{"id":11832,"date":"2014-02-28T00:44:06","date_gmt":"2014-02-28T00:44:06","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=11832"},"modified":"2022-01-12T00:17:15","modified_gmt":"2022-01-12T00:17:15","slug":"as-funcoes-de-heaviside-e-rampa","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=11832","title":{"rendered":"As fun\u00e7\u00f5es de Heaviside e rampa"},"content":{"rendered":"<p><ul id='GTTabs_ul_11832' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_11832' class='GTTabs_curr'><a  id=\"11832_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_11832' ><a  id=\"11832_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_11832'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>\u00a0As fun\u00e7\u00f5es de Heaviside e rampa s\u00e3o definidas, respetivamente, por: \\[\\begin{array}{*{20}{c}}<br \/>\n{H\\left( x \\right) = \\left\\{ {\\begin{array}{*{20}{c}}<br \/>\n0&amp; \\Leftarrow &amp;{x &lt; 0} \\\\<br \/>\n{\\frac{1}{2}}&amp; \\Leftarrow &amp;{x = 0} \\\\<br \/>\n1&amp; \\Leftarrow &amp;{x &gt; 0}<br \/>\n\\end{array}} \\right.}&amp;{\\text{e}}&amp;{R\\left( x \\right) = \\left\\{ {\\begin{array}{*{20}{c}}<br \/>\n0&amp; \\Leftarrow &amp;{x \\leqslant 0} \\\\<br \/>\nx&amp; \\Leftarrow &amp;{x &gt; 0}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}\\]<\/p>\n<p>Mostre que:<\/p>\n<ol>\n<li>$R\\left( x \\right) = x\\,H\\left( x \\right)$<\/li>\n<li>$R\\left( x \\right) = \\frac{{x + \\left| x \\right|}}{2}$<\/li>\n<li>$\\left( {R \\circ R} \\right)\\left( x \\right) = R\\left( x \\right)$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_11832' onClick='GTTabs_show(1,11832)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_11832'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>\\[\\begin{array}{*{20}{c}}<br \/>\n{H\\left( x \\right) = \\left\\{ {\\begin{array}{*{20}{c}}<br \/>\n0&amp; \\Leftarrow &amp;{x &lt; 0} \\\\<br \/>\n{\\frac{1}{2}}&amp; \\Leftarrow &amp;{x = 0} \\\\<br \/>\n1&amp; \\Leftarrow &amp;{x &gt; 0}<br \/>\n\\end{array}} \\right.}&amp;{\\text{e}}&amp;{R\\left( x \\right) = \\left\\{ {\\begin{array}{*{20}{c}}<br \/>\n0&amp; \\Leftarrow &amp;{x \\leqslant 0} \\\\<br \/>\nx&amp; \\Leftarrow &amp;{x &gt; 0}<br \/>\n\\end{array}} \\right.}<br \/>\n\\end{array}\\]<\/p>\n<\/blockquote>\n<ol>\n<li>\\[\\begin{array}{*{20}{c}}<br \/>\n{x\\,H\\left( x \\right)}&amp; = &amp;{x \\times \\left\\{ {\\begin{array}{*{20}{c}}<br \/>\n0&amp; \\Leftarrow &amp;{x &lt; 0} \\\\<br \/>\n{\\frac{1}{2}}&amp; \\Leftarrow &amp;{x = 0} \\\\<br \/>\n1&amp; \\Leftarrow &amp;{x &gt; 0}<br \/>\n\\end{array}} \\right.}&amp; = &amp;{\\left\\{ {\\begin{array}{*{20}{c}}<br \/>\n{x \\times 0}&amp; \\Leftarrow &amp;{x &lt; 0} \\\\<br \/>\n{x \\times \\frac{1}{2}}&amp; \\Leftarrow &amp;{x = 0} \\\\<br \/>\n{x \\times 1}&amp; \\Leftarrow &amp;{x &gt; 0}<br \/>\n\\end{array}} \\right.}&amp; = &amp;{\\left\\{ {\\begin{array}{*{20}{c}}<br \/>\n0&amp; \\Leftarrow &amp;{x &lt; 0} \\\\<br \/>\n0&amp; \\Leftarrow &amp;{x = 0} \\\\<br \/>\nx&amp; \\Leftarrow &amp;{x &gt; 0}<br \/>\n\\end{array}} \\right.}&amp; = &amp;{R\\left( x \\right)}<br \/>\n\\end{array}\\]<\/li>\n<li>\\[\\begin{array}{*{20}{c}}<br \/>\n{\\frac{{x + \\left| x \\right|}}{2}}&amp; = &amp;{\\left\\{ {\\begin{array}{*{20}{c}}<br \/>\n{\\frac{{x + \\left( { &#8211; x} \\right)}}{2}}&amp; \\Leftarrow &amp;{x \\leqslant 0} \\\\<br \/>\n{\\frac{{x + x}}{2}}&amp; \\Leftarrow &amp;{x &gt; 0}<br \/>\n\\end{array}} \\right.}&amp; = &amp;{\\left\\{ {\\begin{array}{*{20}{c}}<br \/>\n0&amp; \\Leftarrow &amp;{x \\leqslant 0} \\\\<br \/>\nx&amp; \\Leftarrow &amp;{x &gt; 0}<br \/>\n\\end{array}} \\right.}&amp; = &amp;{R\\left( x \\right)}<br \/>\n\\end{array}\\]<\/li>\n<li>\\[\\begin{array}{*{20}{c}}<br \/>\n{\\left( {R \\circ R} \\right)\\left( x \\right)}&amp; = &amp;{R\\left( {R\\left( x \\right)} \\right)}&amp; = &amp;{R\\left( {\\left\\{ {\\begin{array}{*{20}{c}}<br \/>\n0&amp; \\Leftarrow &amp;{x \\leqslant 0} \\\\<br \/>\nx&amp; \\Leftarrow &amp;{x &gt; 0}<br \/>\n\\end{array}} \\right.} \\right)}&amp; = &amp;{R\\left( x \\right)}<br \/>\n\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_11832' onClick='GTTabs_show(0,11832)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado \u00a0As fun\u00e7\u00f5es de Heaviside e rampa s\u00e3o definidas, respetivamente, por: \\[\\begin{array}{*{20}{c}} {H\\left( x \\right) = \\left\\{ {\\begin{array}{*{20}{c}} 0&amp; \\Leftarrow &amp;{x &lt; 0} \\\\ {\\frac{1}{2}}&amp; \\Leftarrow &amp;{x = 0} \\\\ 1&amp; \\Leftarrow&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19371,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,133,147],"tags":[422,154,359,148],"series":[],"class_list":["post-11832","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-funcoes-definidas-por-ramos","category-operacoes-com-funcoes","tag-11-o-ano","tag-funcao-composta-2","tag-funcao-definida-por-ramos","tag-operacoes-com-funcoes-2"],"views":3182,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2014\/02\/Rampa-Heaviside.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11832","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=11832"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/11832\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19371"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=11832"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=11832"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=11832"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=11832"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}