Quiz — Exponentielle 2015.pdf

Question 1 sur 10 20:00

[{"id":15249,"question":"Quelle est la dérivée de la fonction f(x) = exp(3x) ?","option_a":"3exp(3x)","option_b":"exp(3x)","option_c":"3xexp(3x)","option_d":"exp(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de exp(u(x)) est u'(x)×exp(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\": \"a\", \"options\": {\"a\": \"3exp(3x)\", \"b\": \"exp(3x)\", \"c\": \"3xexp(3x)\", \"d\": \"exp(x)\"}}","_debug_options_count":4},{"id":15250,"question":"L'équation exp(x) = 0 admet-elle une solution réelle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction exponentielle est toujours strictement positive sur ℝ, donc exp(x)=0 n'a pas de solution.","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":15251,"question":"Quelle est la valeur de exp(ln(5)) ?","option_a":"5","option_b":"ln(5)","option_c":"e^5","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, exp(ln(x)) = x pour tout 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\": \"5\", \"b\": \"ln(5)\", \"c\": \"e^5\", \"d\": \"1\"}}","_debug_options_count":4},{"id":15252,"question":"La fonction exponentielle est-elle toujours croissante ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Sa dérivée exp(x) est toujours positive, donc la fonction exponentielle est strictement croissante sur ℝ.","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":15253,"question":"Résoudre l'équation exp(2x-1) = e^3.","option_a":"x=2","option_b":"x=1","option_c":"x=3","option_d":"x=4","option_e":"","option_f":"","bonne_reponse":"a","explication":"exp(2x-1)=exp(3) ⇒ 2x-1=3 ⇒ 2x=4 ⇒ x=2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x=2\", \"b\": \"x=1\", \"c\": \"x=3\", \"d\": \"x=4\"}}","_debug_options_count":4},{"id":15254,"question":"Quelle est la limite de exp(x)\/x quand x tend vers +∞ ?","option_a":"0","option_b":"+∞","option_c":"1","option_d":"e","option_e":"","option_f":"","bonne_reponse":"b","explication":"Par croissance comparée, exp(x) domine tout polynôme, donc exp(x)\/x tend vers +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"+∞\", \"c\": \"1\", \"d\": \"e\"}}","_debug_options_count":4},{"id":15255,"question":"La fonction f(x) = exp(-x²) est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f(-x) = exp(-(-x)²) = exp(-x²) = f(x), donc f est paire.","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":15256,"question":"Quelle est la solution de l'équation exp(x) + exp(-x) = 2 ?","option_a":"x=0","option_b":"x=1","option_c":"x=-1","option_d":"x=2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Posons y=exp(x), l'équation devient y + 1\/y = 2 ⇒ y²-2y+1=0 ⇒ (y-1)²=0 ⇒ y=1 ⇒ x=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\": \"x=0\", \"b\": \"x=1\", \"c\": \"x=-1\", \"d\": \"x=2\"}}","_debug_options_count":4},{"id":15257,"question":"La fonction exponentielle est-elle bijective de ℝ vers ℝ+* ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Elle est continue, strictement croissante, et ses limites en ±∞ sont 0 et +∞, donc elle est bijective de ℝ vers ℝ+*.","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":15258,"question":"Calculer la dérivée de f(x) = x²exp(x).","option_a":"2xexp(x) + x²exp(x)","option_b":"2xexp(x)","option_c":"xexp(x) + x²exp(x)","option_d":"(2x+x²)exp(x)","option_e":"","option_f":"","bonne_reponse":"d","explication":"Utilisez la formule de dérivation d'un produit : (uv)'=u'v+uv'. Ici u=x² et v=exp(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"2xexp(x) + x²exp(x)\", \"b\": \"2xexp(x)\", \"c\": \"xexp(x) + x²exp(x)","_debug_options_count":4}]