{"id":7866,"date":"2012-04-01T22:01:24","date_gmt":"2012-04-01T21:01:24","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7866"},"modified":"2022-01-30T19:12:44","modified_gmt":"2022-01-30T19:12:44","slug":"considere-a-curva-c","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7866","title":{"rendered":"Considere a curva $C$"},"content":{"rendered":"<p><ul id='GTTabs_ul_7866' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7866' class='GTTabs_curr'><a  id=\"7866_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7866' ><a  id=\"7866_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_7866'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere a curva $C$ de equa\u00e7\u00e3o $$y = \\frac{{3{x^2} + 1}}{{{x^2} + 3}}$$<\/p>\n<ol>\n<li>Determine as abcissas dos pontos da curva de ordenada 1.<\/li>\n<li>Determine uma equa\u00e7\u00e3o de cada uma das retas tangentes \u00e0 curva nos pontos obtidos na al\u00ednea anterior.<\/li>\n<li>Determine as coordenadas do ponto de interse\u00e7\u00e3o das duas tangentes.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7866' onClick='GTTabs_show(1,7866)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7866'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote><p>Considere a curva $C$ de equa\u00e7\u00e3o $$y = \\frac{{3{x^2} + 1}}{{{x^2} + 3}}$$<\/p><\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>Ora, $$\\begin{array}{*{20}{l}}<br \/>\n{\\frac{{3{x^2} + 1}}{{{x^2} + 3}} = 1}&amp; \\Leftrightarrow &amp;{\\frac{{3{x^2} + 1 &#8211; {x^2} &#8211; 3}}{{{x^2} + 3}} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\frac{{2\\left( {{x^2} &#8211; 1} \\right)}}{{{x^2} + 3}} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{\\left( {x =\u00a0 &#8211; 1 \\vee x = 1} \\right)}&amp; \\wedge &amp;{{x^2} + 3 \\ne 0}<br \/>\n\\end{array}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x =\u00a0 &#8211; 1 \\vee x = 1}<br \/>\n\\end{array}$$<br \/>\nOs pontos da curva $C$ de ordenada 1 s\u00e3o $A\\left( { &#8211; 1,1} \\right)$ e $B\\left( {1,1} \\right)$.<br \/>\n\u00ad<\/li>\n<li>Como\u00a0\\[\\begin{array}{*{20}{l}}<br \/>\n{y&#8217;}&amp; = &amp;{{{\\left( {\\frac{{3{x^2} + 1}}{{{x^2} + 3}}} \\right)}^\\prime }} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{6x\\left( {{x^2} + 3} \\right) &#8211; 2x\\left( {3{x^2} + 1} \\right)}}{{{{\\left( {{x^2} + 3} \\right)}^2}}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{6{x^3} + 18x &#8211; 6{x^3} &#8211; 2x}}{{{{\\left( {{x^2} + 3} \\right)}^2}}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{16x}}{{{{\\left( {{x^2} + 3} \\right)}^2}}}}<br \/>\n\\end{array}\\]<br \/>\nent\u00e3o $$y'( &#8211; 1) = \\frac{{ &#8211; 16}}{{{{\\left( {1 + 3} \\right)}^2}}} =\u00a0 &#8211; 1$$ e $$y'(1) = \\frac{{16}}{{{{\\left( {1 + 3} \\right)}^2}}} = 1$$<br \/>\nAs retas pedidas t\u00eam declives:\u00a0\u00a0$${m_{{t_1}}} = y'( &#8211; 1) =\u00a0 &#8211; 1$$ e $${m_{{t_2}}} = y'(1) = 1$$<br \/>\nLogo, as suas equa\u00e7\u00f5es reduzidas s\u00e3o da forma $y =\u00a0 &#8211; x + b$ e $y = x + b$.<br \/>\nComo os pontos A e B pertencem, respetivamente, a cada uma dessas retas, temos: $1 = 1 + b \\Leftrightarrow b = 0$.<\/p>\n<p>Logo, as retas pedidas s\u00e3o: ${t_1}:y =\u00a0 &#8211; x$ e ${t_2}:y = x$.<br \/>\n\u00ad<\/li>\n<li>Essas duas tangentes intersetam-se na origem do referencial $O\\left( {0,0} \\right)$, pois $\\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{y =\u00a0 &#8211; x} \\\\<br \/>\n{y = x}<br \/>\n\\end{array}} \\right. \\Leftrightarrow \\left\\{ {\\begin{array}{*{20}{l}}<br \/>\n{x = 0} \\\\<br \/>\n{y = 0}<br \/>\n\\end{array}} \\right.$.<br \/>\n\u00ad<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag216-53a.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7881\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7881\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag216-53a.png\" data-orig-size=\"791,406\" 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\/12pag216-53a.png\" class=\"aligncenter wp-image-7881 size-full\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag216-53a.png\" alt=\"\" width=\"791\" height=\"406\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag216-53a.png 791w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag216-53a-300x153.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag216-53a-150x76.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag216-53a-400x205.png 400w\" sizes=\"auto, (max-width: 791px) 100vw, 791px\" \/><\/a><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7866' onClick='GTTabs_show(0,7866)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere a curva $C$ de equa\u00e7\u00e3o $$y = \\frac{{3{x^2} + 1}}{{{x^2} + 3}}$$ Determine as abcissas dos pontos da curva de ordenada 1. Determine uma equa\u00e7\u00e3o de cada uma das retas&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21138,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,292],"tags":[427,136],"series":[],"class_list":["post-7866","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-calculo-diferencial","tag-12-o-ano","tag-derivada"],"views":2221,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12V2Pag216-53_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7866","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=7866"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7866\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21138"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7866"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7866"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7866"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7866"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}