Quiz — DME.pdf

Question 1 sur 10 20:00

[{"id":105225,"question":"Quelle est la dérivée de la fonction f(x) = x³ + 2x² - 5x + 1 ?","option_a":"3x² + 4x - 5","option_b":"x² + 2x - 5","option_c":"3x² + 4x + 1","option_d":"x³ + 2x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une somme est la somme des dérivées. On applique la règle de dérivation des puissances : (x^n)' = n*x^(n-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\": \"3x² + 4x - 5\", \"b\": \"x² + 2x - 5\", \"c\": \"3x² + 4x + 1\", \"d\": \"","_debug_options_count":4},{"id":105226,"question":"La limite de f(x) = (x² - 4)\/(x - 2) quand x tend vers 2 est égale à 4.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction n'est pas définie en x=2, mais la limite existe et vaut 4 (simplification par (x-2) possible).","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":105227,"question":"Quelle est l'intégrale de f(x) = 3x² + 2x entre 0 et 2 ?","option_a":"8","option_b":"12","option_c":"16","option_d":"20","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'intégrale se calcule en trouvant une primitive F(x) = x³ + x², puis en évaluant F(2) - F(0) = 8 + 4 = 12.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"8\", \"b\": \"12\", \"c\": \"16\", \"d\": \"20\"}}","_debug_options_count":4},{"id":105228,"question":"La suite définie par uₙ = (n² + 1)\/n a pour limite quand n tend vers l'infini :","option_a":"0","option_b":"1","option_c":"l'infini","option_d":"n","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient uₙ = n + 1\/n, dont la limite est l'infini. Cependant, si la suite est uₙ = (n² + 1)\/n², la limite est 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\": \"l'infini\", \"d\": \"n\"}}","_debug_options_count":4},{"id":105229,"question":"Si f est dérivable en a, alors f est continue en a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème fondamental : la dérivabilité implique la continuité, mais l'inverse n'est pas toujours vrai.","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":105230,"question":"Quelle est la primitive de f(x) = e^(2x) ?","option_a":"(1\/2)e^(2x) + C","option_b":"e^(2x) + C","option_c":"2e^(2x) + C","option_d":"e^(x) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de e^(kx) est (1\/k)e^(kx) + C. Ici, k=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\": \"(1\/2)e^(2x) + C\", \"b\": \"e^(2x) + C\", \"c\": \"2e^(2x) + C\", \"d\": \"e^","_debug_options_count":4},{"id":105231,"question":"La fonction f(x) = 1\/x a pour asymptote verticale :","option_a":"x = 0","option_b":"y = 0","option_c":"x = 1","option_d":"y = 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction 1\/x n'est pas définie en x=0, ce qui crée une asymptote verticale en ce point.","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 = 0\", \"b\": \"y = 0\", \"c\": \"x = 1\", \"d\": \"y = 1\"}}","_debug_options_count":4},{"id":105232,"question":"La suite uₙ = (-1)^n est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette suite alterne entre -1 et 1, elle n'a pas de limite finie : elle est divergente.","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":105233,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (3x² + 2x) dx ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"c","explication":"La primitive est x³ + x². En évaluant entre 0 et 1, on obtient (1 + 1) - (0 + 0) = 2. Cependant, si l'intégrale est ∫₀¹ (3x² + 2x) dx, le résultat est 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\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":105234,"question":"Si f'(x) \u003E 0 sur un intervalle I, alors f est croissante sur I.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale du cours : le signe de la dérivée détermine la monotonie de la fonction.","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}]