Quiz — 62b589d855852_Magazine N°8 - ( Dérivabilité ).pdf

Question 1 sur 10 20:00

[{"id":99705,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"2x + 3","option_b":"x² + 3","option_c":"2x + 5","option_d":"x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme se calcule terme à 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\": \"2x + 5\", \"d\": \"x + 3\"}}","_debug_options_count":4},{"id":99706,"question":"Une fonction peut-elle être dérivable en un point sans y être continue ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si une fonction est dérivable en un point, elle y est automatiquement continue. La continuité est une condition nécessaire (mais non suffisante) à la dérivabilité.","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":99707,"question":"Quelle est la dérivée de f(x) = sin(2x) ?","option_a":"cos(2x)","option_b":"2cos(2x)","option_c":"cos(x)","option_d":"2sin(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la règle de dérivation des fonctions composées, la dérivée de sin(u) est u'cos(u). Ici, u = 2x, donc u' = 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\": \"cos(2x)\", \"b\": \"2cos(2x)\", \"c\": \"cos(x)\", \"d\": \"2sin(x)\"}}","_debug_options_count":4},{"id":99708,"question":"La fonction f(x) = |x| est-elle dérivable en x = 0 ?","option_a":"Oui","option_b":"Non","option_c":"Parfois","option_d":"On ne peut pas savoir","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction valeur absolue n'est pas dérivable en 0 car le taux d'accroissement n'a pas de limite à gauche et à droite 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\": \"Oui\", \"b\": \"Non\", \"c\": \"Parfois\", \"d\": \"On ne peut pas savoir\"}}","_debug_options_count":4},{"id":99709,"question":"Quelle est la dérivée de f(x) = ln(x) ?","option_a":"1\/x","option_b":"x","option_c":"e^x","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction logarithme népérien est 1\/x pour x \u003E 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\": \"1\/x\", \"b\": \"x\", \"c\": \"e^x\", \"d\": \"1\"}}","_debug_options_count":4},{"id":99710,"question":"La dérivée d'une fonction constante est toujours nulle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En effet, une fonction constante f(x) = k a pour dérivée f'(x) = 0 pour tout 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":99711,"question":"Quelle est la dérivée de f(x) = e^(3x) ?","option_a":"e^(3x)","option_b":"3e^(3x)","option_c":"e^x","option_d":"3x e^(3x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la règle de dérivation des fonctions exponentielles composées, la dérivée de e^(u) est u'e^(u). Ici, u = 3x, donc u' = 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\": \"e^(3x)\", \"b\": \"3e^(3x)\", \"c\": \"e^x\", \"d\": \"3x e^(3x)\"}}","_debug_options_count":4},{"id":99712,"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 le taux d'accroissement 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":99713,"question":"Quelle est la dérivée de f(x) = (x² + 1)\/(x - 1) ?","option_a":"(2x(x-1) - (x²+1))\/(x-1)²","option_b":"(2x(x-1) + (x²+1))\/(x-1)²","option_c":"(x²+1)\/(x-1)²","option_d":"2x\/(x-1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la règle 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\": \"(2x(x-1) - (x²+1))\/(x-1)²\", \"b\": \"(2x(x-1) + (x²+1))\/(x-1)²\",","_debug_options_count":4},{"id":99714,"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 du cours : si la dérivée est positive sur un intervalle, la fonction y est 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}]