Quiz — 2. Etude de fonction

Question 1 sur 10 20:00

[{"id":5096,"question":"Quelle est la limite de la fonction f(x) = (x² - 1)\/(x - 1) lorsque x tend vers 1 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"On simplifie d'abord f(x) = (x-1)(x+1)\/(x-1) = x+1 pour x ≠ 1. Ainsi, lim(x→1) f(x) = 1+1 = 2.","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\": \"1\", \"c\": \"2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":5097,"question":"La fonction f(x) = x³ - 3x est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction est polynomiale, donc dérivable en tout point, y compris x = 0. Sa dérivée est f'(x) = 3x² - 3.","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":5098,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"e^(2x)","option_b":"2e^(2x)","option_c":"e^(x)","option_d":"2x e^(2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, f'(x) = 2e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"e^(2x)\", \"b\": \"2e^(2x)\", \"c\": \"e^(x)\", \"d\": \"2x e^(2x)\"}}","_debug_options_count":4},{"id":5099,"question":"La fonction f(x) = 1\/x admet-elle une asymptote verticale en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car lim(x→0+) f(x) = +∞ et lim(x→0-) f(x) = -∞. La droite x = 0 est donc une asymptote verticale.","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":5100,"question":"Quelle est la limite de la fonction f(x) = (sin x)\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est un cas classique de limite remarquable : 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\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"La limite n'existe pas\"}}","_debug_options_count":4},{"id":5101,"question":"La fonction f(x) = x² + 2x + 1 est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Sa dérivée f'(x) = 2x + 2 est négative pour x \u003C -1, donc la fonction est décroissante sur ]-∞, -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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":5102,"question":"Quelle est l'asymptote oblique de la fonction f(x) = (x² + 1)\/x ?","option_a":"y = x","option_b":"y = x + 1","option_c":"y = x - 1","option_d":"Aucune asymptote oblique","option_e":"","option_f":"","bonne_reponse":"a","explication":"En effectuant la division euclidienne, on obtient f(x) = x + 1\/x. Ainsi, l'asymptote oblique est y = 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\": \"y = x\", \"b\": \"y = x + 1\", \"c\": \"y = x - 1\", \"d\": \"Aucune asymptot","_debug_options_count":4},{"id":5103,"question":"La fonction f(x) = |x| est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. La fonction valeur absolue n'est pas dérivable en x = 0 car elle présente un point anguleux (dérivée à gauche ≠ dérivée à droite).","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":5104,"question":"Quelle est la valeur de la dérivée de f(x) = ln(x) en x = 1 ?","option_a":"0","option_b":"1","option_c":"e","option_d":"-1","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(x) est f'(x) = 1\/x. Ainsi, f'(1) = 1\/1 = 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\": \"-1\"}}","_debug_options_count":4},{"id":5105,"question":"La fonction f(x) = x^3 - 3x^2 + 2x admet-elle un extremum local en x = 1 ?","option_a":"Oui, un maximum","option_b":"Oui, un minimum","option_c":"Non","option_d":"Oui, un point d'inflexion","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 3x² - 6x + 2 s'annule en x = 1. En étudiant le signe de f'(x), on voit que f'(x) \u003E 0 pour x \u003C 1 et f'(x) \u003C 0 pour x \u003E 1, donc x = 1 est un maximum 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\": \"Oui, un maximum\", \"b\": \"Oui, un minimum\", \"c\": \"Non\", \"d\": \"Oui, ","_debug_options_count":4}]