Quiz — td_reel.pdf

Question 1 sur 10 20:00

[{"id":53667,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x² + 2x","option_d":"3x + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynomiale se calcule terme à terme : (3x²)' = 6x, (2x)' = 2, et (-5)' = 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\": \"6x + 2\", \"b\": \"3x² + 2\", \"c\": \"6x² + 2x\", \"d\": \"3x + 2\"}}","_debug_options_count":4},{"id":53668,"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 vrai sous réserve que f soit continue sur I. Si f n'est pas continue, elle peut avoir des sauts malgré une dérivée positive.","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":53669,"question":"Quelle est la dérivée de la fonction g(x) = sin(2x) ?","option_a":"cos(2x)","option_b":"2cos(2x)","option_c":"2sin(x)","option_d":"cos(x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la formule de dérivation des fonctions composées : (sin(u))' = 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\": \"2sin(x)\", \"d\": \"cos(x)\"}}","_debug_options_count":4},{"id":53670,"question":"Le théorème de Rolle s'applique à une fonction f continue sur [a,b] et dérivable sur ]a,b[ avec f(a) = f(b).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est l'énoncé exact du théorème de Rolle, qui garantit l'existence d'un point c dans ]a,b[ où f'(c) = 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":53671,"question":"Quelle est la dérivée de la fonction h(x) = e^(3x+1) ?","option_a":"3e^(3x+1)","option_b":"e^(3x+1)","option_c":"(3x+1)e^(3x)","option_d":"e^(3x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction exponentielle composée est (e^u)' = u'e^u, ici u = 3x+1 donc u' = 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\": \"3e^(3x+1)\", \"b\": \"e^(3x+1)\", \"c\": \"(3x+1)e^(3x)\", \"d\": \"e^(3x)\"}}","_debug_options_count":4},{"id":53672,"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 : la dérivabilité implique la continuité. Si f n'est pas continue en a, elle ne peut pas y être 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":53673,"question":"Quelle est la dérivée de la fonction k(x) = ln(5x² + 1) ?","option_a":"10x\/(5x²+1)","option_b":"5x\/(5x²+1)","option_c":"10x","option_d":"ln(10x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la formule de dérivation des fonctions logarithmes : (ln(u))' = u'\/u, ici u = 5x²+1 donc u' = 10x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"10x\/(5x²+1)\", \"b\": \"5x\/(5x²+1)\", \"c\": \"10x\", \"d\": \"ln(10x)\"}}","_debug_options_count":4},{"id":53674,"question":"Le théorème des accroissements finis généralise le théorème de Rolle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Le théorème des accroissements finis affirme qu'il existe c dans ]a,b[ tel que f'(c) = (f(b)-f(a))\/(b-a), ce qui inclut le cas f(a)=f(b) du théorème de Rolle.","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":53675,"question":"Quelle est la dérivée de la fonction m(x) = (2x+3)\/(x-1) ?","option_a":"2\/(x-1)²","option_b":"(2x-5)\/(x-1)²","option_c":"2(x-1)\/(2x+3)","option_d":"2","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la formule de dérivation des fonctions rationnelles : (u\/v)' = (u'v - uv')\/v², ici u=2x+3 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\": \"b\", \"options\": {\"a\": \"2\/(x-1)²\", \"b\": \"(2x-5)\/(x-1)²\", \"c\": \"2(x-1)\/(2x+3)\", \"d\": \"2\"","_debug_options_count":4},{"id":53676,"question":"Si f'(x) = 0 pour tout x dans I, alors f est constante sur I.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai si I est un intervalle. Si f' est nulle sur un intervalle, alors f est constante sur cet intervalle (théorème de Lagrange).","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}]