{"id":7347,"date":"2012-01-20T13:56:36","date_gmt":"2012-01-20T13:56:36","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7347"},"modified":"2022-01-30T16:31:34","modified_gmt":"2022-01-30T16:31:34","slug":"um-deposito-num-banco","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7347","title":{"rendered":"Um dep\u00f3sito num banco"},"content":{"rendered":"<p><ul id='GTTabs_ul_7347' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7347' class='GTTabs_curr'><a  id=\"7347_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7347' ><a  id=\"7347_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_7347'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Se o capital de ${{C}_{0}}$ euros for depositado num banco, numa conta a prazo \u00e0 taxa anual\u00a0$r$ e os juros forem capitalizados $n$ vezes ao ano, o capital $C$ acumulado, ao fim de $t$ anos, ser\u00e1 dado, em euros, pela express\u00e3o<\/p>\n<p>$$C(t)={{C}_{0}}{{\\left( 1+\\frac{r}{n} \\right)}^{nt}}$$<\/p>\n<p>Sabendo que se depositaram 1000 \u20ac \u00e0 taxa anual de 4%, calcule o capital acumulado ap\u00f3s 10 anos se os juros forem capitalizados:<\/p>\n<ol>\n<li>anualmente;<\/li>\n<li>trimestralmente;<\/li>\n<li>mensalmente;<\/li>\n<li>de hora a hora;<\/li>\n<li>de minuto a minuto;<\/li>\n<li>continuamente.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7347' onClick='GTTabs_show(1,7347)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7347'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote><p>Se o capital de ${{C}_{0}}$ euros for depositado num banco, numa conta a prazo \u00e0 taxa anual\u00a0$r$ e os juros forem capitalizados $n$ vezes ao ano, o capital $C$ acumulado, ao fim de $t$ anos, ser\u00e1 dado, em euros, pela express\u00e3o<\/p>\n<p>$$C(t)={{C}_{0}}{{\\left( 1+\\frac{r}{n} \\right)}^{nt}}$$<\/p><\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>Como ${{C}_{0}}=1000$, $r=0,04$, $n=1$ e $t=10$, tem-se: $$C(10)=1000{{\\left( 1+\\frac{0,04}{1} \\right)}^{10}}\\approx 1480,244285$$<br \/>\nNesta situa\u00e7\u00e3o, o capital acumulado ap\u00f3s 10 anos \u00e9, aproximadamente, $1480,244$ \u20ac.<br \/>\n\u00ad<\/li>\n<li>Como ${{C}_{0}}=1000$, $r=0,04$, $n=4$ e $t=10$, tem-se: $$C(10)=1000{{\\left( 1+\\frac{0,04}{4} \\right)}^{40}}\\approx 1488,863734$$<br \/>\nNesta situa\u00e7\u00e3o, o capital acumulado ap\u00f3s 10 anos \u00e9, aproximadamente, $1488,864$ \u20ac.<br \/>\n\u00ad<\/li>\n<li>Como ${{C}_{0}}=1000$, $r=0,04$, $n=12$ e $t=10$, tem-se: $$C(10)=1000{{\\left( 1+\\frac{0,04}{12} \\right)}^{120}}\\approx 1490,832682$$<br \/>\nNesta situa\u00e7\u00e3o, o capital acumulado ap\u00f3s 10 anos \u00e9, aproximadamente, $1490,833$ \u20ac.<br \/>\n\u00ad<\/li>\n<li>Como ${{C}_{0}}=1000$, $r=0,04$,\u00a0$n=24\\times 365=8760$ e $t=10$, tem-se: $$C(10)=1000{{\\left( 1+\\frac{0,04}{8760} \\right)}^{87600}}\\approx 1491,823329$$<br \/>\nNesta situa\u00e7\u00e3o, o capital acumulado ap\u00f3s 10 anos \u00e9, aproximadamente, $1491,823$ \u20ac.<br \/>\n\u00ad<\/li>\n<li>Como ${{C}_{0}}=1000$, $r=0,04$,\u00a0$n=60\\times 24\\times 365=525600$ e $t=10$, tem-se: $$C(10)=1000{{\\left( 1+\\frac{0,04}{525600} \\right)}^{5256000}}\\approx 1491,824669$$<br \/>\nNesta situa\u00e7\u00e3o, o capital acumulado ap\u00f3s 10 anos \u00e9, aproximadamente, $1491,82467$ \u20ac.<br \/>\n\u00ad\u00ad<\/li>\n<li>Como ${{C}_{0}}=1000$, $r=0,04$,\u00a0$n\\to +\\infty $ e $t=10$, tem-se: $$\\begin{array}{*{35}{l}}<br \/>\nC(10) &amp; = &amp; \\lim \\left[ 1000{{\\left( 1+\\frac{0,04}{n} \\right)}^{10n}} \\right]\u00a0 \\\\<br \/>\n{} &amp; = &amp; 1000\\times \\lim {{\\left[ {{\\left( 1+\\frac{1}{25n} \\right)}^{25n}} \\right]}^{\\frac{2}{5}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 1000\\times {{\\left[ \\lim {{\\left( 1+\\frac{1}{25n} \\right)}^{25n}} \\right]}^{\\frac{2}{5}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 1000\\times {{e}^{\\frac{2}{5}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 1000\\times \\sqrt[5]{{{e}^{2}}}\u00a0 \\\\<br \/>\n{} &amp; \\approx\u00a0 &amp; 1491,824698\u00a0 \\\\<br \/>\n\\end{array}$$<br \/>\nNesta situa\u00e7\u00e3o, o capital acumulado ap\u00f3s 10 anos \u00e9, aproximadamente, $1491,82470$ \u20ac.<br \/>\n\u00ad<\/li>\n<\/ol>\n<blockquote><p><strong>RECORDE<\/strong>:<br \/>\n$$\\lim {{\\left( 1+\\frac{1}{n} \\right)}^{n}}=e$$<br \/>\n$$\\lim {{\\left( 1+\\frac{k}{n} \\right)}^{n}}={{e}^{k}}$$<\/p><\/blockquote>\n<p><strong>Prova<\/strong>:<\/p>\n<p>$$\\begin{array}{*{35}{l}}<br \/>\n\\underset{n\\to +\\infty }{\\mathop{\\lim }}\\,{{\\left( 1+\\frac{k}{n} \\right)}^{n}} &amp; = &amp; \\underset{n\\to +\\infty }{\\mathop{\\lim }}\\,{{\\left( 1+\\frac{1}{\\frac{n}{k}} \\right)}^{n}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\underset{n\\to +\\infty }{\\mathop{\\lim }}\\,{{\\left[ {{\\left( 1+\\frac{1}{\\frac{n}{k}} \\right)}^{\\frac{n}{k}}} \\right]}^{k}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; {{\\left[ \\underset{n\\to +\\infty }{\\mathop{\\lim }}\\,{{\\left( 1+\\frac{1}{\\frac{n}{k}} \\right)}^{\\frac{n}{k}}} \\right]}^{k}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; {{\\left[ \\underset{m\\to +\\infty }{\\mathop{\\lim }}\\,{{\\left( 1+\\frac{1}{m} \\right)}^{m}} \\right]}^{k}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; {{e}^{k}}\u00a0 \\\\<br \/>\n\\end{array}$$<\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7347' onClick='GTTabs_show(0,7347)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Se o capital de ${{C}_{0}}$ euros for depositado num banco, numa conta a prazo \u00e0 taxa anual\u00a0$r$ e os juros forem capitalizados $n$ vezes ao ano, o capital $C$ acumulado, ao&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21125,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,267],"tags":[427,268],"series":[],"class_list":["post-7347","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-funcoes-exponenciais-e-logaritmicas","tag-12-o-ano","tag-funcao-exponencial"],"views":1989,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/01\/12V2Pag204-13_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7347","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=7347"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7347\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21125"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7347"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7347"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7347"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7347"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}