{"id":157,"date":"2015-12-09T15:30:05","date_gmt":"2015-12-09T14:30:05","guid":{"rendered":"https:\/\/mathemator.gymgmunden.at\/?p=157"},"modified":"2016-04-16T04:04:47","modified_gmt":"2016-04-16T02:04:47","slug":"heute-im-mo-kurs-3","status":"publish","type":"post","link":"https:\/\/mathemator.gymgmunden.at\/index.php\/2015\/12\/09\/heute-im-mo-kurs-3\/","title":{"rendered":"Heute im MO-Kurs"},"content":{"rendered":"<p style=\"text-align: justify;\"><strong>zum Aufw\u00e4rmen &#8230; Bernoulli-Ungleichung<\/strong><br \/>\nWikipedia informiert: In der Mathematik versteht man unter der bernoullischen Ungleichung (Benannt nach dem Schweizer Mathematiker Jakob Bernoulli) eine einfache, aber wichtige Ungleichung, mit der sich eine Potenzfunktion nach unten absch\u00e4tzen l\u00e4sst. F\u00fcr jede reelle Zahl x gr\u00f6\u00dfer gleich -1 und jede nicht negative ganze Zahl n gilt:<\/p>\n<p><a href=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/bernoulli.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-158 aligncenter\" src=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/bernoulli.png\" alt=\"bernoulli\" width=\"152\" height=\"21\" srcset=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/bernoulli.png 152w, https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/bernoulli-150x21.png 150w\" sizes=\"auto, (max-width: 152px) 100vw, 152px\" \/><\/a><br \/>\nBeweise die Ungleichung!<\/p>\n<p>L\u00f6sungsvorschlag 1: vollst\u00e4ndige Induktion (Stefan)<br \/>\nL\u00f6sungsvorschlag 2: direkt \u00fcber den allgemeinen binomischen Lehrsatz (Lauri)<\/p>\n<div id='gallery-1' class='gallery galleryid-157 gallery-columns-3 gallery-size-thumbnail'><figure class='gallery-item'>\n\t\t\t<div class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/mathemator.gymgmunden.at\/index.php\/2015\/12\/09\/heute-im-mo-kurs-3\/cimg4757\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/CIMG4757-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" \/><\/a>\n\t\t\t<\/div><\/figure><figure class='gallery-item'>\n\t\t\t<div class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/mathemator.gymgmunden.at\/index.php\/2015\/12\/09\/heute-im-mo-kurs-3\/cimg4759\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/CIMG4759-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" \/><\/a>\n\t\t\t<\/div><\/figure><figure class='gallery-item'>\n\t\t\t<div class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/mathemator.gymgmunden.at\/index.php\/2015\/12\/09\/heute-im-mo-kurs-3\/cimg4758\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/CIMG4758-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" \/><\/a>\n\t\t\t<\/div><\/figure><figure class='gallery-item'>\n\t\t\t<div class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/mathemator.gymgmunden.at\/index.php\/2015\/12\/09\/heute-im-mo-kurs-3\/cimg4760\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/CIMG4760-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" \/><\/a>\n\t\t\t<\/div><\/figure><figure class='gallery-item'>\n\t\t\t<div class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/mathemator.gymgmunden.at\/index.php\/2015\/12\/09\/heute-im-mo-kurs-3\/cimg4761\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/CIMG4761-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" \/><\/a>\n\t\t\t<\/div><\/figure><figure class='gallery-item'>\n\t\t\t<div class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/mathemator.gymgmunden.at\/index.php\/2015\/12\/09\/heute-im-mo-kurs-3\/cimg4762\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/CIMG4762-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" \/><\/a>\n\t\t\t<\/div><\/figure><figure class='gallery-item'>\n\t\t\t<div class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/mathemator.gymgmunden.at\/index.php\/2015\/12\/09\/heute-im-mo-kurs-3\/cimg4764\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/CIMG4764-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" \/><\/a>\n\t\t\t<\/div><\/figure><figure class='gallery-item'>\n\t\t\t<div class='gallery-icon landscape'>\n\t\t\t\t<a href='https:\/\/mathemator.gymgmunden.at\/index.php\/2015\/12\/09\/heute-im-mo-kurs-3\/cimg4763\/'><img loading=\"lazy\" decoding=\"async\" width=\"150\" height=\"150\" src=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/CIMG4763-150x150.jpg\" class=\"attachment-thumbnail size-thumbnail\" alt=\"\" \/><\/a>\n\t\t\t<\/div><\/figure>\n\t\t<\/div>\n\n<p><strong>Training: Beweisf\u00fchrung, Einsatz von Ungleichungen<\/strong><br \/>\nMan zeige: Es gibt keine positiven reellen Zalen x, y und z mit<\/p>\n<p style=\"text-align: center;\">(12x\u00b2 + yz)(12y\u00b2 + xz)(12z\u00b2 + xy) = 2014x\u00b2y\u00b2z\u00b2<\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/CIMG4765.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-thumbnail wp-image-171\" src=\"https:\/\/mathemator.gymgmunden.at\/wp-content\/uploads\/2015\/12\/CIMG4765-150x150.jpg\" alt=\"CIMG4765\" width=\"150\" height=\"150\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>zum Aufw\u00e4rmen &#8230; Bernoulli-Ungleichung Wikipedia informiert: In der Mathematik versteht man unter der bernoullischen Ungleichung (Benannt nach dem Schweizer Mathematiker<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,4],"tags":[],"class_list":["post-157","post","type-post","status-publish","format-standard","hentry","category-allgemein","category-mo"],"_links":{"self":[{"href":"https:\/\/mathemator.gymgmunden.at\/index.php\/wp-json\/wp\/v2\/posts\/157","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathemator.gymgmunden.at\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemator.gymgmunden.at\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemator.gymgmunden.at\/index.php\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemator.gymgmunden.at\/index.php\/wp-json\/wp\/v2\/comments?post=157"}],"version-history":[{"count":9,"href":"https:\/\/mathemator.gymgmunden.at\/index.php\/wp-json\/wp\/v2\/posts\/157\/revisions"}],"predecessor-version":[{"id":176,"href":"https:\/\/mathemator.gymgmunden.at\/index.php\/wp-json\/wp\/v2\/posts\/157\/revisions\/176"}],"wp:attachment":[{"href":"https:\/\/mathemator.gymgmunden.at\/index.php\/wp-json\/wp\/v2\/media?parent=157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemator.gymgmunden.at\/index.php\/wp-json\/wp\/v2\/categories?post=157"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemator.gymgmunden.at\/index.php\/wp-json\/wp\/v2\/tags?post=157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}