{"id":7405,"date":"2012-03-08T00:50:06","date_gmt":"2012-03-08T00:50:06","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7405"},"modified":"2022-01-14T14:59:56","modified_gmt":"2022-01-14T14:59:56","slug":"dadas-as-funcoes-reais-de-variavel-real","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7405","title":{"rendered":"Dadas as fun\u00e7\u00f5es reais de vari\u00e1vel real"},"content":{"rendered":"<p><ul id='GTTabs_ul_7405' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7405' class='GTTabs_curr'><a  id=\"7405_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7405' ><a  id=\"7405_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_7405'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Dadas as fun\u00e7\u00f5es reais de vari\u00e1vel real, assim definidas:$$\\begin{array}{*{20}{c}}<br \/>\n{f(x) = {x^2} + 1}&amp;{\\text{e}}&amp;{g(x) = \\frac{1}{x}}<br \/>\n\\end{array}$$<\/p>\n<ol>\n<li>Determine, em fun\u00e7\u00e3o de $h$, a taxa m\u00e9dia de varia\u00e7\u00e3o de cada uma das fun\u00e7\u00f5es no intervalo $\\left[ {1,1 + h} \\right]$, com $h &gt; 0$.<\/li>\n<li>Calcule se existir:\n<p>a) $\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{f(1 + h) &#8211; f(1)}}{h}$<br \/>\n\u00ad<br \/>\nb) $\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{g(1 + h) &#8211; g(1)}}{h}$<br \/>\n\u00ad<br \/>\nc) $\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{f(a + h) &#8211; f(a)}}{h}$<br \/>\n\u00ad<br \/>\nd) $\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{g(a + h) &#8211; g(a)}}{h}$, para ${a \\ne 0}$<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7405' onClick='GTTabs_show(1,7405)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7405'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>$$\\begin{array}{*{20}{c}}<br \/>\n{f(x) = {x^2} + 1}&amp;{\\text{e}}&amp;{g(x) = \\frac{1}{x}}<br \/>\n\\end{array}$$<\/p>\n<\/blockquote>\n<ol>\n<li>Para $f$:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{tm{v_{\\left[ {1,1 + h} \\right]}}}&amp; = &amp;{\\frac{{f(1 + h) &#8211; f(1)}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{{{(1 + h)}^2} + 1 &#8211; 2}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{1 + 2h + {h^2} &#8211; 1}}{h}} \\\\<br \/>\n{}&amp; = &amp;{2 + h}<br \/>\n\\end{array}$$<br \/>\nPara $g$:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{tm{v_{\\left[ {1,1 + h} \\right]}}}&amp; = &amp;{\\frac{{g(1 + h) &#8211; g(1)}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\frac{1}{{1 + h}} &#8211; 1}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\frac{{1 &#8211; 1 &#8211; h}}{{1 + h}}}}{h}} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; \\frac{1}{{1 + h}}}<br \/>\n\\end{array}$$<\/li>\n<li>\n<p>a)<br \/>\nAproveitando o resultado obtido acima, vem:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{f(1 + h) &#8211; f(1)}}{h}}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\left( {2 + h} \\right)} \\\\<br \/>\n{}&amp; = &amp;2<br \/>\n\\end{array}$$<br \/>\nb)<br \/>\nAproveitando o resultado acima, vem:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{g(1 + h) &#8211; g(1)}}{h}}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\left( { &#8211; \\frac{1}{{1 + h}}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; 1}<br \/>\n\\end{array}$$<br \/>\nc)<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{f(a + h) &#8211; f(a)}}{h}}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{{{\\left( {a + h} \\right)}^2} + 1 &#8211; \\left( {{a^2} + 1} \\right)}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{{a^2} + 2ah + {h^2} + 1 &#8211; {a^2} &#8211; 1}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{2ah + {h^2}}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\left( {2a + h} \\right)} \\\\<br \/>\n{}&amp; = &amp;{2a}<br \/>\n\\end{array}$$<br \/>\nd) para ${a \\ne 0}$,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{g(a + h) &#8211; g(a)}}{h}}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{\\frac{1}{{a + h}} &#8211; \\frac{1}{a}}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\frac{{\\frac{{a &#8211; a &#8211; h}}{{a\\left( {a + h} \\right)}}}}{h}} \\\\<br \/>\n{}&amp; = &amp;{\\mathop {\\lim }\\limits_{h \\to 0} \\left( { &#8211; \\frac{1}{{a\\left( {a + h} \\right)}}} \\right)} \\\\<br \/>\n{}&amp; = &amp;{ &#8211; \\frac{1}{{{a^2}}}}<br \/>\n\\end{array}$$<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7405' onClick='GTTabs_show(0,7405)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Dadas as fun\u00e7\u00f5es reais de vari\u00e1vel real, assim definidas:$$\\begin{array}{*{20}{c}} {f(x) = {x^2} + 1}&amp;{\\text{e}}&amp;{g(x) = \\frac{1}{x}} \\end{array}$$ Determine, em fun\u00e7\u00e3o de $h$, a taxa m\u00e9dia de varia\u00e7\u00e3o de cada uma das&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19437,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,285],"tags":[427,287,286,135],"series":[],"class_list":["post-7405","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-indeterminacoes","tag-limites","tag-taxa-media-de-variacao"],"views":3665,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/tmv.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7405","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=7405"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7405\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19437"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7405"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7405"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7405"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7405"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}