{"id":6545,"date":"2011-03-01T02:46:03","date_gmt":"2011-03-01T02:46:03","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6545"},"modified":"2021-12-26T17:50:43","modified_gmt":"2021-12-26T17:50:43","slug":"mais-taxa-media-de-variacao","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6545","title":{"rendered":"Mais taxa m\u00e9dia de varia\u00e7\u00e3o"},"content":{"rendered":"<p><ul id='GTTabs_ul_6545' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6545' class='GTTabs_curr'><a  id=\"6545_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6545' ><a  id=\"6545_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_6545'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Seja $f$ uma fun\u00e7\u00e3o polinomial e $h$ um n\u00famero real positivo.<\/p>\n<p>Calcule a taxa m\u00e9dia de varia\u00e7\u00e3o de $f$ no intervalo $\\left[ x,x+h \\right]$, nos casos seguintes:<\/p>\n<ol>\n<li>$f(x)=-3{{x}^{2}}+7x-5$<\/li>\n<li>$f(x)={{x}^{3}}-3x+1$<\/li>\n<li>$f$ \u00e9 uma fun\u00e7\u00e3o afim<\/li>\n<li>$f$ \u00e9 uma fun\u00e7\u00e3o constante.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6545' onClick='GTTabs_show(1,6545)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6545'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Considerando $f(x)=-3{{x}^{2}}+7x-5$, vem:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\nt.m.{{v}_{\\left[ x,x+h \\right]}} &amp; = &amp; \\frac{f(x+h)-f(x)}{(x+h)-x}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{[-3{{(x+h)}^{2}}+7(x+h)-5]-(-3{{x}^{2}}+7x-5)}{h}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{-3{{x}^{2}}-6hx-3{{h}^{2}}+7x+7h-5+3{{x}^{2}}-7x+5}{h}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{-6hx-3{{h}^{2}}+7h}{h}\u00a0 \\\\<br \/>\n{} &amp; = &amp; -6x-3h+7\u00a0 \\\\<br \/>\n\\end{array}\\]<\/li>\n<li>Considerando $f(x)={{x}^{3}}-3x+1$, vem:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\nt.m.{{v}_{\\left[ x,x+h \\right]}} &amp; = &amp; \\frac{f(x+h)-f(x)}{(x+h)-x}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{[{{(x+h)}^{3}}-3(x+h)+1]-({{x}^{3}}-3x+1)}{h}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{({{x}^{2}}+2hx+{{h}^{2}})(x+h)-3x-3h+1-{{x}^{3}}+3x-1}{h}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{{{x}^{3}}+h{{x}^{2}}+2h{{x}^{2}}+2{{h}^{2}}x+{{h}^{2}}x+{{h}^{3}}-3h-{{x}^{3}}}{h}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{3h{{x}^{2}}+3{{h}^{2}}x+{{h}^{3}}-3h}{h}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 3{{x}^{2}}+3hx+{{h}^{2}}-3\u00a0 \\\\<br \/>\n\\end{array}\\]<\/li>\n<li>Considerando $f(x)=mx+b$, vem:<br \/>\n\\[t.m.{{v}_{\\left[ x,x+h \\right]}}=\\frac{f(x+h)-f(x)}{(x+h)-x}=\\frac{[m(x+h)+b]-(mx+b)}{h}=\\frac{mx+mh+b-mx-b}{h}=m\\]<\/li>\n<li>Considerando $f(x)=k$, vem:<br \/>\n\\[t.m.{{v}_{\\left[ x,x+h \\right]}}=\\frac{f(x+h)-f(x)}{(x+h)-x}=\\frac{k-k}{h}=0\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6545' onClick='GTTabs_show(0,6545)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Seja $f$ uma fun\u00e7\u00e3o polinomial e $h$ um n\u00famero real positivo. Calcule a taxa m\u00e9dia de varia\u00e7\u00e3o de $f$ no intervalo $\\left[ x,x+h \\right]$, nos casos seguintes: $f(x)=-3{{x}^{2}}+7x-5$ $f(x)={{x}^{3}}-3x+1$ $f$ \u00e9&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19438,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[98,97,134],"tags":[135],"series":[],"class_list":["post-6545","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-derivadas","tag-taxa-media-de-variacao"],"views":2356,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/03\/Average_rate_of_change.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6545","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=6545"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6545\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19438"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6545"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6545"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6545"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6545"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}