Quiz — EpreuveII_2002.pdf

Question 1 sur 10 20:00

[{"id":19029,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x + 2","option_c":"6x² + 2","option_d":"3x² + 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle de puissance : f'(x) = 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\": \"6x + 2\", \"b\": \"3x + 2\", \"c\": \"6x² + 2\", \"d\": \"3x² + 2x\"}}","_debug_options_count":4},{"id":19030,"question":"La limite de (sin x)\/x quand x tend vers 0 est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un résultat fondamental en analyse : lim (x→0) (sin x)\/x = 1.","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":19031,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2x e^(2x)","option_d":"e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de dérivation des fonctions exponentielles : (e^u)' = u' e^u, ici u = 2x donc u' = 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\": \"2e^(2x)\", \"b\": \"e^(2x)\", \"c\": \"2x e^(2x)\", \"d\": \"e^x\"}}","_debug_options_count":4},{"id":19032,"question":"La fonction f(x) = 1\/x a une limite finie quand 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 quand x tend vers 0 est +∞ ou -∞ selon le sens d'approche, donc elle n'est pas finie.","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":19033,"question":"Quelle est la dérivée de la fonction f(x) = ln(x) ?","option_a":"1\/x","option_b":"x","option_c":"1\/ln(x)","option_d":"e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(x) est 1\/x, une formule fondamentale en analyse.","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\/ln(x)\", \"d\": \"e^x\"}}","_debug_options_count":4},{"id":19034,"question":"Si f'(x) = 0 pour tout x dans un intervalle, alors f est constante sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est le théorème de Lagrange : si la dérivée est nulle sur un intervalle, la fonction est constante.","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":19035,"question":"Quelle est la limite de (x² - 4)\/(x - 2) quand x tend vers 2 ?","option_a":"0","option_b":"4","option_c":"2","option_d":"indéterminée","option_e":"","option_f":"","bonne_reponse":"c","explication":"En simplifiant l'expression : (x² - 4)\/(x - 2) = (x - 2)(x + 2)\/(x - 2) = x + 2 pour x ≠ 2, donc la limite est 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"4\", \"c\": \"2\", \"d\": \"indéterminée\"}}","_debug_options_count":4},{"id":19036,"question":"La dérivée de la fonction f(x) = cos(x) est f'(x) = -sin(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une formule de dérivation de base : (cos x)' = -sin 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":19037,"question":"Quelle est la dérivée de la fonction f(x) = √x ?","option_a":"1\/(2√x)","option_b":"√x","option_c":"1\/√x","option_d":"x^(-1\/2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de dérivation des racines : (√x)' = 1\/(2√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\/(2√x)\", \"b\": \"√x\", \"c\": \"1\/√x\", \"d\": \"x^(-1\/2)\"}}","_debug_options_count":4},{"id":19038,"question":"Si f est dérivable en a, alors f est continue en a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème fondamental : la dérivabilité implique la continuité.","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}]