{"id":8782,"date":"2012-04-24T19:18:21","date_gmt":"2012-04-24T18:18:21","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8782"},"modified":"2022-01-30T00:27:23","modified_gmt":"2022-01-30T00:27:23","slug":"considere-as-funcoes-reais-de-variavel-real-3","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8782","title":{"rendered":"Considere as fun\u00e7\u00f5es reais de vari\u00e1vel real"},"content":{"rendered":"<p><ul id='GTTabs_ul_8782' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8782' class='GTTabs_curr'><a  id=\"8782_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8782' ><a  id=\"8782_1\" onMouseOver=\"GTTabsShowLinks('R1'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R1<\/a><\/li>\n<li id='GTTabs_li_2_8782' ><a  id=\"8782_2\" onMouseOver=\"GTTabsShowLinks('R2'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R2<\/a><\/li>\n<li id='GTTabs_li_3_8782' ><a  id=\"8782_3\" onMouseOver=\"GTTabsShowLinks('R3'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>R3<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_8782'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere as fun\u00e7\u00f5es reais de vari\u00e1vel real:<\/p>\n<p style=\"text-align: center;\">$\\begin{array}{*{20}{c}}<br \/>\n{f(x) = x + 2\\operatorname{sen} x}&amp;{}&amp;{g(x) = x + \\cos x}&amp;{}&amp;{h(x) = x + \\operatorname{tg} x}<br \/>\n\\end{array}$<\/p>\n<p>Determine, para cada uma das fun\u00e7\u00f5es dadas, as abcissas de todos os pontos do gr\u00e1fico em que a reta tangente \u00e9 horizontal.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8782' onClick='GTTabs_show(1,8782)'>R1 &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8782'>\n<span class='GTTabs_titles'><b>R1<\/b><\/span><!--more--><\/p>\n<blockquote><p>A reta tangente ao gr\u00e1fico de uma fun\u00e7\u00e3o \u00e9 horizontal nos pontos em que a derivada da fun\u00e7\u00e3o \u00e9 nula, pois a derivada (se existir) da fun\u00e7\u00e3o num ponto de abcissa ${x_0}$ \u00e9 igual ao declive da reta tangente ao gr\u00e1fico nesse ponto, isto \u00e9: ${m_t} = f'({x_0})$.<\/p><\/blockquote>\n<p>$${f(x) = x + 2\\operatorname{sen} x}$$<\/p>\n<p>$${D_f} = \\mathbb{R}$$<\/p>\n<p>Ora, $$\\begin{array}{*{20}{l}}<br \/>\n{f'(x)}&amp; = &amp;{\\left( {x + 2\\operatorname{sen} x} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{1 + 2\\cos x}<br \/>\n\\end{array}$$<\/p>\n<p>Logo: $$\\begin{array}{*{20}{l}}<br \/>\n{f&#8217;:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to 1 + 2\\cos x}<br \/>\n\\end{array}$$<\/p>\n<p>Como $$\\begin{array}{*{20}{l}}<br \/>\n{f'(x) = 0}&amp; \\Leftrightarrow &amp;{1 + 2\\cos x = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\cos x =\u00a0 &#8211; \\frac{1}{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x =\u00a0 \\pm \\frac{{2\\pi }}{3} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<p>as abcissas de todos os pontos do gr\u00e1fico de $f$ em que a reta tangente \u00e9 horizontal s\u00e3o dadas por: $${x =\u00a0 \\pm \\frac{{2\\pi }}{3} + 2k\\pi ,k \\in \\mathbb{Z}}$$<\/p>\n<div id=\"attachment_8793\" style=\"width: 686px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10a1.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-8793\" data-attachment-id=\"8793\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8793\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10a1.png\" data-orig-size=\"676,404\" 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\/12pag128-10a1.png\" class=\"size-full wp-image-8793\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10a1.png\" alt=\"\" width=\"676\" height=\"404\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10a1.png 676w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10a1-300x179.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10a1-150x89.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10a1-400x239.png 400w\" sizes=\"auto, (max-width: 676px) 100vw, 676px\" \/><\/a><p id=\"caption-attachment-8793\" class=\"wp-caption-text\">Gr\u00e1fico de f e as suas tangentes horizontais<\/p><\/div>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8782' onClick='GTTabs_show(0,8782)'>&lt;&lt; Enunciado<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8782' onClick='GTTabs_show(2,8782)'>R2 &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_8782'>\n<span class='GTTabs_titles'><b>R2<\/b><\/span><\/p>\n<blockquote><p>A reta tangente ao gr\u00e1fico de uma fun\u00e7\u00e3o \u00e9 horizontal nos pontos em que a derivada da fun\u00e7\u00e3o \u00e9 nula, pois a derivada (se existir) da fun\u00e7\u00e3o num ponto de abcissa ${x_0}$ \u00e9 igual ao declive da reta tangente ao gr\u00e1fico nesse ponto, isto \u00e9: ${m_t} = f'({x_0})$.<\/p><\/blockquote>\n<p>$${g(x) = x + \\cos x}$$<\/p>\n<p>$${D_g} = \\mathbb{R}$$<\/p>\n<p>Ora, $$\\begin{array}{*{20}{l}}<br \/>\n{g'(x)}&amp; = &amp;{\\left( {x + \\cos x} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{1 &#8211; \\sin x}<br \/>\n\\end{array}$$<\/p>\n<p>Logo: $$\\begin{array}{*{20}{l}}<br \/>\n{g&#8217;:}&amp;{\\mathbb{R} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to 1 &#8211; \\sin x}<br \/>\n\\end{array}$$<\/p>\n<p>Como $$\\begin{array}{*{20}{l}}<br \/>\n{g'(x) = 0}&amp; \\Leftrightarrow &amp;{1 &#8211; \\operatorname{sen} x = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x = 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{\\pi }{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<p>as abcissas de todos os pontos do gr\u00e1fico de $g$ em que a reta tangente \u00e9 horizontal s\u00e3o dadas por: $${x = \\frac{\\pi }{2} + 2k\\pi ,k \\in \\mathbb{Z}}$$<\/p>\n<div id=\"attachment_8794\" style=\"width: 686px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10b1.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-8794\" data-attachment-id=\"8794\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8794\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10b1.png\" data-orig-size=\"676,404\" 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\/12pag128-10b1.png\" class=\"size-full wp-image-8794\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10b1.png\" alt=\"\" width=\"676\" height=\"404\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10b1.png 676w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10b1-300x179.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10b1-150x89.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10b1-400x239.png 400w\" sizes=\"auto, (max-width: 676px) 100vw, 676px\" \/><\/a><p id=\"caption-attachment-8794\" class=\"wp-caption-text\">Gr\u00e1fico de g e as suas tangentes horizontais<\/p><\/div>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8782' onClick='GTTabs_show(1,8782)'>&lt;&lt; R1<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8782' onClick='GTTabs_show(3,8782)'>R3 &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_3_8782'>\n<span class='GTTabs_titles'><b>R3<\/b><\/span><\/p>\n<blockquote><p>A reta tangente ao gr\u00e1fico de uma fun\u00e7\u00e3o \u00e9 horizontal nos pontos em que a derivada da fun\u00e7\u00e3o \u00e9 nula, pois a derivada (se existir) da fun\u00e7\u00e3o num ponto de abcissa ${x_0}$ \u00e9 igual ao declive da reta tangente ao gr\u00e1fico nesse ponto, isto \u00e9: ${m_t} = f'({x_0})$.<\/p><\/blockquote>\n<p>$$h(x) = x + \\operatorname{tg} x$$<\/p>\n<p>$${D_h} = \\left\\{ {x \\in \\mathbb{R}:x \\ne \\frac{\\pi }{2} + k\\pi ,k \\in \\mathbb{Z}} \\right\\}$$<\/p>\n<p>Ora, $$\\begin{array}{*{20}{l}}<br \/>\n{h'(x)}&amp; = &amp;{\\left( {x + \\operatorname{tg} x} \\right)&#8217;} \\\\<br \/>\n{}&amp; = &amp;{1 + \\frac{1}{{{{\\cos }^2}x}}}<br \/>\n\\end{array}$$<\/p>\n<p>Logo: $$\\begin{array}{*{20}{l}}<br \/>\n{h&#8217;:}&amp;{\\left\\{ {x \\in \\mathbb{R}:x \\ne \\frac{\\pi }{2} + k\\pi ,k \\in \\mathbb{Z}} \\right\\} \\to \\mathbb{R}} \\\\<br \/>\n{}&amp;{x \\to 1 + \\frac{1}{{{{\\cos }^2}x}}}<br \/>\n\\end{array}$$<\/p>\n<p>Como $$\\begin{array}{*{20}{l}}<br \/>\n{h'(x) = 0}&amp; \\Leftrightarrow &amp;{1 + \\frac{1}{{{{\\cos }^2}x}} = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x \\in \\emptyset }<br \/>\n\\end{array}$$<\/p>\n<p>o gr\u00e1fico de $h$ n\u00e3o possui qualquer tangente horizontal.<\/p>\n<div id=\"attachment_8795\" style=\"width: 646px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10c1.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-8795\" data-attachment-id=\"8795\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8795\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10c1.png\" data-orig-size=\"636,380\" 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\/12pag128-10c1.png\" class=\"size-full wp-image-8795\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10c1.png\" alt=\"\" width=\"636\" height=\"380\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10c1.png 636w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10c1-300x179.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10c1-150x89.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag128-10c1-400x238.png 400w\" sizes=\"auto, (max-width: 636px) 100vw, 636px\" \/><\/a><p id=\"caption-attachment-8795\" class=\"wp-caption-text\">O gr\u00e1fico de h n\u00e3o admite tangentes horizontais<\/p><\/div>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8782' onClick='GTTabs_show(2,8782)'>&lt;&lt; R2<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado R1 Enunciado Considere as fun\u00e7\u00f5es reais de vari\u00e1vel real: $\\begin{array}{*{20}{c}} {f(x) = x + 2\\operatorname{sen} x}&amp;{}&amp;{g(x) = x + \\cos x}&amp;{}&amp;{h(x) = x + \\operatorname{tg} x} \\end{array}$ Determine, para cada uma das fun\u00e7\u00f5es&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21102,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,295],"tags":[427,293,307],"series":[],"class_list":["post-8782","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-funcao-derivada","tag-funcoes-trigonometricas"],"views":3081,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12V3Pag128-10_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8782","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=8782"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8782\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21102"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8782"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8782"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8782"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8782"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}