Quiz — 4. Exponentielle 1 corrections..pdf

Question 1 sur 5 10:00

[{"id":2165,"question":"Quelle est la limite de e^x lorsque x tend vers -∞ ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"-∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle tend vers 0 lorsque x tend vers -∞, car e^x = 1\/e^(-x) et e^(-x) tend vers +∞.","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\": \"1\", \"c\": \"+∞\", \"d\": \"-∞\"}}","_debug_options_count":4},{"id":2166,"question":"Si f(x) = e^(2x), quelle est sa dérivée f'(x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2x·e^(2x)","option_d":"e^(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^u est u'·e^u. Ici, u = 2x, donc u' = 2. Ainsi, f'(x) = 2e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2e^(2x)\", \"b\": \"e^(2x)\", \"c\": \"2x·e^(2x)\", \"d\": \"e^(x)\"}}","_debug_options_count":4},{"id":2167,"question":"Quelle propriété algébrique est vérifiée par e^(a+b) ?","option_a":"e^a + e^b","option_b":"e^a · e^b","option_c":"(e^a)^b","option_d":"e^(a-b)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction exponentielle vérifie e^(a+b) = e^a · e^b, c'est l'une de ses propriétés fondamentales.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"e^a + e^b\", \"b\": \"e^a · e^b\", \"c\": \"(e^a)^b\", \"d\": \"e^(a-b)\"}}","_debug_options_count":4},{"id":2168,"question":"Résolvez l'inéquation e^(x-1) ≤ 1.","option_a":"x ≤ 1","option_b":"x ≥ 1","option_c":"x ≤ 0","option_d":"x ≥ 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"e^(x-1) ≤ 1 ⇔ e^(x-1) ≤ e^0 ⇔ x-1 ≤ 0 (car e^x est croissante) ⇔ x ≤ 1.","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 ≤ 1\", \"b\": \"x ≥ 1\", \"c\": \"x ≤ 0\", \"d\": \"x ≥ 0\"}}","_debug_options_count":4},{"id":2169,"question":"Quelle est la dérivée de la fonction f(x) = x·e^x ?","option_a":"e^x","option_b":"e^x + x·e^x","option_c":"x·e^x","option_d":"e^x - x·e^x","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la formule de dérivation d'un produit : (u·v)' = u'v + uv'. Ici, u = x et v = e^x, donc f'(x) = 1·e^x + x·e^x = e^x(1 + x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"e^x\", \"b\": \"e^x + x·e^x\", \"c\": \"x·e^x\", \"d\": \"e^x - x·e^x\"}}","_debug_options_count":4}]