{"id":8325,"date":"2012-04-16T00:36:03","date_gmt":"2012-04-15T23:36:03","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8325"},"modified":"2022-01-14T00:52:53","modified_gmt":"2022-01-14T00:52:53","slug":"considere-as-funcoes-3","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8325","title":{"rendered":"Considere as fun\u00e7\u00f5es"},"content":{"rendered":"<p><ul id='GTTabs_ul_8325' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8325' class='GTTabs_curr'><a  id=\"8325_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8325' ><a  id=\"8325_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<li id='GTTabs_li_2_8325' ><a  id=\"8325_2\" onMouseOver=\"GTTabsShowLinks('Gr\u00e1ficos'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Gr\u00e1ficos<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_8325'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere as fun\u00e7\u00f5es:<\/p>\n<p>$$f(x) = 2\\operatorname{sen} x$$<\/p>\n<p>$$g(x) =\u00a0 &#8211; 0,5\\operatorname{sen} x$$<\/p>\n<p>$$h(x) =\u00a0 &#8211; 1 + \\operatorname{sen} x$$<\/p>\n<p>$$t(x) =\u00a0 &#8211; 1 + 2\\operatorname{sen} x$$<\/p>\n<p>Determine para cada uma:<\/p>\n<ul style=\"list-style-type: square;\">\n<li>a express\u00e3o geral dos zeros;<\/li>\n<li>os extremos e a express\u00e3o dos minimizantes e maximizantes;<\/li>\n<li>o contradom\u00ednio;<\/li>\n<li>o per\u00edodo positivo m\u00ednimo.<\/li>\n<\/ul>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8325' onClick='GTTabs_show(1,8325)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8325'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>$f(x) = 2\\operatorname{sen} x$<\/p>\n<\/blockquote>\n<ul style=\"list-style-type: square;\">\n<li>a express\u00e3o geral dos zeros<\/li>\n<\/ul>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{f(x) = 0}&amp; \\Leftrightarrow &amp;{2\\operatorname{sen} x = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>os extremos e a express\u00e3o dos minimizantes e maximizantes<\/li>\n<\/ul>\n<p>M\u00ednimo: $- 2$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{f(x) =\u00a0 &#8211; 2}&amp; \\Leftrightarrow &amp;{2\\operatorname{sen} x =\u00a0 &#8211; 2} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x =\u00a0 &#8211; 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{{3\\pi }}{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<p>M\u00e1ximo: $2$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{f(x) = 2}&amp; \\Leftrightarrow &amp;{2\\operatorname{sen} x = 2} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x = 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{\\pi }{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>o contradom\u00ednio<\/li>\n<\/ul>\n<p>$$D{&#8216;_f} = \\left[ { &#8211; 2,2} \\right]$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>o per\u00edodo positivo m\u00ednimo<\/li>\n<\/ul>\n<p>Per\u00edodo positivo m\u00ednimo: $2\\pi $<br \/>\n\u00ad<\/p>\n<blockquote>\n<p>\u00a0$g(x) =\u00a0 &#8211; 0,5\\operatorname{sen} x$<\/p>\n<\/blockquote>\n<ul style=\"list-style-type: square;\">\n<li>\u00a0a express\u00e3o geral dos zeros<\/li>\n<\/ul>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{g(x) = 0}&amp; \\Leftrightarrow &amp;{ &#8211; 0,5\\operatorname{sen} x = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>os extremos e a express\u00e3o dos minimizantes e maximizantes<\/li>\n<\/ul>\n<p>M\u00ednimo: $ &#8211; \\frac{1}{2}$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{g(x) =\u00a0 &#8211; \\frac{1}{2}}&amp; \\Leftrightarrow &amp;{ &#8211; 0,5\\operatorname{sen} x =\u00a0 &#8211; \\frac{1}{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x = 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{\\pi }{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<p>M\u00e1ximo: $\\frac{1}{2}$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{g(x) = \\frac{1}{2}}&amp; \\Leftrightarrow &amp;{ &#8211; 0,5\\operatorname{sen} x = \\frac{1}{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x =\u00a0 &#8211; 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{{3\\pi }}{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>o contradom\u00ednio<\/li>\n<\/ul>\n<p>$$D{&#8216;_g} = \\left[ { &#8211; \\frac{1}{2},\\frac{1}{2}} \\right]$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>o per\u00edodo positivo m\u00ednimo<\/li>\n<\/ul>\n<p>Per\u00edodo positivo m\u00ednimo: $2\\pi $<br \/>\n\u00ad<\/p>\n<blockquote>\n<p>$h(x) =\u00a0 &#8211; 1 + \\operatorname{sen} x$<\/p>\n<\/blockquote>\n<ul style=\"list-style-type: square;\">\n<li>a express\u00e3o geral dos zeros<\/li>\n<\/ul>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{h(x) = 0}&amp; \\Leftrightarrow &amp;{ &#8211; 1 + \\operatorname{sen} x = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x = 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{\\pi }{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>os extremos e a express\u00e3o dos minimizantes e maximizantes<\/li>\n<\/ul>\n<p>M\u00ednimo: $ &#8211; 2$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{h(x) =\u00a0 &#8211; 2}&amp; \\Leftrightarrow &amp;{ &#8211; 1 + \\operatorname{sen} x =\u00a0 &#8211; 2} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x =\u00a0 &#8211; 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{{3\\pi }}{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<p>M\u00e1ximo: $0$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{h(x) = 0}&amp; \\Leftrightarrow &amp;{ &#8211; 1 + \\operatorname{sen} x = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x = 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{\\pi }{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>o contradom\u00ednio<\/li>\n<\/ul>\n<p>$$D{&#8216;_h} = \\left[ { &#8211; 2,0} \\right]$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>o per\u00edodo positivo m\u00ednimo<\/li>\n<\/ul>\n<p>Per\u00edodo positivo m\u00ednimo: $2\\pi $<br \/>\n\u00ad<\/p>\n<blockquote>\n<p>$t(x) =\u00a0 &#8211; 1 + 2\\operatorname{sen} x$<\/p>\n<\/blockquote>\n<ul style=\"list-style-type: square;\">\n<li>a express\u00e3o geral dos zeros<\/li>\n<\/ul>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{t(x) = 0}&amp; \\Leftrightarrow &amp;{ &#8211; 1 + 2\\operatorname{sen} x = 0} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x = \\frac{1}{2}} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\begin{array}{*{20}{l}}<br \/>\n{x = \\frac{\\pi }{6} + 2k\\pi }&amp; \\vee &amp;{x = \\frac{{5\\pi }}{6} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}}<br \/>\n\\end{array}$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>os extremos e a express\u00e3o dos minimizantes e maximizantes<\/li>\n<\/ul>\n<p>M\u00ednimo: $ &#8211; 3$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{t(x) =\u00a0 &#8211; 3}&amp; \\Leftrightarrow &amp;{ &#8211; 1 + 2\\operatorname{sen} x =\u00a0 &#8211; 3} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x =\u00a0 &#8211; 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{{3\\pi }}{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<p>M\u00e1ximo: $1$<\/p>\n<p>$$\\begin{array}{*{20}{l}}<br \/>\n{t(x) = 1}&amp; \\Leftrightarrow &amp;{ &#8211; 1 + 2\\operatorname{sen} x = 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{\\operatorname{sen} x = 1} \\\\<br \/>\n{}&amp; \\Leftrightarrow &amp;{x = \\frac{\\pi }{2} + 2k\\pi ,k \\in \\mathbb{Z}}<br \/>\n\\end{array}$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>o contradom\u00ednio<\/li>\n<\/ul>\n<p>$$D{&#8216;_t} = \\left[ { &#8211; 3,1} \\right]$$<\/p>\n<ul style=\"list-style-type: square;\">\n<li>o per\u00edodo positivo m\u00ednimo<\/li>\n<\/ul>\n<p>Per\u00edodo positivo m\u00ednimo: $2\\pi $<\/p>\n<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8325' onClick='GTTabs_show(0,8325)'>&lt;&lt; Enunciado<\/a><\/span><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8325' onClick='GTTabs_show(2,8325)'>Gr\u00e1ficos &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_2_8325'>\n<span class='GTTabs_titles'><b>Gr\u00e1ficos<\/b><\/span><\/p>\n<p style=\"text-align: center;\"><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"margin: 0 auto;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":951,\r\n\"height\":350,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ 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2\\operatorname{sen} x$$ Determine para cada uma: a&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19641,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,295],"tags":[427,144,296,297,298],"series":[],"class_list":["post-8325","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-funcoes-seno-co-seno-e-tangente","tag-12-o-ano","tag-extremos-relativos","tag-funcao-seno","tag-periodo-positivo-minimo","tag-zeros"],"views":14780,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat220.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8325","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=8325"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8325\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19641"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8325"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8325"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8325"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8325"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}