Quiz — magazine-type-devoir-n01derivabilite-enonce.pdf

Question 1 sur 10 20:00

[{"id":39910,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"A: 2x + 3","option_b":"B: x² + 3","option_c":"C: 2x + 5","option_d":"D: 3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 0. Donc f'(x) = 2x + 3.","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 + 3\", \"b\": \"B: x² + 3\", \"c\": \"C: 2x + 5\", \"d\": \"D: 3x + 2\"","_debug_options_count":4},{"id":39911,"question":"Une fonction peut-elle être continue en un point sans y être dérivable ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, par exemple la fonction f(x) = |x| est continue en 0 mais n'y est pas dérivable.","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":39912,"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":"En utilisant la formule de dérivation des fonctions composées, (sin(2x))' = 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":39913,"question":"La fonction f(x) = x^(1\/3) est-elle dérivable en x = 0 ?","option_a":"A: Oui, f'(0) = 0","option_b":"B: Non, car la tangente est verticale","option_c":"C: Oui, f'(0) = 1\/3","option_d":"D: Non, car la fonction n'est pas continue en 0","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = x^(1\/3) n'est pas dérivable en 0 car sa dérivée tend vers l'infini en ce point (tangente verticale).","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: Oui, f'(0) = 0\", \"b\": \"B: Non, car la tangente est verticale\",","_debug_options_count":4},{"id":39914,"question":"Si f'(x) = 0 pour tout x d'un intervalle, que peut-on dire de f sur cet intervalle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, si la dérivée est nulle sur un intervalle, alors la fonction est constante sur cet intervalle.","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":39915,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"A: 2x\/(x² + 1)","option_b":"B: 1\/(x² + 1)","option_c":"C: 2x","option_d":"D: x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées, f'(x) = (2x)\/(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\": \"A: 2x\/(x² + 1)\", \"b\": \"B: 1\/(x² + 1)\", \"c\": \"C: 2x\", \"d\": \"D: x","_debug_options_count":4},{"id":39916,"question":"La fonction f(x) = x|x| est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car la fonction est égale à x² pour x ≥ 0 et à -x² pour x ≤ 0, donc f'(0) = 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":39917,"question":"Quelle est la dérivée de la fonction f(x) = e^(3x) ?","option_a":"A: e^(3x)","option_b":"B: 3e^(3x)","option_c":"C: e^(x)","option_d":"D: 3e^(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la formule de dérivation des fonctions exponentielles, f'(x) = 3e^(3x).","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^(3x)\", \"b\": \"B: 3e^(3x)\", \"c\": \"C: e^(x)\", \"d\": \"D: 3e^(x)\"}","_debug_options_count":4},{"id":39918,"question":"Si f est dérivable en a, alors f est-elle nécessairement continue en a ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, la dérivabilité en un point implique la continuité 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":39919,"question":"Quelle est la dérivée de la fonction f(x) = (x² + 1)\/(x - 1) ?","option_a":"A: (2x)\/(x - 1)","option_b":"B: (x² - 2x - 1)\/(x - 1)²","option_c":"C: (2x - 1)\/(x - 1)","option_d":"D: (x² + 1)\/(x - 1)²","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la formule de dérivation d'un quotient, f'(x) = [(2x)(x - 1) - (x² + 1)(1)] \/ (x - 1)² = (x² - 2x - 1)\/(x - 1)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: (2x)\/(x - 1)\", \"b\": \"B: (x² - 2x - 1)\/(x - 1)²\", \"c\": \"C: (2","_debug_options_count":4}]