Quiz — fn2vardiff.pdf

Question 1 sur 10 20:00

[{"id":49950,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"2x + 3","option_b":"x² + 3","option_c":"2x² + 3x","option_d":"x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : la dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 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\": \"2x + 3\", \"b\": \"x² + 3\", \"c\": \"2x² + 3x\", \"d\": \"x + 3\"}}","_debug_options_count":4},{"id":49951,"question":"Si f'(x) = 0 pour tout x dans un intervalle, alors la fonction f est constante sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un résultat fondamental du théorème de Rolle : si la dérivée est nulle sur un intervalle, la fonction y est constante.","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":49952,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"e^(2x)","option_b":"2e^(2x)","option_c":"e^x","option_d":"2e^x","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée d'une fonction exponentielle e^(u(x)) est u'(x)e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 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\": \"e^(2x)\", \"b\": \"2e^(2x)\", \"c\": \"e^x\", \"d\": \"2e^x\"}}","_debug_options_count":4},{"id":49953,"question":"La fonction f(x) = |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 valeur absolue n'est pas dérivable en x = 0 car sa courbe présente un point anguleux à cet endroit.","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":49954,"question":"Quelle est la dérivée de la fonction f(x) = ln(x) ?","option_a":"1\/x","option_b":"x","option_c":"1","option_d":"ln(x)\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction logarithme népérien ln(x) est 1\/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\": \"1\/x\", \"b\": \"x\", \"c\": \"1\", \"d\": \"ln(x)\/x\"}}","_debug_options_count":4},{"id":49955,"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":"La dérivabilité implique la continuité : si une fonction est dérivable en un point, elle y est nécessairement continue.","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":49956,"question":"Quelle est la dérivée de la fonction f(x) = sin(3x) ?","option_a":"cos(3x)","option_b":"3cos(3x)","option_c":"sin(3x)","option_d":"cos(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de sin(u(x)) est u'(x)cos(u(x)). Ici, u(x) = 3x, donc u'(x) = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"cos(3x)\", \"b\": \"3cos(3x)\", \"c\": \"sin(3x)\", \"d\": \"cos(x)\"}}","_debug_options_count":4},{"id":49957,"question":"La fonction f(x) = x^(1\/3) 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 f(x) = x^(1\/3) n'est pas dérivable en x = 0 car sa dérivée 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":49958,"question":"Quelle est la dérivée de la fonction f(x) = 1\/x ?","option_a":"-1\/x","option_b":"-1\/x²","option_c":"1\/x²","option_d":"x","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de la fonction f(x) = 1\/x est f'(x) = -1\/x².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"-1\/x\", \"b\": \"-1\/x²\", \"c\": \"1\/x²\", \"d\": \"x\"}}","_debug_options_count":4},{"id":49959,"question":"Si f'(x) \u003E 0 sur un intervalle, alors f est croissante sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un résultat fondamental : si la dérivée est positive sur un intervalle, la fonction y est strictement croissante.","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}]