{"id":7414,"date":"2012-03-09T14:04:37","date_gmt":"2012-03-09T14:04:37","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7414"},"modified":"2022-01-25T11:58:31","modified_gmt":"2022-01-25T11:58:31","slug":"considere-a-funcao-2","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7414","title":{"rendered":"Considere a fun\u00e7\u00e3o"},"content":{"rendered":"<p><ul id='GTTabs_ul_7414' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7414' class='GTTabs_curr'><a  id=\"7414_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7414' ><a  id=\"7414_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_7414'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere a fun\u00e7\u00e3o $$f:x \\to 4{x^3} &#8211; 7x + 1$$<\/p>\n<ol>\n<li>Complete o quadro com as imagens dos valores assinalados.<br \/>\n<table class=\" aligncenter\" style=\"width: 70%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$x$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$-2$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$1$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$2$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$f(x)$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Justifique a seguinte afirma\u00e7\u00e3o:<br \/>\n&#8220;A equa\u00e7\u00e3o $f(x) = 0$ tem tr\u00eas e s\u00f3 tr\u00eas ra\u00edzes: uma pertencente ao intervalo ]-2, 0[, outra pertencente ao intervalo ]0, 1[ e a terceira pertencente ao intervalo ]1, 2[.&#8221;<\/li>\n<li>Determine, a menos de $0,1$, a maior das ra\u00edzes.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7414' onClick='GTTabs_show(1,7414)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7414'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>$$f:x \\to 4{x^3} &#8211; 7x + 1$$<\/p>\n<\/blockquote>\n<ol>\n<li>\n<table class=\" aligncenter\" style=\"width: 70%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$x$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$-2$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$1$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$2$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$f(x)$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$-17$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$1$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$-2$<\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #2f4f4f 1px solid;\">$19$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>A fun\u00e7\u00e3o $f$ \u00e9 cont\u00ednua em $\\mathbb{R}$, consequentemente \u00e9 cont\u00ednua em qualquer intervalo fechado contido no seu dom\u00ednio.\n<p>Ora, $f( &#8211; 2) &lt; 0 &lt; f(0)$, $f(1) &lt; 0 &lt; f(0)$ e $f(1) &lt; 0 &lt; f(2)$.<\/p>\n<p>Logo, de acordo com o teorema de Bolzano, a fun\u00e7\u00e3o admite pelo menos um zero em cada um desses intervalos.<\/p>\n<p>Dado que $f$ \u00e9 uma fun\u00e7\u00e3o polinomial de grau tr\u00eas, admite tr\u00eas zeros no m\u00e1ximo.<\/p>\n<p>Logo, a equa\u00e7\u00e3o $f(x) = 0$ tem tr\u00eas e s\u00f3 tr\u00eas ra\u00edzes: uma pertencente ao intervalo ]-2, 0[, outra pertencente ao intervalo ]0, 1[ e a terceira pertencente ao intervalo ]1, 2[.<\/p>\n<\/li>\n<li>A maior das ra\u00edzes pertence ao intervalo $\\left] {1,2} \\right[$.\n<p>Calculando o sinal de $f(x)$ para $x = 1,1$, para $x = 1,2$, etc., conclui-se que a raiz procurada pertence ao intervalo $\\left] {1,2;1,3} \\right[$, pois $f(1,2) &lt; 0 &lt; f(1,3)$.<\/p>\n<p style=\"text-align: center;\" data-tadv-p=\"keep\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12pag208-27a.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7511\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7511\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12pag208-27a.png\" data-orig-size=\"198,134\" 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=\"Tabela\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12pag208-27a.png\" class=\"size-full wp-image-7511 alignnone\" title=\"Tabela\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12pag208-27a.png\" alt=\"\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12pag208-27a.png 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12pag208-27a-150x101.png 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a>\u00a0 <a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12pag208-27b.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7512\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7512\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12pag208-27b.png\" data-orig-size=\"198,134\" 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\/03\/12pag208-27b.png\" class=\"size-full wp-image-7512 alignnone\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12pag208-27b.png\" alt=\"\" width=\"198\" height=\"134\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12pag208-27b.png 198w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12pag208-27b-150x101.png 150w\" sizes=\"auto, (max-width: 198px) 100vw, 198px\" \/><\/a><\/p>\n<p>Um valor aproximado \u00e0s d\u00e9cimas da\u00a0raiz procurada \u00e9 $x = 1,2$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7414' onClick='GTTabs_show(0,7414)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere a fun\u00e7\u00e3o $$f:x \\to 4{x^3} &#8211; 7x + 1$$ Complete o quadro com as imagens dos valores assinalados. $x$ $-2$ $0$ $1$ $2$ $f(x)$ Justifique a seguinte afirma\u00e7\u00e3o: &#8220;A equa\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20951,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,285],"tags":[427,286,289],"series":[],"class_list":["post-7414","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-teoria-de-limites","tag-12-o-ano","tag-limites","tag-teorema-de-bolzano"],"views":1526,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/03\/12V2Pag208-27_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7414","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=7414"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7414\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20951"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7414"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7414"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7414"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7414"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}