Quiz — Lycée Pilote Ariana - Fonction Ln.pdf

Question 1 sur 10 20:00

[{"id":87283,"question":"Quelle est la dérivée de f(x) = ln(3x + 2) ?","option_a":"3\/(3x+2)","option_b":"1\/(3x+2)","option_c":"3x\/(3x+2)","option_d":"1\/(x+2\/3)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u) est u'\/u. Ici u = 3x + 2, donc u' = 3. Ainsi f'(x) = 3\/(3x+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\": \"3\/(3x+2)\", \"b\": \"1\/(3x+2)\", \"c\": \"3x\/(3x+2)\", \"d\": \"1\/(x+2\/3)\"}}","_debug_options_count":4},{"id":87284,"question":"La fonction ln est-elle définie pour x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln est définie uniquement pour x \u003E 0. ln(0) n'existe pas.","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":87285,"question":"Quelle est la limite de ln(x)\/x lorsque x tend vers +∞ ?","option_a":"0","option_b":"+∞","option_c":"1","option_d":"e","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par croissance comparée, ln(x) croît plus lentement que x. Donc ln(x)\/x → 0 quand x → +∞.","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\": \"1\", \"d\": \"e\"}}","_debug_options_count":4},{"id":87286,"question":"Si ln(x) = 2, quelle est la valeur de x ?","option_a":"e^2","option_b":"2e","option_c":"ln(2)","option_d":"e\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(x) = 2 ⇒ x = e^2, car ln est la fonction réciproque de l'exponentielle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"e^2\", \"b\": \"2e\", \"c\": \"ln(2)\", \"d\": \"e\/2\"}}","_debug_options_count":4},{"id":87287,"question":"La fonction ln est-elle croissante sur son ensemble de définition ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(x) est 1\/x \u003E 0 pour x \u003E 0, donc ln est strictement croissante.","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":87288,"question":"Quelle propriété permet d'écrire ln(a^b) = b·ln(a) ?","option_a":"Propriété de l'exponentielle","option_b":"Propriété de la croissance","option_c":"Propriété logarithmique","option_d":"Propriété de la dérivée","option_e":"","option_f":"","bonne_reponse":"c","explication":"C'est une propriété fondamentale des logarithmes : ln(a^b) = b·ln(a) pour a \u003E 0 et b réel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"Propriété de l'exponentielle\", \"b\": \"Propriété de la croissan","_debug_options_count":4},{"id":87289,"question":"Résoudre l'équation ln(x-1) + ln(x+1) = ln(3).","option_a":"x = 2","option_b":"x = -2","option_c":"x = 1","option_d":"x = 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln((x-1)(x+1)) = ln(3) ⇒ (x-1)(x+1) = 3 ⇒ x² - 1 = 3 ⇒ x² = 4 ⇒ x = 2 (car x \u003E 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 = 2\", \"b\": \"x = -2\", \"c\": \"x = 1\", \"d\": \"x = 0\"}}","_debug_options_count":4},{"id":87290,"question":"La fonction ln est-elle paire ou impaire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"ln(-x) n'est pas défini pour x \u003E 0, donc ln n'est ni paire ni impaire.","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":87291,"question":"Quelle est la dérivée de f(x) = x·ln(x) ?","option_a":"ln(x) + 1","option_b":"1\/x","option_c":"x + ln(x)","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Utiliser la dérivée d'un produit : f'(x) = ln(x) + x·(1\/x) = ln(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\": \"ln(x) + 1\", \"b\": \"1\/x\", \"c\": \"x + ln(x)\", \"d\": \"1\"}}","_debug_options_count":4},{"id":87292,"question":"Si ln(2x) = ln(5) + ln(3), quelle est la valeur de x ?","option_a":"x = 15\/2","option_b":"x = 6","option_c":"x = 7.5","option_d":"x = 5\/3","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(2x) = ln(15) ⇒ 2x = 15 ⇒ x = 15\/2. Utilisation de ln(ab) = ln(a) + ln(b).","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 = 15\/2\", \"b\": \"x = 6\", \"c\": \"x = 7.5\", \"d\": \"x = 5\/3\"}}","_debug_options_count":4}]