{"id":8126,"date":"2012-04-09T18:31:26","date_gmt":"2012-04-09T17:31:26","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8126"},"modified":"2022-01-30T19:57:22","modified_gmt":"2022-01-30T19:57:22","slug":"a-representacao-grafica-de-uma-funcao-real-de-variavel-real","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8126","title":{"rendered":"A representa\u00e7\u00e3o gr\u00e1fica de uma fun\u00e7\u00e3o real de vari\u00e1vel real"},"content":{"rendered":"<p><ul id='GTTabs_ul_8126' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8126' class='GTTabs_curr'><a  id=\"8126_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8126' ><a  id=\"8126_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_8126'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8129\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8129\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b.jpg\" data-orig-size=\"353,315\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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\/12pag226-85b.jpg\" class=\"alignright wp-image-8129 size-full\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b.jpg\" alt=\"\" width=\"353\" height=\"315\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b.jpg 353w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b-300x267.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b-150x133.jpg 150w\" sizes=\"auto, (max-width: 353px) 100vw, 353px\" \/><\/a>Segue-se a representa\u00e7\u00e3o gr\u00e1fica de uma fun\u00e7\u00e3o $f$ real de dom\u00ednio $\\mathbb{R}$.<\/p>\n<p>O eixo das ordenadas e a reta\u00a0de equa\u00e7\u00e3o $y = mx + b$, representada a tra\u00e7o-ponto, s\u00e3o as \u00fanicas ass\u00edntotas do gr\u00e1fico.<\/p>\n<p>As retas tangentes ao gr\u00e1fico de $f$, nos pontos de abcissas -2 e 1, s\u00e3o horizontais.<\/p>\n<ol>\n<li>Determine o contradom\u00ednio de $f$.<\/li>\n<li>Calcule o valor de $\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{f(x)}}{x}$.<br \/>\nEscreva uma equa\u00e7\u00e3o da ass\u00edntota obl\u00edqua.<\/li>\n<li>Indique, justificando, quais os extremos da fun\u00e7\u00e3o.<\/li>\n<li>Determine os valores de $x$ que satisfazem a condi\u00e7\u00e3o: $f(x) \\times f'(x) &gt; 0$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8126' onClick='GTTabs_show(1,8126)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8126'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8129\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8129\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b.jpg\" data-orig-size=\"353,315\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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\/12pag226-85b.jpg\" class=\"alignright wp-image-8129 size-full\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b.jpg\" alt=\"\" width=\"353\" height=\"315\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b.jpg 353w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b-300x267.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag226-85b-150x133.jpg 150w\" sizes=\"auto, (max-width: 353px) 100vw, 353px\" \/><\/a>O contradom\u00ednio da fun\u00e7\u00e3o \u00e9 \\({{D&#8217;}_f} = \\left] { &#8211; \\infty ,0} \\right] \\cup \\left[ {2, + \\infty } \\right[\\).<br \/>\n\u00ad<\/li>\n<li>Ora, $\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{f(x)}}{x} = m = \\frac{{2 &#8211; 0}}{{2 &#8211; 1}} = 2$, sendo $m$ o declive da reta representada a tra\u00e7o-ponto, \u00fanica ass\u00edntota obl\u00edqua do gr\u00e1fico da fun\u00e7\u00e3o.Como o ponto de coordenadas $\\left( {1,0} \\right)$ pertence a essa reta, as suas coordenadas t\u00eam de verificar a equa\u00e7\u00e3o $y = 2x + b$, donde $0 = 2 \\times 1 + b \\Leftrightarrow b =\u00a0 &#8211; 2$.<br \/>\nLogo, $y = 2x &#8211; 2$ \u00e9 a equa\u00e7\u00e3o reduzida da ass\u00edntota obl\u00edqua.<br \/>\n\u00ad<\/li>\n<li><span style=\"text-decoration: underline;\">M\u00e1ximo relativo<\/span>: $0$, para $x=-2$, pois $f'(-2)=0$ e $f&#8217;$ \u00e9 positiva \u00e0 esquerda e negativa \u00e0 direita de $x=-2$;<br \/>\n<span style=\"text-decoration: underline;\">M\u00ednimo relativo<\/span>: $-2$, para $x=0$, pois, ainda que n\u00e3o exista $f'(0)$, temos $f'({0^ &#8211; }) &lt; 0$ e $f'({0^ + }) &gt; 0$;<br \/>\n<span style=\"text-decoration: underline;\">M\u00ednimo relativo<\/span>: $2$, para $x=1$, pois $f'(1)=0$ e $f&#8217;$ \u00e9 negativa \u00e0 esquerda e positiva \u00e0 direita de $x=1$.<br \/>\n\u00ad<\/li>\n<li>Ora,<br \/>\n<table class=\" aligncenter\" style=\"width: 80%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$x$<\/td>\n<td style=\"text-align: left; border: #00008b 1px solid;\">${ &#8211; \\infty }$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-2$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$1$<\/td>\n<td style=\"text-align: right; border: #00008b 1px solid;\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 ${ + \\infty }$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$f(x)$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-2$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$2$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$f'(x)$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">n.d.<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$f(x) \\times f'(x)$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">n.d.<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Portanto, $f(x) \\times f'(x) &gt; 0 \\Leftrightarrow x \\in \\left] { &#8211; 2,0} \\right[ \\cup \\left] {1, + \\infty } \\right[$.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8126' onClick='GTTabs_show(0,8126)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Segue-se a representa\u00e7\u00e3o gr\u00e1fica de uma fun\u00e7\u00e3o $f$ real de dom\u00ednio $\\mathbb{R}$. O eixo das ordenadas e a reta\u00a0de equa\u00e7\u00e3o $y = mx + b$, representada a tra\u00e7o-ponto, s\u00e3o as \u00fanicas&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21149,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,292],"tags":[427,288,145,144],"series":[],"class_list":["post-8126","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-calculo-diferencial","tag-12-o-ano","tag-assintota","tag-derivadas-2","tag-extremos-relativos"],"views":4407,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12V2Pag226-85_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8126","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=8126"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8126\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21149"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8126"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8126"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8126"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8126"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}