Quiz — 6537d6fe504e2_Corrigé_Série d'exercices n 11-12.pdf

Question 1 sur 10 20:00

[{"id":31640,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"f'(x) = 6x - 5","option_b":"f'(x) = 3x - 5","option_c":"f'(x) = 6x² - 5x","option_d":"f'(x) = 6x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle de dérivation terme à terme : f'(x) = 6x - 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f'(x) = 6x - 5\", \"b\": \"f'(x) = 3x - 5\", \"c\": \"f'(x) = 6x² - 5x\",","_debug_options_count":4},{"id":31641,"question":"La limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 4) lorsque x tend vers l'infini est égale à 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En divisant numérateur et dénominateur par x², on obtient f(x) = (2 + 3\/x - 1\/x²)\/(1 - 4\/x²). La limite est donc 2\/1 = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":31642,"question":"Quelle est la limite de la fonction f(x) = (x - 1)\/(x² - 1) lorsque x tend vers 1 ?","option_a":"0","option_b":"1","option_c":"Indéterminée","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La limite est indéterminée car on obtient 0\/0. Il faut factoriser : f(x) = (x - 1)\/((x - 1)(x + 1)) = 1\/(x + 1) pour x ≠ 1. La limite est donc 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\": \"Indéterminée\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":31643,"question":"La dérivée de la fonction f(x) = ln(x) est f'(x) = 1\/x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(x) est bien 1\/x, une propriété fondamentale en analyse.","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":31644,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x³ - 2x + 1 au point d'abscisse x = 1 ?","option_a":"y = x","option_b":"y = x + 1","option_c":"y = x - 1","option_d":"y = 3x - 3","option_e":"","option_f":"","bonne_reponse":"d","explication":"La tangente a pour équation y = f'(1)(x - 1) + f(1). f'(x) = 3x² - 2, donc f'(1) = 1. f(1) = 0. L'équation est y = 1(x - 1) + 0 = x - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"y = x\", \"b\": \"y = x + 1\", \"c\": \"y = x - 1\", \"d\": \"y = 3x - 3\"}}","_debug_options_count":4},{"id":31645,"question":"La suite définie par uₙ = (n² + 1)\/n converge vers 0 lorsque n tend vers l'infini.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La suite uₙ = (n² + 1)\/n = n + 1\/n tend vers l'infini, donc elle ne converge pas vers 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":31646,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"f'(x) = e^(2x)","option_b":"f'(x) = 2e^(2x)","option_c":"f'(x) = e^x","option_d":"f'(x) = 2e^x","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\": \"f'(x) = e^(2x)\", \"b\": \"f'(x) = 2e^(2x)\", \"c\": \"f'(x) = e^x\", \"d\":","_debug_options_count":4},{"id":31647,"question":"La fonction f(x) = 1\/x est 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":31648,"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":"Indéterminée","option_e":"","option_f":"","bonne_reponse":"b","explication":"La limite de (sin(x))\/x lorsque x tend vers 0 est une limite fondamentale en analyse, égale à 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":31649,"question":"La dérivée de la fonction f(x) = √x est f'(x) = 1\/(2√x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de √x est bien 1\/(2√x), une propriété essentielle des fonctions racines carrées.","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}]