Quiz — سلسلة النهايات ودراسة الدوال 2 ع ت 1.pdf

Question 1 sur 10 20:00

[{"id":18019,"question":"Quelle est la limite de la fonction f(x) = (x² - 4)\/(x - 2) lorsque x tend vers 2 ?","option_a":"0","option_b":"4","option_c":"Indéterminée","option_d":"2","option_e":"","option_f":"","bonne_reponse":"b","explication":"Factorisez le numérateur : (x² - 4) = (x - 2)(x + 2). La limite devient alors (x + 2) = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"4\", \"c\": \"Indéterminée\", \"d\": \"2\"}}","_debug_options_count":4},{"id":18020,"question":"Une fonction continue sur un intervalle est toujours dérivable sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Exemple : f(x) = |x| est continue en 0 mais non dérivable en ce point.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":18021,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 3x² + 2x + 1 ?","option_a":"3x² - 6x + 2","option_b":"3x² - 6x + 1","option_c":"x² - 3x + 2","option_d":"3x - 6","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée se calcule terme à terme : f'(x) = 3x² - 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3x² - 6x + 2\", \"b\": \"3x² - 6x + 1\", \"c\": \"x² - 3x + 2\", \"d\": \"","_debug_options_count":4},{"id":18022,"question":"La fonction f(x) = 1\/x admet une limite finie lorsque x tend vers 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite de 1\/x lorsque x tend vers 0 est +∞ ou -∞ selon le signe de x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":18023,"question":"Quel est le nombre dérivé de f(x) = sin(x) en x = π\/2 ?","option_a":"0","option_b":"1","option_c":"-1","option_d":"π\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin(x) est cos(x), donc f'(π\/2) = cos(π\/2) = 0 ? Non, cos(π\/2)=0, mais sin'(x)=cos(x). Correction : f'(π\/2)=0. Réponse correcte : 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"-1\", \"d\": \"π\/2\"}}","_debug_options_count":4},{"id":18024,"question":"La fonction f(x) = x² est-elle dérivable en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = x² est dérivable partout, car sa dérivée f'(x) = 2x existe pour tout x réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":18025,"question":"Quelle est la limite de la fonction f(x) = (e^x - 1)\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"e","option_d":"Indéterminée","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une forme indéterminée 0\/0. En utilisant la règle de l'Hôpital ou le développement limité, on trouve que la limite vaut 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"e\", \"d\": \"Indéterminée\"}}","_debug_options_count":4},{"id":18026,"question":"Le théorème des valeurs intermédiaires s'applique-t-il à toute fonction continue sur un intervalle fermé ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le théorème des valeurs intermédiaires s'applique bien à toute fonction continue sur un intervalle fermé [a, b].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":18027,"question":"Quelle est la dérivée de la fonction f(x) = ln(x) ?","option_a":"1\/x","option_b":"x","option_c":"1","option_d":"e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction logarithme népérien ln(x) est 1\/x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1\/x\", \"b\": \"x\", \"c\": \"1\", \"d\": \"e^x\"}}","_debug_options_count":4},{"id":18028,"question":"La fonction f(x) = x³ - 3x + 2 admet-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 3 s'annule en x = 1. Le changement de signe de f' confirme qu'il s'agit d'un extremum local.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]