{"id":8482,"date":"2012-04-17T23:07:00","date_gmt":"2012-04-17T22:07:00","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8482"},"modified":"2022-01-29T23:51:09","modified_gmt":"2022-01-29T23:51:09","slug":"considere-as-funcoes-f-e-g","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8482","title":{"rendered":"Considere as fun\u00e7\u00f5es $f$ e $g$"},"content":{"rendered":"<p><ul id='GTTabs_ul_8482' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8482' class='GTTabs_curr'><a  id=\"8482_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8482' ><a  id=\"8482_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_8482'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere as fun\u00e7\u00f5es $f$ e $g$ de dom\u00ednio $\\mathbb{R}$, definidas por:<\/p>\n<p>$$f(x) = \\frac{4}{3} + 3{e^{(1 &#8211; x)}}$$<\/p>\n<p>$$g(x) = 2\\operatorname{sen} x &#8211; \\cos x$$<\/p>\n<p>Utilize m\u00e9todos exclusivamente anal\u00edticos para responder \u00e0s duas primeiras quest\u00f5es.<\/p>\n<ol>\n<li>Estude a fun\u00e7\u00e3o $f$ quanto \u00e0 exist\u00eancia de ass\u00edntotas paralelas aos eixos coordenados.<\/li>\n<li>Resolva a equa\u00e7\u00e3o $f(x) = g\\left( {\\frac{{5\\pi }}{2}} \\right)$ e apresente as solu\u00e7\u00f5es na forma $\\ln \\left( {ke} \\right)$, em que $k$ \u00e9 um n\u00famero real positivo.<\/li>\n<li>Recorrendo \u00e0 calculadora, determine as solu\u00e7\u00f5es inteiras da inequa\u00e7\u00e3o $f(x) &gt; g(x)$ no intervalo $\\left[ {0,12} \\right]$.<br \/>\nExplique como procedeu e apresente, na sua resposta, os elementos recolhidos na utiliza\u00e7\u00e3o da calculadora: gr\u00e1ficos e coordenadas de alguns pontos (coordenadas arredondadas \u00e0s d\u00e9cimas).<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8482' onClick='GTTabs_show(1,8482)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8482'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote><p>Considere as fun\u00e7\u00f5es $f$ e $g$ de dom\u00ednio $\\mathbb{R}$, definidas por:<\/p>\n<p>$$f(x) = \\frac{4}{3} + 3{e^{(1 &#8211; x)}}$$<\/p>\n<p>$$g(x) = 2\\operatorname{sen} x &#8211; \\cos x$$<\/p><\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>Como a fun\u00e7\u00e3o $f$ \u00e9 cont\u00ednua\u00a0em $\\mathbb{R}$, o seu gr\u00e1fico n\u00e3o admite qualquer ass\u00edntota vertical.<br \/>\nOra, $$\\begin{array}{*{20}{l}}<br \/>\n{\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } f(x)}&amp; = &amp;{\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } \\left( {\\frac{4}{3} + 3{e^{(1 &#8211; x)}}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{ + \\infty }<br \/>\n\\end{array}$$<br \/>\ne $$\\begin{array}{*{20}{l}}<br \/>\n{\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } f(x)}&amp; = &amp;{\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\left( {\\frac{4}{3} + 3{e^{(1 &#8211; x)}}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{4}{3} + 3 \\times \\underbrace {\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } {e^{(1 &#8211; x)}}}_0} \\\\<br \/>\n{}&amp; = &amp;{\\frac{4}{3}}<br \/>\n\\end{array}$$<br \/>\nLogo, o gr\u00e1fico de $f$ apenas admite uma ass\u00edntota horizontal, de equa\u00e7\u00e3o $y = \\frac{4}{3}$, quando ${x \\to\u00a0 + \\infty }$.<br \/>\n\u00ad<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{f(x) = g\\left( {\\frac{{5\\pi }}{2}} \\right)}&amp; \\Leftrightarrow &amp;{\\frac{4}{3} + 3{e^{(1 &#8211; x)}} = 2\\operatorname{sen} \\frac{{5\\pi }}{2} &#8211; \\cos \\frac{{5\\pi }}{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\frac{4}{3} + 3{e^{(1 &#8211; x)}} = 2 \\times 1 &#8211; 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{4 + 9{e^{(1 &#8211; x)}} = 6} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{{e^{(1 &#8211; x)}} = \\frac{2}{9}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{1 &#8211; x = \\ln \\frac{2}{9}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = 1 &#8211; \\ln \\frac{2}{9}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\ln e + \\ln \\frac{9}{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\ln \\left( {\\frac{9}{2}e} \\right)}<br \/>\n\\end{array}$$<br \/>\n\u00ad<\/li>\n<li>Como $f(x) &gt; g(x) \\Leftrightarrow f(x) &#8211; g(x) &gt; 0$, optou-se por definir ${f_3}(x) = f(x) &#8211; g(x)$ e considerar a inequa\u00e7\u00e3o ${f_3}(x) &gt; 0$. Estabelecida a correspondente representa\u00e7\u00e3o gr\u00e1fica e a determina\u00e7\u00e3o das coordenadas de alguns pontos significativos, conclui-se que as solu\u00e7\u00f5es inteiras da inequa\u00e7\u00e3o $f(x) &gt; g(x)$ no intervalo $\\left[ {0,12} \\right]$ s\u00e3o: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11 e 12.<br \/>\n\u00ad<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/17-04-2012Ecra001.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8494\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8494\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/17-04-2012Ecra001.jpg\" data-orig-size=\"682,462\" 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\/17-04-2012Ecra001.jpg\" class=\"aligncenter wp-image-8494 size-full\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/17-04-2012Ecra001.jpg\" alt=\"\" width=\"682\" height=\"462\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/17-04-2012Ecra001.jpg 682w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/17-04-2012Ecra001-300x203.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/17-04-2012Ecra001-150x101.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/17-04-2012Ecra001-400x270.jpg 400w\" sizes=\"auto, (max-width: 682px) 100vw, 682px\" \/><\/a><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8482' onClick='GTTabs_show(0,8482)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere as fun\u00e7\u00f5es $f$ e $g$ de dom\u00ednio $\\mathbb{R}$, definidas por: $$f(x) = \\frac{4}{3} + 3{e^{(1 &#8211; x)}}$$ $$g(x) = 2\\operatorname{sen} x &#8211; \\cos x$$ Utilize m\u00e9todos exclusivamente anal\u00edticos para responder&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21095,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,295],"tags":[427,288,299,268,296],"series":[],"class_list":["post-8482","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-funcoes-seno-co-seno-e-tangente","tag-12-o-ano","tag-assintota","tag-funcao-co-seno","tag-funcao-exponencial","tag-funcao-seno"],"views":3025,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12V3Pag126-4_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8482","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=8482"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8482\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21095"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8482"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8482"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8482"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8482"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}