{"id":8149,"date":"2012-04-09T19:49:38","date_gmt":"2012-04-09T18:49:38","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8149"},"modified":"2022-01-30T20:19:15","modified_gmt":"2022-01-30T20:19:15","slug":"a-representacao-grafica-da-derivada-de-f","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8149","title":{"rendered":"A representa\u00e7\u00e3o gr\u00e1fica da derivada de $f$"},"content":{"rendered":"<p><ul id='GTTabs_ul_8149' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8149' class='GTTabs_curr'><a  id=\"8149_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8149' ><a  id=\"8149_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_8149'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8155\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8155\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86.jpg\" data-orig-size=\"551,620\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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\/12pag227-86.jpg\" class=\"alignright wp-image-8155\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86-266x300.jpg\" alt=\"\" width=\"300\" height=\"338\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86-266x300.jpg 266w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86-133x150.jpg 133w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86-400x450.jpg 400w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86.jpg 551w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>A curva $C$ \u00e9 a representa\u00e7\u00e3o gr\u00e1fica da fun\u00e7\u00e3o derivada $f&#8217;$ de uma fun\u00e7\u00e3o $f$ deriv\u00e1vel em $\\left[ {1,5} \\right]$.<\/p>\n<p>A tangente \u00e0 curva no ponto de abcissa 4 \u00e9 horizontal.<\/p>\n<ol>\n<li>Diga, justificando, se \u00e9 verdadeira ou falsa cada uma das seguintes afirma\u00e7\u00f5es:<br \/>\na) $f$ \u00e9 cont\u00ednua em $\\left[ {1,5} \\right]$;<br \/>\nb) $f(1)&lt;f(5)$.<\/li>\n<li>Sabendo que $f(2)=3$, escreva uma equa\u00e7\u00e3o da reta tangente ao gr\u00e1fico de $f$ no ponto de abcissa 2.<\/li>\n<li>Como varia o sinal da segunda derivada de $f$ no intervalo $\\left[ {1,5} \\right]$? Justifique a resposta.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8149' onClick='GTTabs_show(1,8149)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8149'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8155\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8155\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86.jpg\" data-orig-size=\"551,620\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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\/12pag227-86.jpg\" class=\"alignright wp-image-8155\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86.jpg\" alt=\"\" width=\"300\" height=\"338\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86.jpg 551w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86-266x300.jpg 266w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86-133x150.jpg 133w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag227-86-400x450.jpg 400w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><br \/>\na) A afirma\u00e7\u00e3o \u00e9 verdadeira, pois ${D_{f&#8217;}} = \\left[ {1,5} \\right]$ e $f$ tem derivada\u00a0finita nesse intervalo (toda a fun\u00e7\u00e3o com derivada finita num ponto \u00e9 cont\u00ednua nesse ponto);<\/p>\n<p>b) Como $f'(x)&lt;0$ no intervalo $\\left[ {1,5} \\right]$, ent\u00e3o $f$ \u00e9 decrescente nesse intervalo.<br \/>\nLogo, $f(1)&gt;f(5)$ e, portanto, a afirma\u00e7\u00e3o \u00e9 falsa.<br \/>\n\u00ad<\/li>\n<li>O ponto de tang\u00eancia \u00e9 $T\\left( {2,3} \\right)$ e o declive dessa tangente \u00e9 $m = f'(2) =\u00a0 &#8211; 4$.Como o ponto T pertence a essa reta, tem-se para ordenada na origem: $3 =\u00a0 &#8211; 4 \\times 2 + b \\Leftrightarrow b = 11$.\n<p>Logo, $y =\u00a0 &#8211; 4x + 11$ \u00e9 a equa\u00e7\u00e3o reduzida da reta pedida.<br \/>\n\u00ad<\/li>\n<li>\\(f&#8221;(x) &gt; 0\\) se $x \\in \\left[ {1,4} \\right[$, pois neste intervalo $f&#8217;$ \u00e9 estritamente crescente;<br \/>\n\\(f&#8221;(x) &lt; 0\\) se $x \\in \\left] {4,5} \\right]$, pois neste intervalo $f&#8217;$ \u00e9 estritamente decrescente;<br \/>\n\\(f&#8221;(4) = 0\\), pois a tangente \u00e0 curva no ponto de abcissa 4 \u00e9 horizontal.<\/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;\">$1$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$4$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$5$<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Varia\u00e7\u00e3o de $f&#8217;$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\" colspan=\"2\">$ \\nearrow $<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$-2$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\" colspan=\"2\">$ \\searrow $<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Sinal de $f&#8221;$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\" colspan=\"2\">$+$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">$0$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\" colspan=\"2\">$-$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8149' onClick='GTTabs_show(0,8149)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A curva $C$ \u00e9 a representa\u00e7\u00e3o gr\u00e1fica da fun\u00e7\u00e3o derivada $f&#8217;$ de uma fun\u00e7\u00e3o $f$ deriv\u00e1vel em $\\left[ {1,5} \\right]$. A tangente \u00e0 curva no ponto de abcissa 4 \u00e9 horizontal.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21150,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,292],"tags":[427,136,294],"series":[],"class_list":["post-8149","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-calculo-diferencial","tag-12-o-ano","tag-derivada","tag-monotonia"],"views":3777,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12V2Pag227-86_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8149","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=8149"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8149\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21150"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8149"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8149"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8149"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}