Quiz — MethodiX analyse.pdf

Question 1 sur 10 20:00

[{"id":118381,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"A. 0","option_b":"B. 3","option_c":"C. -1","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite d'un polynôme de même degré au numérateur et au dénominateur est le rapport des coefficients dominants. Ici, 3\/1 = 3.","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. 0\", \"b\": \"B. 3\", \"c\": \"C. -1\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":118382,"question":"La fonction f(x) = x³ - 3x² + 2x est-elle continue sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les fonctions polynômes sont continues sur ℝ, car elles sont dérivables partout.","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":118383,"question":"Quelle est la dérivée de la fonction f(x) = ln(2x + 1) ?","option_a":"A. 1\/(2x + 1)","option_b":"B. 2\/(2x + 1)","option_c":"C. 1\/(x + 1)","option_d":"D. 2\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 2x + 1, donc u' = 2. Ainsi, f'(x) = 2\/(2x + 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\": \"A. 1\/(2x + 1)\", \"b\": \"B. 2\/(2x + 1)\", \"c\": \"C. 1\/(x + 1)\", \"d\": \"","_debug_options_count":4},{"id":118384,"question":"La fonction f(x) = x² - 4x + 3 admet-elle un minimum ? Si oui, quelle est sa valeur ?","option_a":"A. Non","option_b":"B. Oui, -1","option_c":"C. Oui, 1","option_d":"D. Oui, -4","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction est une parabole ouverte vers le haut, elle admet un minimum en x = -b\/(2a) = 2. f(2) = -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\": \"A. Non\", \"b\": \"B. Oui, -1\", \"c\": \"C. Oui, 1\", \"d\": \"D. Oui, -4\"}}","_debug_options_count":4},{"id":118385,"question":"La fonction f(x) = 1\/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":"La fonction 1\/x n'est pas définie en x = 0, donc elle n'est pas 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":118386,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x² en x = 1 ?","option_a":"A. y = 2x - 1","option_b":"B. y = x + 1","option_c":"C. y = 2x + 1","option_d":"D. y = x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente en x = a a pour équation y = f'(a)(x - a) + f(a). Ici, f'(1) = 2 et f(1) = 1, donc y = 2(x - 1) + 1 = 2x - 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\": \"A. y = 2x - 1\", \"b\": \"B. y = x + 1\", \"c\": \"C. y = 2x + 1\", \"d\": \"","_debug_options_count":4},{"id":118387,"question":"La fonction f(x) = e^x est-elle toujours positive sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle e^x est strictement positive 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":118388,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"A. cos(2x)","option_b":"B. 2cos(2x)","option_c":"C. -2cos(2x)","option_d":"D. sin(2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de sin(u) est u'cos(u). Ici, u = 2x, donc u' = 2. Ainsi, f'(x) = 2cos(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\": \"A. cos(2x)\", \"b\": \"B. 2cos(2x)\", \"c\": \"C. -2cos(2x)\", \"d\": \"D. si","_debug_options_count":4},{"id":118389,"question":"La fonction f(x) = x³ - 3x + 2 admet-elle un point d'inflexion ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un point d'inflexion existe si f''(x) change de signe. Ici, f''(x) = 6x, qui change de signe 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},{"id":118390,"question":"Quelle est la limite de la fonction f(x) = (sin x)\/x lorsque x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. -1","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une limite classique : 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. -1\", \"d\": \"D. +∞\"}}","_debug_options_count":4}]