{"id":7932,"date":"2012-04-02T17:15:55","date_gmt":"2012-04-02T16:15:55","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7932"},"modified":"2021-12-31T00:56:25","modified_gmt":"2021-12-31T00:56:25","slug":"determine-uma-equacao-da-reta-tangente-ao-grafico","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7932","title":{"rendered":"Determine uma equa\u00e7\u00e3o da reta tangente ao gr\u00e1fico"},"content":{"rendered":"<p><ul id='GTTabs_ul_7932' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7932' class='GTTabs_curr'><a  id=\"7932_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7932' ><a  id=\"7932_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7932'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<ol>\n<li>Determine uma equa\u00e7\u00e3o da reta tangente ao gr\u00e1fico de $y = {x^3} + \\ln \\left( {2x &#8211; 3} \\right)$ no ponto $T\\left( {2,8} \\right)$.<\/li>\n<li>Determine uma equa\u00e7\u00e3o da reta tangente ao gr\u00e1fico de\u00a0$y = 2x + \\ln x$, perpendicular \u00e0 reta de equa\u00e7\u00e3o $x + 3y + 1 = 0$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7932' onClick='GTTabs_show(1,7932)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7932'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, $$\\begin{array}{*{20}{l}}<br \/>\n{y&#8217;}&amp; = &amp;{\\left( {{x^3} + \\ln \\left( {2x &#8211; 3} \\right)} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{3{x^2} + \\frac{2}{{2x &#8211; 3}}}<br \/>\n\\end{array}$$<br \/>\nA derivada de uma fun\u00e7\u00e3o num ponto, caso exista, \u00e9 igual ao declive da reta tangente ao gr\u00e1fico da fun\u00e7\u00e3o nesse ponto. Logo, ${m_t} = y'(2) = 3 \\times {2^2} + \\frac{2}{{2 \\times 2 &#8211; 3}} = 14$.<\/p>\n<p>Como $T\\left( {2,8} \\right)$ \u00e9 o ponto de tang\u00eancia, as suas coordenadas t\u00eam de verificar a equa\u00e7\u00e3o $y = 14x + b$, donde $8 = 14 \\times 2 + b \\Leftrightarrow b =\u00a0 &#8211; 20$.<\/p>\n<p style=\"text-align: left;\">Assim, uma equa\u00e7\u00e3o da reta pedida \u00e9 $y = 14x &#8211; 20$.<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89a2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7948\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7948\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89a2.png\" data-orig-size=\"685,389\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Gr\u00e1fico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89a2.png\" class=\"aligncenter wp-image-7948 size-full\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89a2.png\" alt=\"\" width=\"685\" height=\"389\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89a2.png 685w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89a2-300x170.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89a2-150x85.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89a2-400x227.png 400w\" sizes=\"auto, (max-width: 685px) 100vw, 685px\" \/>\u00ad<\/a><\/p>\n<\/li>\n<li>Ora, $$\\begin{array}{*{20}{l}}<br \/>\n{y&#8217;}&amp; = &amp;{\\left( {2x + \\ln x} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{2 + \\frac{1}{x}}<br \/>\n\\end{array}$$<br \/>\nComo $x + 3y + 1 = 0 \\Leftrightarrow y =\u00a0 &#8211; \\frac{1}{3}x &#8211; \\frac{1}{3}$, a reta dada tem declive $ &#8211; \\frac{1}{3}$. Logo a reta pedida tem declive ${m_t} =\u00a0 &#8211; \\frac{1}{{{m_r}}} = 3$.<\/p>\n<p>Como $$\\begin{array}{*{20}{l}}<br \/>\n{y&#8217; = 3}&amp; \\Leftrightarrow &amp;{2 + \\frac{1}{x} = 3} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = 1}<br \/>\n\\end{array}$$<br \/>\no ponto de tang\u00eancia \u00e9 $T\\left( {1,2} \\right)$, pelo que as suas coordenadas t\u00eam de verificar a equa\u00e7\u00e3o $y = 3x + b$, donde $2 = 3 \\times 1 + b \\Leftrightarrow b =\u00a0 &#8211; 1$.<\/p>\n<p>Assim, uma equa\u00e7\u00e3o da reta pedida \u00e9 $y = 3x &#8211; 1$.<\/p>\n<\/li>\n<\/ol>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89b2.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7953\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7953\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89b2.png\" data-orig-size=\"548,312\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Gr\u00e1fico\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89b2.png\" class=\"aligncenter wp-image-7953 size-full\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89b2.png\" alt=\"\" width=\"548\" height=\"312\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89b2.png 548w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89b2-300x170.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89b2-150x85.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89b2-400x227.png 400w\" sizes=\"auto, (max-width: 548px) 100vw, 548px\" \/><\/a><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7932' onClick='GTTabs_show(0,7932)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Determine uma equa\u00e7\u00e3o da reta tangente ao gr\u00e1fico de $y = {x^3} + \\ln \\left( {2x &#8211; 3} \\right)$ no ponto $T\\left( {2,8} \\right)$. Determine uma equa\u00e7\u00e3o da reta tangente ao&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":7953,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,292],"tags":[427,145],"series":[],"class_list":["post-7932","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-calculo-diferencial","tag-12-o-ano","tag-derivadas-2"],"views":7177,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag228-89b2.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7932","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7932"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7932\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/7953"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7932"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7932"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7932"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7932"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}