Quiz — 26_EtudeDeFonctions.pdf

Question 1 sur 10 20:00

[{"id":83562,"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. x + 3","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. x + 3\"}","_debug_options_count":4},{"id":83563,"question":"Une fonction dérivable est toujours continue.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction dérivable est continue, mais la réciproque n'est pas toujours vraie.","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":83564,"question":"Quelle est la limite de f(x) = (x² - 1)\/(x - 1) lorsque x tend vers 1 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. Indéterminée","option_e":"","option_f":"","bonne_reponse":"c","explication":"En simplifiant f(x) = (x-1)(x+1)\/(x-1) = x+1 pour x ≠ 1. Donc lim(x→1) f(x) = 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. 2\", \"d\": \"D. Indéterminée\"}}","_debug_options_count":4},{"id":83565,"question":"Quelle est la dérivée de la fonction f(x) = sin(x) ?","option_a":"A. cos(x)","option_b":"B. -cos(x)","option_c":"C. sin(x)","option_d":"D. -sin(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin(x) est cos(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\": \"A. cos(x)\", \"b\": \"B. -cos(x)\", \"c\": \"C. sin(x)\", \"d\": \"D. -sin(x)","_debug_options_count":4},{"id":83566,"question":"Une fonction peut avoir une asymptote verticale si elle n'est pas définie en un point.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si une fonction n'est pas définie en un point, elle peut avoir 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":83567,"question":"Quelle est la dérivée de la fonction f(x) = ln(x) ?","option_a":"A. 1\/x","option_b":"B. x","option_c":"C. e^x","option_d":"D. 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de 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\": \"A. 1\/x\", \"b\": \"B. x\", \"c\": \"C. e^x\", \"d\": \"D. 1\"}}","_debug_options_count":4},{"id":83568,"question":"Quelle est la limite de f(x) = (x³ - 8)\/(x - 2) lorsque x tend vers 2 ?","option_a":"A. 0","option_b":"B. 4","option_c":"C. 12","option_d":"D. Indéterminée","option_e":"","option_f":"","bonne_reponse":"c","explication":"En factorisant, f(x) = (x-2)(x²+2x+4)\/(x-2) = x²+2x+4 pour x ≠ 2. Donc lim(x→2) f(x) = 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\": \"A. 0\", \"b\": \"B. 4\", \"c\": \"C. 12\", \"d\": \"D. Indéterminée\"}}","_debug_options_count":4},{"id":83569,"question":"Une fonction peut avoir une asymptote horizontale si sa limite en l'infini est finie.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si lim(x→±∞) f(x) = L (L fini), alors la droite y = L est une asymptote horizontale.","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":83570,"question":"Quelle est la dérivée de la fonction f(x) = e^x ?","option_a":"A. e^x","option_b":"B. x*e^(x-1)","option_c":"C. 1","option_d":"D. 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^x est e^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\": \"A. e^x\", \"b\": \"B. x*e^(x-1)\", \"c\": \"C. 1\", \"d\": \"D. 0\"}}","_debug_options_count":4},{"id":83571,"question":"Quelle est la limite de f(x) = (√(x+1) - 1)\/x lorsque x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1\/2","option_c":"C. 1","option_d":"D. Indéterminée","option_e":"","option_f":"","bonne_reponse":"b","explication":"En multipliant par le conjugué, on obtient lim(x→0) (x+1-1)\/(x(√(x+1)+1)) = lim(x→0) 1\/(√(x+1)+1) = 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. 0\", \"b\": \"B. 1\/2\", \"c\": \"C. 1\", \"d\": \"D. Indéterminée\"}}","_debug_options_count":4}]