Quiz — Série 6_ dervabilite pdfDONE(DONE).pdf

Question 1 sur 10 20:00

[{"id":77166,"question":"Soit la fonction f(x) = x² + 3x - 5. Quelle est sa dérivée f'(x) ?","option_a":"f'(x) = 2x + 3","option_b":"f'(x) = x² + 3","option_c":"f'(x) = 2x - 5","option_d":"f'(x) = 2x² + 3x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la formule (ax^n)' = n·a·x^(n-1). Ici, 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\": \"f'(x) = 2x + 3\", \"b\": \"f'(x) = x² + 3\", \"c\": \"f'(x) = 2x - 5\", \"","_debug_options_count":4},{"id":77167,"question":"La fonction f(x) = |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 valeur absolue n'est pas dérivable en 0 car elle présente un point anguleux : la dérivée à gauche (-1) diffère de la dérivée à droite (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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":77168,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"f'(x) = cos(2x)","option_b":"f'(x) = 2cos(2x)","option_c":"f'(x) = sin(2x)","option_d":"f'(x) = 2sin(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la règle de dérivation des fonctions composées, f'(x) = cos(2x) * 2 = 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\": \"f'(x) = cos(2x)\", \"b\": \"f'(x) = 2cos(2x)\", \"c\": \"f'(x) = sin(2x)\"","_debug_options_count":4},{"id":77169,"question":"Soit f(x) = (x² + 1)\/(x - 1). Quelle est sa dérivée ?","option_a":"f'(x) = (2x(x-1) - (x²+1))\/(x-1)²","option_b":"f'(x) = (2x)\/(x-1)","option_c":"f'(x) = (x²+1)\/(x-1)²","option_d":"f'(x) = 2x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v². Ici, u = x² + 1 et v = x - 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\": \"f'(x) = (2x(x-1) - (x²+1))\/(x-1)²\", \"b\": \"f'(x) = (2x)\/(x-1)\", ","_debug_options_count":4},{"id":77170,"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 : toute fonction dérivable en un point est nécessairement continue 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":77171,"question":"Quelle est la dérivée de la fonction f(x) = e^(3x) ?","option_a":"f'(x) = e^(3x)","option_b":"f'(x) = 3e^(3x)","option_c":"f'(x) = 3x·e^(3x)","option_d":"f'(x) = e^(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En appliquant la règle de dérivation des fonctions exponentielles composées, f'(x) = 3·e^(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\": \"f'(x) = e^(3x)\", \"b\": \"f'(x) = 3e^(3x)\", \"c\": \"f'(x) = 3x·e^(3x)","_debug_options_count":4},{"id":77172,"question":"Soit f(x) = ln(x² + 1). Quelle est sa dérivée ?","option_a":"f'(x) = 2x\/(x² + 1)","option_b":"f'(x) = 1\/(x² + 1)","option_c":"f'(x) = 2x·ln(x² + 1)","option_d":"f'(x) = x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la règle de dérivation des fonctions composées : f'(x) = (1\/(x² + 1)) * 2x = 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\": \"f'(x) = 2x\/(x² + 1)\", \"b\": \"f'(x) = 1\/(x² + 1)\", \"c\": \"f'(x) = ","_debug_options_count":4},{"id":77173,"question":"La fonction f(x) = x^(1\/3) 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 f(x) = x^(1\/3) n'est pas dérivable en 0 car sa dérivée f'(x) = (1\/3)x^(-2\/3) tend vers l'infini lorsque x tend 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":77174,"question":"Quelle est la dérivée de la fonction f(x) = tan(x) ?","option_a":"f'(x) = 1 + tan²(x)","option_b":"f'(x) = -1 - tan²(x)","option_c":"f'(x) = 1 - tan²(x)","option_d":"f'(x) = cos²(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de tan(x) est 1 + tan²(x), ce qui équivaut aussi à 1\/cos²(x).","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) = 1 + tan²(x)\", \"b\": \"f'(x) = -1 - tan²(x)\", \"c\": \"f'(x) ","_debug_options_count":4},{"id":77175,"question":"Si f'(x) \u003E 0 sur un intervalle I, alors f est strictement croissante sur I.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une conséquence directe du théorème de la dérivée : si f'(x) \u003E 0 sur I, alors f est strictement croissante sur I.","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}]