Quiz — تمارين محلولة الدوال الأسية - مصطفاي عبد العزيز.pdf

Question 1 sur 10 20:00

[{"id":6271,"question":"Quelle est la dérivée de la fonction f(x) = e^(3x) ?","option_a":"3e^(3x)","option_b":"e^(3x)","option_c":"3x e^(3x-1)","option_d":"e^(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u'(x) * e^(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\": \"3e^(3x)\", \"b\": \"e^(3x)\", \"c\": \"3x e^(3x-1)\", \"d\": \"e^(x)\"}}","_debug_options_count":4},{"id":6272,"question":"L'équation e^(x) = 5 admet-elle une solution réelle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle est bijective de ℝ vers ]0, +∞[, donc elle admet une unique solution réelle.","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":6273,"question":"Quelle est la limite de e^(x) quand x tend vers -∞ ?","option_a":"0","option_b":"+∞","option_c":"-∞","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, lim(x→-∞) e^(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\": \"0\", \"b\": \"+∞\", \"c\": \"-∞\", \"d\": \"1\"}}","_debug_options_count":4},{"id":6274,"question":"La fonction f(x) = e^(-x) est-elle décroissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Sa dérivée est f'(x) = -e^(-x) \u003C 0 pour tout x, donc la fonction est décroissante.","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":6275,"question":"Quelle propriété des exponentielles est illustrée par l'égalité e^(a+b) = e^a * e^b ?","option_a":"Additivité","option_b":"Multiplicativité","option_c":"Commutativité","option_d":"Associativité","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette égalité montre que l'exponentielle transforme une addition en multiplication.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Additivité\", \"b\": \"Multiplicativité\", \"c\": \"Commutativité\", \"d","_debug_options_count":4},{"id":6276,"question":"La fonction f(x) = 2e^(x) - 3 est-elle toujours positive ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Elle est positive si 2e^(x) - 3 \u003E 0, soit x \u003E ln(1.5). Sinon, elle est négative.","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":6277,"question":"Quelle est la solution de l'équation e^(2x) = 7 ?","option_a":"x = ln(7)\/2","option_b":"x = 7\/2","option_c":"x = ln(3.5)","option_d":"x = 2ln(7)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En passant au logarithme, on obtient 2x = ln(7), donc x = ln(7)\/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 = ln(7)\/2\", \"b\": \"x = 7\/2\", \"c\": \"x = ln(3.5)\", \"d\": \"x = 2ln(7","_debug_options_count":4},{"id":6278,"question":"La fonction exponentielle est-elle définie pour tout nombre réel ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle est définie sur tout ℝ et à valeurs dans ]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":6279,"question":"Quelle est la dérivée de f(x) = e^(x^2 + 1) ?","option_a":"(2x + 1)e^(x^2 + 1)","option_b":"2x e^(x^2 + 1)","option_c":"e^(2x)","option_d":"x^2 e^(x^2 + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^(u(x)) est u'(x) * e^(u(x)). Ici, u(x) = x^2 + 1, donc u'(x) = 2x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"(2x + 1)e^(x^2 + 1)\", \"b\": \"2x e^(x^2 + 1)\", \"c\": \"e^(2x)\", \"d\": ","_debug_options_count":4},{"id":6280,"question":"La fonction f(x) = e^(x) + e^(-x) est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f(-x) = e^(-x) + e^(x) = f(x), donc la fonction 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}]