Quiz — Fonctions exponentielles.pdf

Question 1 sur 5 10:00

[{"id":507,"question":"Quelle est la dérivée de la fonction f(x) = e^(3x) ?","option_a":"A) 3e^(3x)","option_b":"B) e^(3x)","option_c":"C) 3x e^(3x-1)","option_d":"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. Ainsi, f'(x) = 3e^(3x).","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) 3e^(3x)\", \"b\": \"B) e^(3x)\", \"c\": \"C) 3x e^(3x-1)\", \"d\": \"D) e^","_debug_options_count":4},{"id":508,"question":"Résolvez l'équation : 2^(x+1) = 8.","option_a":"A) x = 1","option_b":"B) x = 2","option_c":"C) x = 3","option_d":"D) x = 0","option_e":"","option_f":"","bonne_reponse":"b","explication":"8 peut s'écrire 2^3. L'équation devient 2^(x+1) = 2^3. Comme les bases sont égales, on a x+1 = 3, donc x = 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) x = 1\", \"b\": \"B) x = 2\", \"c\": \"C) x = 3\", \"d\": \"D) x = 0\"}}","_debug_options_count":4},{"id":509,"question":"Quelle est la limite de (e^x - 1)\/x lorsque x tend vers 0 ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) e","option_d":"D) +∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une forme indéterminée 0\/0. En utilisant la règle de l'Hôpital ou le développement limité de e^x, on trouve que la limite vaut 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\": \"A) 0\", \"b\": \"B) 1\", \"c\": \"C) e\", \"d\": \"D) +∞\"}}","_debug_options_count":4},{"id":510,"question":"Soit f(x) = 5 * e^(-2x). Quelle est la valeur de f'(0) ?","option_a":"A) -10","option_b":"B) 10","option_c":"C) -5","option_d":"D) 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x) = 5e^(-2x) est f'(x) = 5 * (-2) * e^(-2x) = -10e^(-2x). En x=0, f'(0) = -10.","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) -10\", \"b\": \"B) 10\", \"c\": \"C) -5\", \"d\": \"D) 5\"}}","_debug_options_count":4},{"id":511,"question":"La fonction g(x) = (1\/2)^x est-elle croissante ou décroissante ?","option_a":"A) Croissante","option_b":"B) Décroissante","option_c":"C) Constante","option_d":"D) Non définie","option_e":"","option_f":"","bonne_reponse":"b","explication":"La base de l'exponentielle est 1\/2, qui est comprise entre 0 et 1. Les fonctions exponentielles de base a \u003C 1 sont décroissantes sur ℝ.","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) Croissante\", \"b\": \"B) Décroissante\", \"c\": \"C) Constante\", \"d\"","_debug_options_count":4}]