{"id":7981,"date":"2012-04-03T17:39:13","date_gmt":"2012-04-03T16:39:13","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7981"},"modified":"2022-01-30T19:20:41","modified_gmt":"2022-01-30T19:20:41","slug":"considere-a-funcao-real-de-variavel-real-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7981","title":{"rendered":"Considere a fun\u00e7\u00e3o real de vari\u00e1vel real"},"content":{"rendered":"<p><ul id='GTTabs_ul_7981' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7981' class='GTTabs_curr'><a  id=\"7981_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7981' ><a  id=\"7981_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_7981'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere a fun\u00e7\u00e3o real de vari\u00e1vel real $$f:x \\to \\ln \\left( {{e^x} &#8211; 1} \\right)$$<\/p>\n<ol>\n<li>Determine o dom\u00ednio e zeros de $f$.<\/li>\n<li>Determine as equa\u00e7\u00f5es das ass\u00edntotas ao gr\u00e1fico de $f$.<\/li>\n<li>Estude a monotonia da fun\u00e7\u00e3o.<\/li>\n<li>Esboce o gr\u00e1fico de $f$.<\/li>\n<li>Determine uma equa\u00e7\u00e3o da reta tangente ao gr\u00e1fico de $f$ no ponto de abcissa $\\ln 2$.<\/li>\n<li>A partir do gr\u00e1fico obtido, construa os gr\u00e1ficos de $f( &#8211; x)$, $\\left| {f(x)} \\right|$, $2\\,f(x)$ e $f(x &#8211; 2)$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7981' onClick='GTTabs_show(1,7981)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7981'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote><p>Considere a fun\u00e7\u00e3o real de vari\u00e1vel real $$f:x \\to \\ln \\left( {{e^x} &#8211; 1} \\right)$$<\/p><\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>O dom\u00ednio da fun\u00e7\u00e3o \u00e9 ${D_f} = \\left\\{ {x \\in \\mathbb{R}:{e^x} &#8211; 1 &gt; 0} \\right\\} = {\\mathbb{R}^ + }$.<br \/>\nA fun\u00e7\u00e3o apenas tem um zero: $$f(x) = 0 \\Leftrightarrow \\ln \\left( {{e^x} &#8211; 1} \\right) = 0 \\Leftrightarrow {e^x} &#8211; 1 = 1 \\Leftrightarrow {e^x} = 2 \\Leftrightarrow x = \\ln 2$$<br \/>\n\u00ad<\/li>\n<li>A fun\u00e7\u00e3o \u00e9 cont\u00ednua no seu dom\u00ednio, pois \u00e9 uma fun\u00e7\u00e3o composta de fun\u00e7\u00f5es cont\u00ednuas.<br \/>\nPortanto, o seu gr\u00e1fico apenas pode admitir uma ass\u00edntota vertical em $x=0$.<br \/>\nComo $$\\mathop {\\lim }\\limits_{x \\to {0^ + }} f(x) = \\mathop {\\lim }\\limits_{x \\to {0^ + }} \\ln \\left( {{e^x} &#8211; 1} \\right) = \\ln \\left( {\\underbrace {\\mathop {\\lim }\\limits_{x \\to {0^ + }} \\left( {{e^x} &#8211; 1} \\right)}_{{0^ + }}} \\right) =\u00a0 &#8211; \\infty $$<br \/>\no gr\u00e1fico da fun\u00e7\u00e3o admite uma ass\u00edntota vertical de equa\u00e7\u00e3o $x=0$.<\/p>\n<p>Como $\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } f(x) = \\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\ln \\left( {{e^x} &#8211; 1} \\right) =\u00a0 + \\infty $, o gr\u00e1fico da fun\u00e7\u00e3o n\u00e3o tem ass\u00edntota horizontal.<\/p>\n<p>Ora, $$\\begin{array}{*{20}{l}}<br \/>\nm&amp; = &amp;{\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{f(x)}}{x}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{\\ln \\left( {{e^x} &#8211; 1} \\right)}}{x}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{\\ln \\left( {{e^x}\\left( {1 &#8211; {e^{ &#8211; x}}} \\right)} \\right)}}{x}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{x + \\ln \\left( {1 &#8211; {e^{ &#8211; x}}} \\right)}}{x}} \\\\<br \/>\n{}&amp; = &amp;{1 + \\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{{\\ln \\left( {1 &#8211; {e^{ &#8211; x}}} \\right)}}{x}} \\\\<br \/>\n{}&amp; = &amp;{1 + \\underbrace {\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\frac{1}{x}}_{`0} \\times \\underbrace {\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\left( {\\ln \\left( {1 &#8211; {e^{ &#8211; x}}} \\right)} \\right)}_0} \\\\<br \/>\n{}&amp; = &amp;1<br \/>\n\\end{array}$$\u00a0e $$b = \\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\left( {f(x) &#8211; 1x} \\right) = \\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\left( {\\ln \\left( {{e^x} &#8211; 1} \\right) &#8211; x} \\right) = \\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\left( {x + \\ln \\left( {1 &#8211; {e^{ &#8211; x}}} \\right) &#8211; x} \\right) = 0$$<br \/>\nPortanto, a reta de equa\u00e7\u00e3o $y = x$ \u00e9 ass\u00edntota obl\u00edqua do gr\u00e1fico da fun\u00e7\u00e3o.<br \/>\n\u00ad<\/li>\n<li>Ora, $$f'(x) = \\left( {\\ln \\left( {{e^x} &#8211; 1} \\right)} \\right)&#8217; = \\frac{{\\left( {{e^x} &#8211; 1} \\right)&#8217;}}{{{e^x} &#8211; 1}} = \\frac{{{e^x}}}{{{e^x} &#8211; 1}}$$<br \/>\nComo ${e^x} &#8211; 1 &gt; 0,\\forall x \\in {\\mathbb{R}^ + }$, ent\u00e3o $f'(x) = \\frac{{{e^x}}}{{{e^x} &#8211; 1}} &gt; 0,\\forall x \\in {\\mathbb{R}^ + }$.<\/p>\n<p>Logo, a fun\u00e7\u00e3o \u00e9 estritamente crescente no seu dom\u00ednio.<br \/>\n\u00ad<\/li>\n<li>Ora,\u00a0\\[f&#8221;\\left( x \\right) = {\\left( {\\frac{{{e^x}}}{{{e^x} &#8211; 1}}} \\right)^\\prime } = \\frac{{{e^x}\\left( {{e^x} &#8211; 1} \\right) &#8211; {e^x} \\times {e^x}}}{{{{\\left( {{e^x} &#8211; 1} \\right)}^2}}} =\u00a0 &#8211; \\frac{{{e^x}}}{{{{\\left( {{e^x} &#8211; 1} \\right)}^2}}}\\]<br \/>\nO gr\u00e1fico da fun\u00e7\u00e3o possui a concavidade\u00a0voltada para baixo em todo o seu dom\u00ednio, pois $f&#8221;(x) =\u00a0 &#8211; \\frac{{{e^x}}}{{{{\\left( {{e^x} &#8211; 1} \\right)}^2}}} &lt; 0,\\forall x \\in {\\mathbb{R}^ + }$.<br \/>\n\u00ad<\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag177-108.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8000\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8000\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag177-108.png\" data-orig-size=\"751,700\" 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\/12pag177-108.png\" class=\"aligncenter wp-image-8000\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag177-108.png\" alt=\"\" width=\"500\" height=\"466\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag177-108.png 751w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag177-108-300x279.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag177-108-150x139.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag177-108-400x372.png 400w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a>\u00ad<\/li>\n<li>O ponto de tang\u00eancia \u00e9 $T\\left( {\\ln 2,f(\\ln 2)} \\right) = \\left( {\\ln 2,0} \\right)$.O declive dessa reta tangente \u00e9 $${m_t} = f'(\\ln 2) = \\frac{{{e^{\\ln 2}}}}{{{e^{\\ln 2}} &#8211; 1}} = \\frac{2}{{2 &#8211; 1}} = 2$$<br \/>\nComo o ponto $T$ pertence a essa reta, as suas coordenadas t\u00eam de verificar a equa\u00e7\u00e3o $y = 2x + b$, donde $0 = 2 \\times \\ln 2 + b \\Leftrightarrow b =\u00a0 &#8211; \\ln 4$.<\/p>\n<p>Logo, $y = 2x &#8211; \\ln 4$ \u00e9 a equa\u00e7\u00e3o reduzida da reta pedida.<\/li>\n<li><\/li>\n<\/ol>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":744,\r\n\"height\":429,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 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