Quiz — 6717930110c36_énoncé_Trigonométrie et Généralités sur les fonctions.pdf

Question 1 sur 10 20:00

[{"id":6659,"question":"Quelle est la dérivée de la fonction f(x) = sin(3x) ?","option_a":"3cos(3x)","option_b":"cos(3x)","option_c":"3sin(3x)","option_d":"cos(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin(u) est u'cos(u). Ici, u = 3x, donc u' = 3. Ainsi, f'(x) = 3cos(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3cos(3x)\", \"b\": \"cos(3x)\", \"c\": \"3sin(3x)\", \"d\": \"cos(x)\"}}","_debug_options_count":4},{"id":6660,"question":"La fonction f(x) = x² + 1 est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction est paire si f(-x) = f(x). Ici, f(-x) = (-x)² + 1 = x² + 1 = f(x). Donc la fonction est paire.","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":6661,"question":"Quelle est la solution principale de l'équation cos(x) = -1\/2 dans l'intervalle [0, π] ?","option_a":"π\/3","option_b":"2π\/3","option_c":"π\/6","option_d":"5π\/6","option_e":"","option_f":"","bonne_reponse":"b","explication":"Dans [0, π], cos(x) = -1\/2 pour x = 2π\/3 (car cos(2π\/3) = -1\/2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"π\/3\", \"b\": \"2π\/3\", \"c\": \"π\/6\", \"d\": \"5π\/6\"}}","_debug_options_count":4},{"id":6662,"question":"La fonction f(x) = tan(x) est-elle définie pour x = π\/2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction tangente n'est pas définie pour x = π\/2 + kπ (k ∈ ℤ), car cos(x) = 0 à ces points, ce qui rend le dénominateur de tan(x) = sin(x)\/cos(x) nul.","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":6663,"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 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\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"La limite n'existe pas\"}}","_debug_options_count":4},{"id":6664,"question":"La fonction f(x) = x³ - 3x est-elle dérivable en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction polynôme est dérivable sur ℝ. Ici, f'(x) = 3x² - 3, donc f est dérivable en 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":6665,"question":"Quelle est la période de la fonction f(x) = sin(2x) + cos(3x) ?","option_a":"2π","option_b":"π","option_c":"6π","option_d":"La fonction n'est pas périodique","option_e":"","option_f":"","bonne_reponse":"c","explication":"La période de sin(2x) est π et celle de cos(3x) est 2π\/3. La période de la somme est le PPCM de π et 2π\/3, soit 2π.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"2π\", \"b\": \"π\", \"c\": \"6π\", \"d\": \"La fonction n'est pas périodi","_debug_options_count":4},{"id":6666,"question":"L'équation sin(x) = 2 a-t-elle des solutions réelles ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction sinus prend des valeurs dans l'intervalle [-1, 1]. Comme 2 ∉ [-1, 1], l'équation n'a pas de solution réelle.","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":6667,"question":"Quelle est la dérivée de la fonction f(x) = ln(cos(x)) ?","option_a":"-sin(x)\/cos(x)","option_b":"tan(x)","option_c":"1\/cos(x)","option_d":"-tan(x)","option_e":"","option_f":"","bonne_reponse":"d","explication":"En utilisant la dérivée des fonctions composées : f'(x) = (1\/cos(x)) * (-sin(x)) = -tan(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"-sin(x)\/cos(x)\", \"b\": \"tan(x)\", \"c\": \"1\/cos(x)\", \"d\": \"-tan(x)\"}}","_debug_options_count":4},{"id":6668,"question":"La fonction f(x) = e^(-x²) admet-elle un maximum en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = -2x e^(-x²). f'(0) = 0 et f''(0) = -2 \u003C 0, donc f admet un maximum local en x = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]