{"id":8221,"date":"2012-04-11T17:28:15","date_gmt":"2012-04-11T16:28:15","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8221"},"modified":"2022-01-30T21:58:17","modified_gmt":"2022-01-30T21:58:17","slug":"reproducao-de-duas-especies-vegetais","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8221","title":{"rendered":"Reprodu\u00e7\u00e3o de duas esp\u00e9cies vegetais"},"content":{"rendered":"<p><ul id='GTTabs_ul_8221' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8221' class='GTTabs_curr'><a  id=\"8221_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8221' ><a  id=\"8221_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_8221'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Um bi\u00f3logo provoca, em laborat\u00f3rio, a reprodu\u00e7\u00e3o de duas esp\u00e9cies vegetais, A e B. O n\u00famero de exemplares de cada uma das esp\u00e9cies, ao fim de $t$ meses, ap\u00f3s o in\u00edcio da experi\u00eancia, \u00e9 dado por:<\/p>\n<p>$$\\begin{array}{*{20}{c}}<br \/>\n{{\\text{Esp}}{\\text{. A:}}}&amp;{A(t) = 40 + \\ln \\left( {{t^2} + 1} \\right)}<br \/>\n\\end{array}$$<\/p>\n<p>$$\\begin{array}{*{20}{c}}<br \/>\n{{\\text{Esp}}{\\text{. B:}}}&amp;{B(t) = \\frac{{50}}{{1 + 8{e^{ &#8211; 0,5t}}}}}<br \/>\n\\end{array}\\,\\,\\,\\left( {t &gt; 0} \\right)$$<\/p>\n<ol>\n<li>Determine o n\u00famero de plantas de cada esp\u00e9cie utilizadas no in\u00edcio do processo.<\/li>\n<li>Verifique, analiticamente, que o n\u00famero de exemplares de cada esp\u00e9cie aumenta \u00e0 medida que o tempo passa, mas apresentando comportamentos diferentes. Explique essa diferen\u00e7a.<\/li>\n<li>Recorrendo \u00e0 sua calculadora gr\u00e1fica, comprove que h\u00e1 momentos em que os exemplares de cada esp\u00e9cie s\u00e3o em igual n\u00famero.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8221' onClick='GTTabs_show(1,8221)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8221'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote><p>Um bi\u00f3logo provoca, em laborat\u00f3rio, a reprodu\u00e7\u00e3o de duas esp\u00e9cies vegetais, A e B. O n\u00famero de exemplares de cada uma das esp\u00e9cies, ao fim de $t$ meses, ap\u00f3s o in\u00edcio da experi\u00eancia, \u00e9 dado por:<\/p>\n<p>$$\\begin{array}{*{20}{c}}<br \/>\n{{\\text{Esp}}{\\text{. A:}}}&amp;{A(t) = 40 + \\ln \\left( {{t^2} + 1} \\right)}<br \/>\n\\end{array}$$<\/p>\n<p>$$\\begin{array}{*{20}{c}}<br \/>\n{{\\text{Esp}}{\\text{. B:}}}&amp;{B(t) = \\frac{{50}}{{1 + 8{e^{ &#8211; 0,5t}}}}}<br \/>\n\\end{array}\\,\\,\\,\\left( {t &gt; 0} \\right)$$<\/p><\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>Como $$A(0) = 40 + \\ln \\left( {0 + 1} \\right) = 40$$ e $$B(0) = \\frac{{50}}{{1 + 8{e^0}}} = \\frac{{50}}{9} \\approx 5$$ conclui-se que foram utilizadas 40 plantas da esp\u00e9cie A e 5 plantas da esp\u00e9cie B no in\u00edcio do processo.<br \/>\n\u00ad<\/li>\n<li>Como\u00a0\\[\\begin{array}{*{20}{l}}<br \/>\n{A&#8217;\\left( t \\right)}&amp; = &amp;{{{\\left( {40 + \\ln \\left( {{t^2} + 1} \\right)} \\right)}^\\prime }} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{2t}}{{{t^2} + 1}}}<br \/>\n\\end{array}\\]<br \/>\ne<br \/>\n\\[\\begin{array}{*{20}{l}}<br \/>\n{B&#8217;\\left( t \\right)}&amp; = &amp;{{{\\left( {\\frac{{50}}{{1 + 8{e^{ &#8211; 0,5t}}}}} \\right)}^\\prime }} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{200{e^{ &#8211; 0,5t}}}}{{{{\\left( {1 + 8{e^{ &#8211; 0,5t}}} \\right)}^2}}}}<br \/>\n\\end{array}\\]<br \/>\nconclui-se que $$A'(t) &gt; 0,\\forall t \\in \\mathbb{R}_0^ + $$ e $$B'(t) &gt; 0,\\forall t \\in \\mathbb{R}_0^ + $$<br \/>\nPortanto, ambas as fun\u00e7\u00f5es s\u00e3o estritamente crescentes no seu dom\u00ednio.<\/p>\n<p>Por outro lado, $$\\mathop {\\lim }\\limits_{t \\to\u00a0 + \\infty } A(t) = \\mathop {\\lim }\\limits_{t \\to\u00a0 + \\infty } \\left[ {40 + \\ln \\left( {{t^2} + 1} \\right)} \\right] =\u00a0 + \\infty $$ e $$\\mathop {\\lim }\\limits_{t \\to\u00a0 + \\infty } B(t) = \\mathop {\\lim }\\limits_{t \\to\u00a0 + \\infty } \\frac{{50}}{{1 + 8{e^{ &#8211; 0,5t}}}} = 50$$<br \/>\nAinda que ambas as fun\u00e7\u00f5es sejam estritamente crescentes, a esp\u00e9cie B n\u00e3o ultrapassa os 50 exemplares, enquanto esse n\u00famero ser\u00e1 largamente ultrapassado, se a experi\u00eancia tiver a dura\u00e7\u00e3o suficiente (!).<br \/>\n\u00ad<\/li>\n<li>Existem dois (!) momentos em que os exemplares de cada esp\u00e9cie s\u00e3o em igual n\u00famero:<br \/>\n\u00ad<\/li>\n<\/ol>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag229-93.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"8233\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=8233\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag229-93.png\" data-orig-size=\"676,354\" 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\/12pag229-93.png\" class=\"aligncenter wp-image-8233 size-full\" title=\"Gr\u00e1fico\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag229-93.png\" alt=\"\" width=\"676\" height=\"354\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag229-93.png 676w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag229-93-300x157.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag229-93-150x78.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12pag229-93-400x209.png 400w\" sizes=\"auto, (max-width: 676px) 100vw, 676px\" \/><\/a><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8221' onClick='GTTabs_show(0,8221)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Um bi\u00f3logo provoca, em laborat\u00f3rio, a reprodu\u00e7\u00e3o de duas esp\u00e9cies vegetais, A e B. O n\u00famero de exemplares de cada uma das esp\u00e9cies, ao fim de $t$ meses, ap\u00f3s o in\u00edcio&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21156,"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-8221","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":2018,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/04\/12V2Pag229-93_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8221","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=8221"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8221\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21156"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8221"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8221"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8221"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8221"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}