b>tangentă</b> la o <a href="/wiki/Curb%C4%83" title="Curbă">curbă</a> într-un <a href="/wiki/Punct_(geometrie)" title="Punct (geometrie)">punct</a> dat este o <a href="/wiki/Dreapt%C4%83" title="Dreaptă">dreaptă</a> care „doar atinge” curba în acel punct. <a href="/wiki/Leibniz" class="mw-redirect" title="Leibniz">Leibniz</a> a definit-o ca dreapta definită de o pereche de puncte de pe curbă infinit de apropiate.<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> Mai exact, se spune că o dreaptă este tangentă la curba <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 y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span> în punctul <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 x=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3642398a43ac55e2d858529b48f8e113682801d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.435ex; height:1.676ex;" alt="{\displaystyle x=c}"></span> dacă dreapta trece prin punctul <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 (c,f(c))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c,f(c))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2504a7999581fca339fa102a4f7fdecfb54253c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.945ex; height:2.843ex;" alt="{\displaystyle (c,f(c))}"></span> de pe curbă iar <a href="/wiki/Pant%C4%83" title="Pantă">panta</a> sa este <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^{\prime }(c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{\prime }(c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8f9da487b2346e7c486899bf6a6ee6e11d6b30" 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^{\prime }(c)}"></span>, unde <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^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4ba7eed864135ec5838c81b7692e19556580d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f^{\prime }}"></span> este <a href="/wiki/Derivat%C4%83" title="Derivată">derivata</a> lui <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>. O definiție similară este valabilă pentru curbe în spațiu și curbe în <a href="/wiki/Spa%C8%9Biu_euclidian" title="Spațiu euclidian">spațiul euclidian</a> <i>n</i>-<a href="/wiki/Dimensiune_(matematic%C4%83)" class="mw-redirect" title="Dimensiune (matematică)">dimensional</a>. </p><p>Pe măsură ce trece prin punctul în care se întâlnesc dreapta tangentă și curba, denumit <b>punctul de tangență</b>, tangenta „merge în aceeași direcție” ca și curba și este astfel cea mai bună dreaptă de aproximare a curbei în acel punct. </p><p>Dreapta tangentă la un punct de pe o curbă poate fi considerată, de asemenea, ca <a href="/wiki/Graficul_unei_func%C8%9Bii" title="Graficul unei funcții">graficul</a> <a href="/wiki/Func%C8%9Bie_afin%C4%83" class="mw-redirect" title="Funcție afină">funcției afine</a> care aproximează cel mai bine funcția originală în punctul dat.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Similar, <b>planul tangent</b> la o <a href="/wiki/Suprafa%C8%9B%C4%83" title="Suprafață">suprafață</a> într-un punct dat este <a href="/wiki/Plan_(geometrie)" title="Plan (geometrie)">planul</a> care „doar atinge” suprafața în acel punct. Conceptul de tangentă este una dintre noțiunile fundamentale din <a href="/wiki/Geometrie_diferen%C8%9Bial%C4%83" title="Geometrie diferențială">geometria diferențială</a> și a fost larg generalizat; vezi <a href="/w/index.php?title=Spa%C8%9Biu_tangent&action=edit&redlink=1" class="new" title="Spațiu tangent — pagină inexistentă">spațiu tangent</a>. </p><p>Cuvântul „tangentă” vine din <a href="/wiki/Limba_latin%C4%83" title="Limba latină">latină</a> <span lang="la" style="font-style:italic">tangere</span>, „a atinge”. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Istoric">Istoric</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&veaction=edit&section=1" title="Modifică secțiunea: Istoric" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=edit&section=1" title="Edit section's source code: Istoric"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Euclid" title="Euclid">Euclid</a> face mai multe referiri la tangentă (în <a href="/wiki/Limba_greac%C4%83" title="Limba greacă">greacă</a> <span lang="grk" style="font-style:italic">ἐφαπτομένη</span> <i>efaptomeni</i>) la un <a href="/wiki/Cerc" title="Cerc">cerc</a> din cartea a III-a din <i><a href="/wiki/Elementele" title="Elementele">Elementele</a></i> (c. 300 î.Hr.).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> În lucrarea sa, <i>Conicele</i> (c. 225 î.Hr.), <a href="/wiki/Apoloniu_din_Perga" title="Apoloniu din Perga">Apoloniu din Perga</a> definește tangenta drept <i>o dreaptă astfel încât nici o altă dreaptă să nu poată cădea între ea și curbă</i>.<sup id="cite_ref-Shenk_4-0" class="reference"><a href="#cite_note-Shenk-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Arhimede" title="Arhimede">Arhimede</a> a găsit tangenta la o <a href="/wiki/Spirala_lui_Arhimede" title="Spirala lui Arhimede">spirală arhimedică</a> studiind mișcarea unui punct de-a lungul curbei.<sup id="cite_ref-Shenk_4-1" class="reference"><a href="#cite_note-Shenk-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>În anii 1630 <a href="/wiki/Fermat" class="mw-redirect" title="Fermat">Fermat</a> a dezvoltat tehnica „adecvării” pentru a calcula tangentele și alte probleme din analiză și a folosit-o pentru a calcula tangentele la parabolă. Tehnica adecvării este similară cu luarea diferenței dintre <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(x+h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x+h)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d79d112f0e80b791b42f289ae9b9b520ca2edd28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.597ex; height:2.843ex;" alt="{\displaystyle f(x+h)}"></span> și <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> și împărțirea la o putere a lui <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>. Independent, <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a> și-a folosit „metoda normalelor”, bazată pe observația că raza unui cerc este întotdeauna <a href="/wiki/Normal%C4%83" title="Normală">normală</a> pe cercul în sine.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Aceste metode au dus în <a href="/wiki/Secolul_al_XVII-lea" title="Secolul al XVII-lea">secolul al XVII-lea</a> la dezvoltarea <a href="/wiki/Calcul_diferen%C8%9Bial" title="Calcul diferențial">calculului diferențial</a>. Mulți oameni au contribuit, de exemplu <a href="/w/index.php?title=Gilles_de_Roberval&action=edit&redlink=1" class="new" title="Gilles de Roberval — pagină inexistentă">Gilles de Roberval</a> a descoperit o metodă generală de trasare a tangentelor, luând în considerare o curbă așa cum este descrisă de un punct în mișcare a cărui mișcare este rezultanta mai multor mișcări mai simple.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <a href="/w/index.php?title=Ren%C3%A9-Fran%C3%A7ois_de_Sluse&action=edit&redlink=1" class="new" title="René-François de Sluse — pagină inexistentă">René-François de Sluse</a> și <a href="/w/index.php?title=Johannes_Hudde&action=edit&redlink=1" class="new" title="Johannes Hudde — pagină inexistentă">Johannes Hudde</a> au găsit algoritmi algebrici pentru aflarea tangentelor.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> În continuare <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a> și <a href="/wiki/Isaac_Barrow" title="Isaac Barrow">Isaac Barrow</a> au făcut progrese care au condus la teoriile lui <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> și Leibniz. </p><p>O definiție a tangentei din 1828 era „o dreaptă care atinge o curbă, dar nu o taie”.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Această definiție veche exclude existența tangentei într-un <a href="/wiki/Punct_de_inflexiune" title="Punct de inflexiune">punct de inflexiune</a>. A fost respinsă, iar definițiile moderne sunt echivalente cu cea a lui Leibniz, care a definit dreapta tangentă ca dreapta printr-o pereche de puncte infinit de aproape pe curbă. </p> <div class="mw-heading mw-heading2"><h2 id="Tangenta_la_o_curbă"><span id="Tangenta_la_o_curb.C4.83"></span>Tangenta la o curbă</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&veaction=edit&section=2" title="Modifică secțiunea: Tangenta la o curbă" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=edit&section=2" title="Edit section's source code: Tangenta la o curbă"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/w/index.php?title=Fi%C8%99ier:CIRCLE_LINES-en.svg&lang=ro" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/langro-220px-CIRCLE_LINES-en.svg.png" decoding="async" width="220" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/langro-330px-CIRCLE_LINES-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/CIRCLE_LINES-en.svg/langro-440px-CIRCLE_LINES-en.svg.png 2x" data-file-width="612" data-file-height="618" /></a><figcaption>O tangentă, o <a href="/wiki/Coard%C4%83_(geometrie)" title="Coardă (geometrie)">coardă</a> și o <a href="/wiki/Secant%C4%83" title="Secantă">secantă</a> ale unui <a href="/wiki/Cerc" title="Cerc">cerc</a></figcaption></figure> <p>Noțiunea intuitivă că o tangentă „atinge” o curbă poate fi făcută mai explicită luând în considerare succesiunea dreptelor (<a href="/wiki/Secant%C4%83" title="Secantă">secante</a>) care trec prin două puncte, <i>A</i> și <i>B</i>, care se află pe curbă. Tangenta în <i>A</i> este limita când punctul <i>B</i> aproximează sau tinde spre <i>A</i>. Existența și unicitatea tangentei depinde de un anumit tip de netezime matematică, cunoscută sub numele de „diferențiabilitate”. De exemplu, dacă două arce de cerc se întâlnesc într-un punct ascuțit (un vârf), atunci nu există o tangentă definită în mod unic la vârf, deoarece limita progresiei dreptelor secante depinde de direcția în care se apropie vârf („punctul <i>B</i>”). </p><p>În majoritatea punctelor tangenta atinge curba fără a o traversa (deși poate, atunci când este prelungită, să traverseze curba în alte locuri, departe de punctul de tangență). Un punct în care o tangentă (în acel punct) traversează curba se numește <i>punct de inflexiune</i>. <a href="/wiki/Cerc" title="Cerc">Cercurile</a>, <a href="/wiki/Parabol%C4%83" title="Parabolă">parabolele</a>, <a href="/wiki/Hiperbol%C4%83" title="Hiperbolă">hiperbolele</a> și <a href="/wiki/Elips%C4%83" title="Elipsă">elipsele</a> nu au niciun punct de inflexiune, dar curbele mai complicate pot avea, la fel ca graficul unei <a href="/wiki/Func%C8%9Bie_algebric%C4%83_de_gradul_al_treilea" title="Funcție algebrică de gradul al treilea">funcții algebrice de gradul al treilea</a>, care are exact un punct de inflexiune, sau o <a href="/wiki/Sinusoid%C4%83" title="Sinusoidă">sinusoidă</a>, care are câte două puncte de inflexiune la fiecare <a href="/wiki/Func%C8%9Bie_periodic%C4%83" title="Funcție periodică">perioadă</a>. </p><p>Se poate întâmpla ca curba să se afle în întregime pe o parte a unei drepte care trece printr-un punct de pe ea și totuși această dreaptă nu este o tangentă. Acesta este cazul, de exemplu, pentru o dreaptă care trece prin vârful unui <a href="/wiki/Triunghi" title="Triunghi">triunghi</a> și nu-l intersectează în alt punct, vârf unde tangenta nu există din motivele explicate mai sus. </p> <div class="mw-heading mw-heading2"><h2 id="Abordarea_analitică"><span id="Abordarea_analitic.C4.83"></span>Abordarea analitică</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&veaction=edit&section=3" title="Modifică secțiunea: Abordarea analitică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=edit&section=3" title="Edit section's source code: Abordarea analitică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Graph_of_sliding_derivative_line.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7a/Graph_of_sliding_derivative_line.gif" decoding="async" width="400" height="400" class="mw-file-element" data-file-width="400" data-file-height="400" /></a><figcaption>În fiecare punct dreapta în mișcare este întotdeauna tangentă la curbă. Panta sa este derivata; derivatele pozitive sunt colorate în verde, cele negative în roșu, iar cele zero în negru. Punctul <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 (x,\,y)=(0,\,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,\,y)=(0,\,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6212f4505f4b7649217c14aa2bad46f9ecda5ae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.369ex; height:2.843ex;" alt="{\displaystyle (x,\,y)=(0,\,1)}"></span> în care tangenta intersectează curba nu este un <a href="/wiki/Maxim_%C8%99i_minim" title="Maxim și minim">maxim sau minim</a>, ci este un punct de inflexiune.</figcaption></figure> <p>Ideea geometrică a tangentei ca limită a secantelor servește drept model pentru metodele analitice care sunt folosite pentru a calcula explicit tangentele. Problema obținerii tangentei la un grafic a fost una dintre chestiunile esențiale care au condus în secolul al XVII-lea la dezvoltarea <a href="/wiki/Calcul_diferen%C8%9Bial" title="Calcul diferențial">calculului diferențial</a>. În cea de-a doua sa carte, <i>La Geometrie</i>, Descartes<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> a zis cu privire la problema construirii tangentei la o curbă: „Și îndrăznesc să spun că aceasta nu este doar cea mai utilă și mai generală problemă în geometrie pe care o cunosc, ci chiar pe care am dorit vreodată să o cunosc”.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Descriere_intuitivă"><span id="Descriere_intuitiv.C4.83"></span>Descriere intuitivă</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&veaction=edit&section=4" title="Modifică secțiunea: Descriere intuitivă" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=edit&section=4" title="Edit section's source code: Descriere intuitivă"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Să presupunem că o curbă este graficul unei funcții <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 y=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311a6a75c54b0ea085a381ba472c31d59321514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}"></span>. Pentru a afla tangenta în punctul <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 p=(a,\,f(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=(a,\,f(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93a07cfb04c477b5e7509b09ed4bb11016fc9936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.23ex; height:2.843ex;" alt="{\displaystyle p=(a,\,f(a)}"></span> se ia în considerare un alt punct din apropiere, <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 q=a+h,\,f(a+h'))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=a+h,\,f(a+h'))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f77b1dde41503de2a647a968b635d571676cd241" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.085ex; height:3.009ex;" alt="{\displaystyle q=a+h,\,f(a+h'))}"></span> pe curbă. Panta dreptei secante care trece prin <span class="texhtml mvar" style="font-style:italic;">p</span> și <span class="texhtml mvar" style="font-style:italic;">q</span> este <i>raportul diferențelor</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 {\frac {f(a+h)-f(a)}{h}}.}"> <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>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(a+h)-f(a)}{h}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c924ec6626c572300b256a28bbb285cc98fe040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.138ex; height:5.843ex;" alt="{\displaystyle {\frac {f(a+h)-f(a)}{h}}.}"></span></dd></dl> <p>Pe măsură ce punctul <span class="texhtml mvar" style="font-style:italic;">q</span> se apropie de <span class="texhtml mvar" style="font-style:italic;">p</span>, ceea ce corespunde faptului că <span class="texhtml mvar" style="font-style:italic;">h</span> devine din ce în ce mai mic, raportul diferențelor ar trebui să se apropie de o anumită <a href="/wiki/Valoare_(matematic%C4%83)" title="Valoare (matematică)">valoare</a> limită <span class="texhtml mvar" style="font-style:italic;">k</span>, care este panta tangentei în punctul <span class="texhtml mvar" style="font-style:italic;">p</span>. Dacă se cunoaște <span class="texhtml mvar" style="font-style:italic;">k</span>, ecuația tangentei poate fi obținută din punct și pantă: </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 y-f(a)=k(x-a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y-f(a)=k(x-a).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648aee7ec817bb89852bbf43fad1bf4aeacce37a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.479ex; height:2.843ex;" alt="{\displaystyle y-f(a)=k(x-a).}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Descriere_mai_riguroasă"><span id="Descriere_mai_riguroas.C4.83"></span>Descriere mai riguroasă</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&veaction=edit&section=5" title="Modifică secțiunea: Descriere mai riguroasă" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=edit&section=5" title="Edit section's source code: Descriere mai riguroasă"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pentru ca <a href="/wiki/Ra%C8%9Bionament" class="mw-redirect" title="Raționament">raționamentul</a> precedent să fie riguros trebuie explicat ce se înțelege prin raportul diferențelor, care se apropie de o anumită valoare limită <span class="texhtml mvar" style="font-style:italic;">k</span>. Formularea matematică precisă a fost dată de <a href="/wiki/Augustin-Louis_Cauchy" class="mw-redirect" title="Augustin-Louis Cauchy">Cauchy</a> în secolul al XIX-lea și se bazează pe noțiunea de <a href="/wiki/Limit%C4%83_(matematic%C4%83)" title="Limită (matematică)">limită</a>. Se presupune că graficul nu are o discontinuitate sau un vârf ascuțit în <span class="texhtml mvar" style="font-style:italic;">p</span> și că nu este nici prea vertical, nici nu variază foarte rapid în apropierea lui <span class="texhtml mvar" style="font-style:italic;">p</span>. Atunci există o valoare unică a <span class="texhtml mvar" style="font-style:italic;">k</span> astfel încât pe măsură ce <span class="texhtml mvar" style="font-style:italic;">h</span> se apropie de 0, diferența se apropie din ce în ce mai mult de <span class="texhtml mvar" style="font-style:italic;">k</span>, iar variația sa devine neglijabilă în comparație cu dimensiunea <span class="texhtml mvar" style="font-style:italic;">h</span>, <span class="texhtml mvar" style="font-style:italic;">h</span> fiind suficient de mic. Acest lucru duce la definirea pantei tangentei la graficul funcției <span class="texhtml mvar" style="font-style:italic;">f</span> ca limită a raporturilor diferențelor. Această limită este derivata funcției <span class="texhtml mvar" style="font-style:italic;">f</span> în <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 x=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaae23950e96a955ab5b07015a168fd931d4d82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}"></span>, notată <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^{\prime }(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{\prime }(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9acf16f3bf1a1ec9aedec4e71f3946d3cf897ad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.044ex; height:3.009ex;" alt="{\displaystyle f^{\prime }(a)}"></span>. Folosind derivate, ecuația tangentei poate fi scrisă după cum urmează: </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 y=f(a)+f^{\prime }(a)(x-a).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(a)+f^{\prime }(a)(x-a).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b526a05f7a323981d22cdb4916e6e18a978e24ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.312ex; height:3.009ex;" alt="{\displaystyle y=f(a)+f^{\prime }(a)(x-a).}"></span></dd></dl> <p><a href="/wiki/Calcul_diferen%C8%9Bial" title="Calcul diferențial">Calculul diferențial</a> oferă reguli pentru calcularea derivatelor funcțiilor care sunt date de formule, cum ar fi <a href="/wiki/Polinom" title="Polinom">funcțiile polinomiale</a>, <a href="/wiki/Func%C8%9Bie_trigonometric%C4%83" title="Funcție trigonometrică">funcțiile trigonometrice</a>, <a href="/wiki/Func%C8%9Bie_exponen%C8%9Bial%C4%83" title="Funcție exponențială">funcția exponențială</a>, <a href="/wiki/Logaritm" title="Logaritm">logaritmul</a> și diversele combinații ale acestora. Astfel, ecuațiile tangențelor la graficele tuturor acestor funcții, precum și multe altele, pot fi găsite prin metodele calculului diferențial. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&veaction=edit&section=6" title="Modifică secțiunea: Note" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=edit&section=6" title="Edit section's source code: Note"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small columns references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text">Leibniz, G., "Nova Methodus pro Maximis et Minimis", <i>Acta Eruditorum</i>, Oct. 1684.</span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> Dan Sloughter, <a rel="nofollow" class="external text" href="https://math.dartmouth.edu/opencalc2/dcsbook/c3pdf/sec31.pdf"><i>Best Affine Approximations</i></a>, dartmouth.edu, 2000, accesat 2021-07-22</span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite class="citation web">Euclid. <a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/elements/bookIII/bookIII.html">„Euclid's Elements”</a><span class="reference-accessdate">. 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Accesat în <time datetime="2015-06-01">1 iunie 2015</time></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=e-CALCULUS+Section++2.8&rft.pages=2.8&rft.aulast=Shenk&rft.aufirst=Al&rft_id=http%3A%2F%2Fmath.ucsd.edu%2F~ashenk%2FSection2_8.pdf&rfr_id=info%3Asid%2Fro.wikipedia.org%3ATangent%C4%83+%28geometrie%29" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></span> </li> <li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite class="citation book">Katz, Victor J. (<time datetime="2008">2008</time>). <i>A History of Mathematics</i> (ed. 3rd). 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(<time datetime="2001">2001</time>). „The Crooked Made Straight: Roberval and Newton on Tangents”. <i>The American Mathematical Monthly</i>. <b>108</b> (3): 206–216. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2695381">10.2307/2695381</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=The+Crooked+Made+Straight%3A+Roberval+and+Newton+on+Tangents&rft.volume=108&rft.issue=3&rft.pages=206-216&rft.date=2001&rft_id=info%3Adoi%2F10.2307%2F2695381&rft.aulast=Wolfson&rft.aufirst=Paul+R.&rfr_id=info%3Asid%2Fro.wikipedia.org%3ATangent%C4%83+%28geometrie%29" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></span> </li> <li id="cite_note-7"><b><a href="#cite_ref-7">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite class="citation book">Katz, Victor J. 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Courier. p. 95. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/0-486-60068-8" title="Special:Referințe în cărți/0-486-60068-8">0-486-60068-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+geometry+of+Ren%C3%A9+Descartes&rft.pages=95&rft.pub=Courier&rft.date=1954&rft.isbn=0-486-60068-8&rft.aulast=Descartes&rft.aufirst=Ren%C3%A9&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeometryofrenede00rend%2Fpage%2F95&rfr_id=info%3Asid%2Fro.wikipedia.org%3ATangent%C4%83+%28geometrie%29" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></span> </li> <li id="cite_note-10"><b><a href="#cite_ref-10">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite class="citation journal">R. 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Langer (octombrie 1937). „Rene Descartes”. <i>American Mathematical Monthly</i>. <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>. <b>44</b> (8): 495–512. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2301226">10.2307/2301226</a>. <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a> <a rel="nofollow" class="external text" href="//www.jstor.org/stable/2301226">2301226</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Rene+Descartes&rft.volume=44&rft.issue=8&rft.pages=495-512&rft.date=1937-10&rft_id=info%3Adoi%2F10.2307%2F2301226&rft_id=%2F%2Fwww.jstor.org%2Fstable%2F2301226&rft.au=R.+E.+Langer&rfr_id=info%3Asid%2Fro.wikipedia.org%3ATangent%C4%83+%28geometrie%29" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografie">Bibliografie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&veaction=edit&section=7" title="Modifică secțiunea: Bibliografie" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=edit&section=7" title="Edit section's source code: Bibliografie"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation book">J. Edwards (<time datetime="1892">1892</time>). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.109607"><i>Differential Calculus</i></a>. London: MacMillan and Co. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.109607/page/n161">143</a> ff.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Calculus&rft.place=London&rft.pages=143+ff.&rft.pub=MacMillan+and+Co.&rft.date=1892&rft.au=J.+Edwards&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.109607&rfr_id=info%3Asid%2Fro.wikipedia.org%3ATangent%C4%83+%28geometrie%29" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li></ul> <div class="mw-heading mw-heading2"><h2 id="Legături_externe"><span id="Leg.C4.83turi_externe"></span>Legături externe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&veaction=edit&section=8" title="Modifică secțiunea: Legături externe" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Tangent%C4%83_(geometrie)&action=edit&section=8" title="Edit section's source code: Legături externe"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Commons-logo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Materiale media legate de <span class="plainlinks"><b><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Tangency?uselang=ro">tangentă</a></b></span> la <a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commons</a></li> <li><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite id="CITEREFHazewinkel2001" class="citation">Hazewinkel, Michiel, ed. (<time datetime="2001">2001</time>), <a rel="nofollow" class="external text" href="http://eom.springer.de/p/t092170.htm">„Tangent line”</a>, <i><a href="/wiki/Encyclopaedia_of_Mathematics" class="mw-redirect" title="Encyclopaedia of Mathematics">Encyclopaedia of Mathematics</a></i>, Kluwer Academic Publishers, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-1556080104" title="Special:Referințe în cărți/978-1556080104">978-1556080104</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Tangent+line&rft.btitle=Encyclopaedia+of+Mathematics&rft.pub=Kluwer+Academic+Publishers&rft.date=2001&rft.isbn=978-1556080104&rft_id=http%3A%2F%2Feom.springer.de%2Fp%2Ft092170.htm&rfr_id=info%3Asid%2Fro.wikipedia.org%3ATangent%C4%83+%28geometrie%29" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite id="Reference-Mathworld-Tangent_Line"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <i><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/TangentLine.html">Tangent Line</a></i> la <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite></li> <li><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <a rel="nofollow" class="external text" href="http://www.mathopenref.com/tangent.html">Tangent to a circle</a> cu o animație interactivă</li> <li><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <a rel="nofollow" class="external text" href="http://www.vias.org/simulations/simusoft_difftangent.html">Tangent and first derivative</a> — o simulare interactivă</li></ul> <div class="noprint tright portal" style="border:solid #aaa 1px; 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