{"id":8672,"date":"2012-04-23T01:12:56","date_gmt":"2012-04-23T00:12:56","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8672"},"modified":"2022-01-14T01:22:59","modified_gmt":"2022-01-14T01:22:59","slug":"considere-a-funcao-f","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8672","title":{"rendered":"Considere a fun\u00e7\u00e3o $f$"},"content":{"rendered":"<p><ul id='GTTabs_ul_8672' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8672' class='GTTabs_curr'><a  id=\"8672_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8672' ><a  id=\"8672_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_8672'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere a fun\u00e7\u00e3o $$f:x \\to 2x &#8211; \\operatorname{sen} x$$<\/p>\n<ol>\n<li>Estude a paridade da fun\u00e7\u00e3o $f$ e exprima $f(x + 2\\pi )$ em fun\u00e7\u00e3o de $f(x)$.<br \/>\nVerifique que se pode estudar $f$ em $\\left[ {0,\\pi } \\right]$ e obter toda a curva ${C_f}$, recorrendo a transforma\u00e7\u00f5es adequadas.<\/li>\n<li>Estude a varia\u00e7\u00e3o de $f$ em $\\left[ {0,\\pi } \\right]$.<\/li>\n<li>Mostre que $2x &#8211; 1 \\leqslant f(x) \\leqslant 2x + 1,\\forall x \\in \\mathbb{R}$ e conjeture os limites de $f$ em $ + \\infty $ e em $ &#8211; \\infty $.<\/li>\n<li>Sejam ${d_1}$ e ${d_2}$ as retas de equa\u00e7\u00e3o $y = 2x &#8211; 1$ e $y = 2x\u00a0+ 1$, respetivamente.<br \/>\nDetermine os pontos comuns \u00e0 curva e a ${d_1}$, e depois \u00e0 curva e a ${d_2}$, indicando as equa\u00e7\u00f5es das tangentes nesses pontos.<\/li>\n<li>Estude a posi\u00e7\u00e3o de ${C_f}$ em rela\u00e7\u00e3o \u00e0 tangente no ponto de abcissa 0.<\/li>\n<li>Trace um esbo\u00e7o do gr\u00e1fico de $f$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8672' onClick='GTTabs_show(1,8672)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8672'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Considere a fun\u00e7\u00e3o $$f:x \\to 2x &#8211; \\operatorname{sen} x$$<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>A fun\u00e7\u00e3o $f$ \u00e9 \u00edmpar, pois $$f( &#8211; x) = 2 \\times ( &#8211; x) &#8211; \\operatorname{sen} ( &#8211; x) =\u00a0 &#8211; 2x + \\operatorname{sen} x =\u00a0 &#8211; f(x),\\forall x \\in R$$<br \/>\nOra, $$f(x + 2\\pi ) = 2\\left( {x + 2\\pi } \\right) &#8211; \\operatorname{sen} \\left( {x + 2\\pi } \\right) = 4\\pi\u00a0 + 2x &#8211; \\operatorname{sen} x = f(x) + 4\\pi ,\\forall x \\in \\mathbb{R}$$<br \/>\nComo a fun\u00e7\u00e3o $f$ \u00e9 \u00edmpar, o gr\u00e1fico em $\\left[ { &#8211; \\pi ,0} \\right]$ \u00e9 sim\u00e9trico relativamente \u00e0 origem do gr\u00e1fico em $\\left[ {0,\\pi } \\right]$. Basta, ent\u00e3o representar $f$ em $\\left[ {0,\\pi } \\right]$.<\/p>\n<p>Como $f(x + 2\\pi ) = f(x) + 4\\pi ,\\forall x \\in \\mathbb{R}$, ent\u00e3o o gr\u00e1fico de $f$ nos intervalos $\\left[ {\\left( {2k &#8211; 1} \\right)\\pi ,\\left( {2k + 1} \\right)\\pi } \\right]$ pode obter-se do gr\u00e1fico de $f$ em $\\left[ { &#8211; \\pi ,\\pi } \\right]$ por uma transla\u00e7\u00e3o associada ao vetor $\\overrightarrow v \\left( {2k\\pi ,4k\\pi } \\right)$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{f'(x)}&amp; = &amp;{\\left( {2x &#8211; \\operatorname{sen} x} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{2 &#8211; \\cos x}<br \/>\n\\end{array}$$<br \/>\nComo $f'(x) &gt; 0,\\forall x \\in \\mathbb{R}$, a fun\u00e7\u00e3o \u00e9 estritamente crescente no seu dom\u00ednio e, consequentemente, tamb\u00e9m crescente em $\\left[ {0,\\pi } \\right]$.<br \/>\n\u00ad<\/li>\n<li>Para todo o $x$ real, temos: $$\\begin{array}{*{20}{c}}<br \/>\n{ &#8211; 1}&amp; \\leqslant &amp;{\\operatorname{sen} x}&amp; \\leqslant &amp;1 \\\\<br \/>\n{ &#8211; 1}&amp; \\leqslant &amp;{ &#8211; \\operatorname{sen} x}&amp; \\leqslant &amp;1 \\\\<br \/>\n{2x &#8211; 1}&amp; \\leqslant &amp;{2x &#8211; \\operatorname{sen} x}&amp; \\leqslant &amp;{2x + 1}<br \/>\n\\end{array}$$<br \/>\nPortanto, $2x &#8211; 1 \\leqslant f(x) \\leqslant 2x + 1,\\forall x \\in \\mathbb{R}$.<\/p>\n<p>Como $\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\left( {2x &#8211; 1} \\right) =\u00a0 + \\infty $ e $\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } \\left( {2x + 1} \\right) =\u00a0 + \\infty $, ent\u00e3o, pelo teorema das sucess\u00f5es enquadradas, conclui-se que $\\mathop {\\lim }\\limits_{x \\to\u00a0 + \\infty } f(x) =\u00a0 + \\infty $.<\/p>\n<p>Como $\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } \\left( {2x &#8211; 1} \\right) =\u00a0 &#8211; \\infty $ e $\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } \\left( {2x + 1} \\right) =\u00a0 &#8211; \\infty $, ent\u00e3o, pelo teorema das sucess\u00f5es enquadradas, conclui-se que $\\mathop {\\lim }\\limits_{x \\to\u00a0 &#8211; \\infty } f(x) =\u00a0 &#8211; \\infty $.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Ora, $$\\begin{array}{*{20}{l}}<br \/>\n{f(x) = 2x &#8211; 1}&amp; \\Leftrightarrow &amp;{2x &#8211; \\operatorname{sen} x = 2x &#8211; 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x = 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{\\pi }{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<br \/>\nPortanto, os pontos comuns a ${C_f}$ e a ${d_1}$ s\u00e3o os pontos de coordenadas $$\\left( {\\frac{\\pi }{2} + 2k\\pi ,2\\left( {\\frac{\\pi }{2} + 2k\\pi } \\right) &#8211; 1} \\right) = \\left( {\\frac{\\pi }{2} + 2k\\pi ,\\pi\u00a0 + 4k\\pi\u00a0 &#8211; 1} \\right),k \\in \\mathbb{Z}$$<br \/>\nOra, $$\\begin{array}{*{20}{l}}<br \/>\n{f(x) = 2x + 1}&amp; \\Leftrightarrow &amp;{2x &#8211; \\operatorname{sen} x = 2x + 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x =\u00a0 &#8211; 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{{3\\pi }}{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<br \/>\nPortanto, os pontos comuns a ${C_f}$ e a ${d_2}$ s\u00e3o os pontos de coordenadas $$\\left( {\\frac{{3\\pi }}{2} + 2k\\pi ,2\\left( {\\frac{{3\\pi }}{2} + 2k\\pi } \\right) + 1} \\right) = \\left( {\\frac{{3\\pi }}{2} + 2k\\pi ,3\\pi\u00a0 + 4k\\pi\u00a0 + 1} \\right),k \\in \\mathbb{Z}$$<br \/>\nComo $$f&#8217;\\left( {\\frac{\\pi }{2} + 2k\\pi } \\right) = 2 &#8211; \\cos \\left( {\\frac{\\pi }{2} + 2k\\pi } \\right) = 2,\\forall k \\in \\mathbb{Z}$$ e $$f&#8217;\\left( {\\frac{{3\\pi }}{2} + 2k\\pi } \\right) = 2 &#8211; \\cos \\left( {\\frac{{3\\pi }}{2} + 2k\\pi } \\right) = 2,\\forall k \\in \\mathbb{Z}$$ as retas tangentes a ${C_f}$ nesses pontos s\u00e3o, respetivamente, ${d_1}:y = 2x &#8211; 1$ e ${d_2}:y = 2x + 1$.<br \/>\n\u00ad<\/li>\n<li>O declive da reta tangente a ${C_f}$ no ponto de abcissa $0$ \u00e9 ${m_0} = f'(0) = 2 &#8211; \\cos 0 = 1$ e o ponto de tang\u00eancia \u00e9 a origem do referencial, pois $\\left( {0,f(0)} \\right) = \\left( {0,0} \\right)$.<br \/>\nLogo,\u00a0$y = x$ \u00e9 a equa\u00e7\u00e3o da reta pedida.<\/p>\n<p>Como $f&#8221;(x) = \\left( {2 &#8211; \\cos x} \\right)&#8217; = \\operatorname{sen} x,\\forall x \\in \\mathbb{R}$, temos em $\\left[ { &#8211; \\pi ,\\pi } \\right]$:<\/p>\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: center; border: #00008b 1px solid;\">$ &#8211; \\pi $<\/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;\">$ + \\pi $<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$f'(x) = 2 &#8211; \\cos x$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$+$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Varia\u00e7\u00e3o de $f$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ &#8211; 2\\pi $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\nearrow $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\nearrow $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$2\\pi $<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$f&#8221;(x) = \\operatorname{sen} x$<\/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;\">$0$<\/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<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">\u00a0Sentido das concavidades de $f$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">P.I.<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\cap $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">P.I.<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$ \\cup $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">P.I.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>O gr\u00e1fico de $f$ nas proximidades de $0$ est\u00e1 acima do gr\u00e1fico de $y=x$, \u00e0 direita de $0$, e abaixo, \u00e0 esquerda de $0$, pois a concavidade do gr\u00e1fico de $f$ est\u00e1 voltada para cima \u00e0 direita de $0$ e voltada para baixo \u00e0 esquerda.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Segue um esbo\u00e7o do gr\u00e1fico de $f$:<\/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\":698,\r\n\"height\":536,\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 , 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Verifique que se pode estudar $f$&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19175,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,295],"tags":[427,307,286],"series":[],"class_list":["post-8672","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-funcoes-trigonometricas","tag-limites"],"views":2762,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat66.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8672","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=8672"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8672\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8672"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8672"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8672"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8672"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}