Quiz — سلسلة تمارين في الدالة الوغاريتم النيبيري.docx

Question 1 sur 10 20:00

[{"id":91972,"question":"Quelle est la valeur de ln(e^5) ?","option_a":"5","option_b":"e^5","option_c":"1","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(e^5) = 5·ln(e) = 5·1 = 5, car ln(e) = 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\": \"5\", \"b\": \"e^5\", \"c\": \"1\", \"d\": \"0\"}}","_debug_options_count":4},{"id":91973,"question":"La fonction ln(x) est définie pour x \u003C 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln(x) n'est définie que pour x \u003E 0.","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":91974,"question":"Quelle est la dérivée de f(x) = ln(3x + 1) ?","option_a":"3\/(3x+1)","option_b":"1\/(3x+1)","option_c":"3x\/(3x+1)","option_d":"1\/(x+1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées : f'(x) = 3\/(3x+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\": \"3\/(3x+1)\", \"b\": \"1\/(3x+1)\", \"c\": \"3x\/(3x+1)\", \"d\": \"1\/(x+1)\"}}","_debug_options_count":4},{"id":91975,"question":"Résoudre l'inéquation ln(x) \u003E 2.","option_a":"x \u003E e^2","option_b":"x \u003E 2","option_c":"x \u003C e^2","option_d":"x \u003C 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(x) \u003E 2 ⇨ x \u003E e^2, car la fonction 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\": \"x \u003E e^2\", \"b\": \"x \u003E 2\", \"c\": \"x \u003C e^2\", \"d\": \"x \u003C 2\"}}","_debug_options_count":4},{"id":91976,"question":"La limite de ln(x) quand x tend vers 0+ est -∞.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En effet, ln(x) tend vers -∞ lorsque x tend vers 0 par la droite.","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":91977,"question":"Quelle est la valeur de ln(1) ?","option_a":"0","option_b":"1","option_c":"e","option_d":"-1","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(1) = 0, car e^0 = 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\": \"0\", \"b\": \"1\", \"c\": \"e\", \"d\": \"-1\"}}","_debug_options_count":4},{"id":91978,"question":"La fonction ln(x) est-elle convexe sur son domaine de définition ?","option_a":"Oui","option_b":"Non","option_c":"Parfois","option_d":"On ne peut pas savoir","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde de ln(x) est -1\/x² \u003C 0, donc la fonction est concave (et non convexe).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Oui\", \"b\": \"Non\", \"c\": \"Parfois\", \"d\": \"On ne peut pas savoir\"}}","_debug_options_count":4},{"id":91979,"question":"Résoudre l'équation ln(x^2 - 5) = 0.","option_a":"x = ±√6","option_b":"x = 6","option_c":"x = √6","option_d":"x = -√6","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(x^2 - 5) = 0 ⇨ x^2 - 5 = 1 ⇨ x^2 = 6 ⇨ x = ±√6.","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 = ±√6\", \"b\": \"x = 6\", \"c\": \"x = √6\", \"d\": \"x = -√6\"}}","_debug_options_count":4},{"id":91980,"question":"La fonction ln(x) est-elle périodique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln(x) n'est pas périodique, elle est strictement croissante.","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":91981,"question":"Quelle est la limite de (ln(x))\/x quand x tend vers +∞ ?","option_a":"0","option_b":"1","option_c":"+∞","option_d":"-∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par croissance comparée, (ln(x))\/x tend vers 0 quand 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}]