Quiz — série22-etude_des_fonctions corrigé.pdf

Question 1 sur 10 20:00

[{"id":30920,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5 ?","option_a":"3x² - 4x","option_b":"x² - 4x + 5","option_c":"3x² - 4x + 5","option_d":"x³ - 4x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la formule (x^n)' = n*x^(n-1). Ici, f'(x) = 3x² - 4x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3x² - 4x\", \"b\": \"x² - 4x + 5\", \"c\": \"3x² - 4x + 5\", \"d\": \"x³ ","_debug_options_count":4},{"id":30921,"question":"La fonction f(x) = (2x + 1)\/(x - 3) admet-elle une asymptote verticale ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une asymptote verticale existe si le dénominateur s'annule sans que le numérateur ne s'annule au même point. Ici, x = 3 est 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":30922,"question":"Quelle est la limite de f(x) = (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, f(x) = (x - 2)(x + 2)\/(x - 2) = x + 2 pour x ≠ 2. Donc lim(x→2) f(x) = 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":30923,"question":"La fonction f(x) = x³ - 3x² + 2 admet-elle un extremum local ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En étudiant le signe de la dérivée f'(x) = 3x² - 6x, on trouve des points critiques où la dérivée s'annule. L'analyse du tableau de variations montre un extremum 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":30924,"question":"Quelle est la dérivée de la fonction f(x) = √(x² + 1) ?","option_a":"x\/√(x² + 1)","option_b":"(x² + 1)\/√(x² + 1)","option_c":"2x\/√(x² + 1)","option_d":"x\/2√(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées, f'(x) = (1\/2)(x² + 1)^(-1\/2) * 2x = x\/√(x² + 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\": \"x\/√(x² + 1)\", \"b\": \"(x² + 1)\/√(x² + 1)\", \"c\": \"2x\/√(x² ","_debug_options_count":4},{"id":30925,"question":"La fonction f(x) = e^x est-elle toujours croissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = e^x est f'(x) = e^x, qui est toujours positive. Donc la fonction est strictement croissante sur ℝ.","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":30926,"question":"Quelle est la limite de f(x) = (sin x)\/x quand x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"indéterminée","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une limite classique 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\": \"indéterminée\"}}","_debug_options_count":4},{"id":30927,"question":"La fonction f(x) = ln(x) est-elle définie pour x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction logarithme népérien ln(x) est définie uniquement pour x \u003E 0. Donc f(0) n'existe pas.","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":30928,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x + 2) ?","option_a":"3\/(3x + 2)","option_b":"1\/(3x + 2)","option_c":"3x\/(3x + 2)","option_d":"1\/(x + 2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées, f'(x) = (1\/(3x + 2)) * 3 = 3\/(3x + 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\": \"3\/(3x + 2)\", \"b\": \"1\/(3x + 2)\", \"c\": \"3x\/(3x + 2)\", \"d\": \"1\/(x + ","_debug_options_count":4},{"id":30929,"question":"La fonction f(x) = x^4 - 4x³ + 6x² 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 la dérivée seconde s'annule en changeant de signe. Ici, f''(x) = 12x² - 24x + 12 = 12(x - 1)², qui ne change pas de signe. Donc pas de 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}]