{"id":803,"date":"2019-11-06T14:32:23","date_gmt":"2019-11-06T13:32:23","guid":{"rendered":"https:\/\/blogs.futura-sciences.com\/lehning\/?p=803"},"modified":"2019-11-14T10:02:28","modified_gmt":"2019-11-14T09:02:28","slug":"le-tvi-a-la-grande-ruine","status":"publish","type":"post","link":"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/11\/06\/le-tvi-a-la-grande-ruine\/","title":{"rendered":"Le TVI \u00e0 la Grande Ruine"},"content":{"rendered":"

Pour monter sur un sommet \u00e0 3712 m\u00e8tres d’altitude, comme la Grande Ruine dans le massif de Ecrins (image mise en avant @Herv\u00e9 Lehning), en partant d’un refuge situ\u00e9 \u00e0 3169 m\u00e8tres d’altitude (le refuge Ad\u00e8le Planchard), il est n\u00e9cessaire de passer par toutes les altitudes interm\u00e9diaires.<\/p>\n

\"\"
Quelque soit la voie prise pour monter au sommet, nous passerons au moins une fois \u00e0 chaque altitude entre le point de d\u00e9part et celui d’arriv\u00e9e. La photo repr\u00e9sente un sommet proche de l’Everest, dans l’Himalaya. @Herv\u00e9 Lehning<\/figcaption><\/figure>\n

Le Th\u00e9or\u00e8me des Valeurs Interm\u00e9diaires (ou TVI)<\/h2>\n

Ce r\u00e9sultat de bon sens correspond \u00e0 un th\u00e9or\u00e8me de math\u00e9matiques concernant les fonctions continues sur un intervalle r\u00e9el \u00e0 valeurs r\u00e9elles. La plupart des fonctions qu’on rencontre en math\u00e9matiques sont continues sauf, \u00e9ventuellement, en quelques points exceptionnels, appel\u00e9s pour cela points de discontinuit\u00e9. Physiquement, dans la pratique, ces points correspondent souvent \u00e0 des sauts.<\/p>\n

\"\"
Exemple de discontinuit\u00e9 : en un point x<\/em>, les limites \u00e0 droite et \u00e0 gauche\u00a0 diff\u00e8rent de la valeur en x. Dans ce cas, la fonction pr\u00e9sente un saut en x<\/em>.<\/figcaption><\/figure>\n

Le th\u00e9or\u00e8me des valeurs interm\u00e9diaires peut s’illustrer ainsi :<\/p>\n

\"\"
Si f est une fonction continue sur [a, b] et m est une valeur entre f (a) et f (b), il existe x tel que f (x) = m.<\/figcaption><\/figure>Ce th\u00e9or\u00e8me est donc un th\u00e9or\u00e8me existentiel: il affirme l’existence d’un nombre sans permettre pour autant de le calculer.<\/p>\n

Utilit\u00e9<\/h2>\n

L’utilit\u00e9 pratique\u00a0 essentielle de ce th\u00e9or\u00e8me est de montrer l’existence de racines d’\u00e9quations : si une fonction continue change de signe entre deux points\u00a0a<\/em> et\u00a0b<\/em>, <\/em>elle s’annule entre ces deux points.<\/p>\n

On en d\u00e9duit, par exemple, qu’une fonction continue sur un segment [a<\/em>, b<\/em>] \u00e0 valeurs dans lui-m\u00eame admet au moins un point fixe, c’est-\u00e0-dire un point\u00a0x<\/em> tel que\u00a0f<\/em> (x<\/em>) =\u00a0x<\/em>. Pour le d\u00e9montrer, il suffit de remarquer que la fonction\u00a0g<\/em> d\u00e9finie par\u00a0g\u00a0<\/em>(x<\/em>) =\u00a0f \u00a0(x)\u00a0 –\u00a0 <\/em>x change de signe entre\u00a0a<\/em> et\u00a0b<\/em>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Pour monter sur un sommet \u00e0 3712 m\u00e8tres d’altitude, comme la Grande Ruine dans le massif de Ecrins (image mise en avant @Herv\u00e9 Lehning), en partant d’un refuge situ\u00e9 \u00e0 3169 m\u00e8tres d’altitude (le refuge Ad\u00e8le Planchard), il est n\u00e9cessaire de passer par toutes les altitudes interm\u00e9diaires. Le Th\u00e9or\u00e8me des Valeurs Interm\u00e9diaires (ou TVI) Ce … Continuer la lecture de Le TVI \u00e0 la Grande Ruine<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":12,"featured_media":804,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"aside","meta":{"footnotes":"","_links_to":"","_links_to_target":""},"categories":[112,29,11,1,13,152,98],"tags":[318,317,316,319,315],"class_list":["post-803","post","type-post","status-publish","format-aside","has-post-thumbnail","hentry","category-humour","category-maths-2","category-nature","category-non-classe","category-photographies","category-sport","category-viequotidienne","tag-alpinisme","tag-contiuite","tag-fonction","tag-marche","tag-theoreme-des-valeurs-intermediaires","post_format-post-format-aside"],"yoast_head":"\nLe TVI \u00e0 la Grande Ruine, par Herv\u00e9 Lehning<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/11\/06\/le-tvi-a-la-grande-ruine\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Le TVI \u00e0 la Grande Ruine, par Herv\u00e9 Lehning\" \/>\n<meta property=\"og:description\" content=\"Pour monter sur un sommet \u00e0 3712 m\u00e8tres d’altitude, comme la Grande Ruine dans le massif de Ecrins (image mise en avant @Herv\u00e9 Lehning), en partant d’un refuge situ\u00e9 \u00e0 3169 m\u00e8tres d’altitude (le refuge Ad\u00e8le Planchard), il est n\u00e9cessaire de passer par toutes les altitudes interm\u00e9diaires. Le Th\u00e9or\u00e8me des Valeurs Interm\u00e9diaires (ou TVI) Ce … Continuer la lecture de Le TVI \u00e0 la Grande Ruine →\" \/>\n<meta property=\"og:url\" content=\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/11\/06\/le-tvi-a-la-grande-ruine\/\" \/>\n<meta property=\"og:site_name\" content=\"MATH'MONDE, le blog d'Herv\u00e9 LEHNING, agr\u00e9g\u00e9 de math\u00e9matiques\" \/>\n<meta property=\"article:published_time\" content=\"2019-11-06T13:32:23+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-11-14T09:02:28+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/06\/GrandeRuine-01.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"850\" \/>\n\t<meta property=\"og:image:height\" content=\"567\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/11\/06\/le-tvi-a-la-grande-ruine\/\",\"url\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/11\/06\/le-tvi-a-la-grande-ruine\/\",\"name\":\"Le TVI \u00e0 la Grande Ruine, par Herv\u00e9 Lehning\",\"isPartOf\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/11\/06\/le-tvi-a-la-grande-ruine\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/11\/06\/le-tvi-a-la-grande-ruine\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/06\/GrandeRuine-01.jpg\",\"datePublished\":\"2019-11-06T13:32:23+00:00\",\"dateModified\":\"2019-11-14T09:02:28+00:00\",\"author\":{\"@id\":\"\"},\"breadcrumb\":{\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/11\/06\/le-tvi-a-la-grande-ruine\/#breadcrumb\"},\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/11\/06\/le-tvi-a-la-grande-ruine\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/11\/06\/le-tvi-a-la-grande-ruine\/#primaryimage\",\"url\":\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/06\/GrandeRuine-01.jpg\",\"contentUrl\":\"https:\/\/blogs.futura-sciences.com\/lehning\/wp-content\/uploads\/sites\/13\/2019\/06\/GrandeRuine-01.jpg\",\"width\":850,\"height\":567},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/2019\/11\/06\/le-tvi-a-la-grande-ruine\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Accueil\",\"item\":\"https:\/\/blogs.futura-sciences.com\/lehning\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Le TVI \u00e0 la Grande Ruine\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/blogs.futura-sciences.com\/lehning\/#website\",\"url\":\"https:\/\/blogs.futura-sciences.com\/lehning\/\",\"name\":\"MATH'MONDE, le blog d'Herv\u00e9 LEHNING, agr\u00e9g\u00e9 de math\u00e9matiques\",\"description\":\"Comment comprendre le monde moderne sans culture math\u00e9matique ? 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