Quiz — cours_fctusuelles.pdf

Question 1 sur 10 20:00

[{"id":10883,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"A. e^(2x)","option_b":"B. 2e^(2x)","option_c":"C. e^x","option_d":"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.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. e^(2x)\", \"b\": \"B. 2e^(2x)\", \"c\": \"C. e^x\", \"d\": \"D. 2x e^(2x)\"","_debug_options_count":4},{"id":10884,"question":"La fonction f(x) = ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln(x) est définie uniquement pour x \u003E 0.","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":10885,"question":"Quelle est la limite de (x² - 1)\/(x - 1) quand x tend vers 1 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"En simplifiant (x² - 1)\/(x - 1) = (x - 1)(x + 1)\/(x - 1) = x + 1 pour x ≠ 1. La limite est donc 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":10886,"question":"La fonction f(x) = x³ - 3x + 2 admet un extremum local en x = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 3x² - 3. f'(1) = 0, mais f''(1) = 6 \u003E 0, donc c'est un minimum 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},{"id":10887,"question":"Quelle est la dérivée de f(x) = sin(x) * cos(x) ?","option_a":"A. cos²(x) - sin²(x)","option_b":"B. cos(x) - sin(x)","option_c":"C. sin(x) + cos(x)","option_d":"D. 2sin(x)cos(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule (uv)' = u'v + uv', on obtient cos²(x) - sin²(x) = cos(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. cos²(x) - sin²(x)\", \"b\": \"B. cos(x) - sin(x)\", \"c\": \"C. sin(","_debug_options_count":4},{"id":10888,"question":"La fonction f(x) = 1\/x est continue sur son domaine de définition.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction 1\/x est continue sur ℝ*, mais pas en x = 0 (non définie).","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":10889,"question":"Quelle est la solution de l'équation e^(2x) = 5 ?","option_a":"A. x = ln(5)\/2","option_b":"B. x = 5\/2","option_c":"C. x = ln(2)\/5","option_d":"D. x = 2ln(5)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En prenant le logarithme des deux côtés : 2x = ln(5), donc x = ln(5)\/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\": \"A. x = ln(5)\/2\", \"b\": \"B. x = 5\/2\", \"c\": \"C. x = ln(2)\/5\", \"d\": \"","_debug_options_count":4},{"id":10890,"question":"La fonction f(x) = x^4 - 4x³ + 6x² admet un point d'inflexion en x = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f''(x) = 12x² - 24x + 12. f''(1) = 0 et f'''(1) ≠ 0, donc c'est un point d'inflexion.","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":10891,"question":"Quelle est la dérivée de f(x) = ln(x² + 1) ?","option_a":"A. 2x\/(x² + 1)","option_b":"B. 2\/(x² + 1)","option_c":"C. x\/(x² + 1)","option_d":"D. 1\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la dérivée de ln(u(x)) = u'(x)\/u(x), avec u(x) = x² + 1, u'(x) = 2x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. 2x\/(x² + 1)\", \"b\": \"B. 2\/(x² + 1)\", \"c\": \"C. x\/(x² + 1)\", \"","_debug_options_count":4},{"id":10892,"question":"La fonction f(x) = e^x est toujours croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^x est e^x, toujours positive, donc la fonction est strictement croissante.","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}]