Quiz — 65d628eeb2daf_Corrigé-Fonction ln.pdf

Question 1 sur 10 20:00

[{"id":13784,"question":"Quelle est la valeur de ln(e^3) ?","option_a":"3","option_b":"e^3","option_c":"1\/3","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(e^3) = 3 car ln et exp sont des fonctions réciproques.","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\", \"b\": \"e^3\", \"c\": \"1\/3\", \"d\": \"0\"}}","_debug_options_count":4},{"id":13785,"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.","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":13786,"question":"Quelle est la dérivée de f(x) = ln(2x + 1) ?","option_a":"1\/(2x + 1)","option_b":"2\/(2x + 1)","option_c":"1\/x","option_d":"2x\/(2x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"Par la règle de dérivation des fonctions composées : f'(x) = (2)\/(2x + 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\": \"1\/(2x + 1)\", \"b\": \"2\/(2x + 1)\", \"c\": \"1\/x\", \"d\": \"2x\/(2x + 1)\"}}","_debug_options_count":4},{"id":13787,"question":"Si ln(x) = 2, alors x est égal à :","option_a":"2","option_b":"e^2","option_c":"1\/e^2","option_d":"ln(2)","option_e":"","option_f":"","bonne_reponse":"b","explication":"ln(x) = 2 ⇨ x = e^2 (propriété d'injectivité de ln).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\", \"b\": \"e^2\", \"c\": \"1\/e^2\", \"d\": \"ln(2)\"}}","_debug_options_count":4},{"id":13788,"question":"La fonction ln est-elle convexe sur son domaine de définition ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde de ln(x) est -1\/x^2 \u003C 0, donc ln est concave.","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":13789,"question":"Quelle est l'expression simplifiée de ln(8) - ln(2) ?","option_a":"ln(4)","option_b":"ln(6)","option_c":"ln(16)","option_d":"4","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(8) - ln(2) = ln(8\/2) = ln(4) (propriété des logarithmes).","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(4)\", \"b\": \"ln(6)\", \"c\": \"ln(16)\", \"d\": \"4\"}}","_debug_options_count":4},{"id":13790,"question":"La limite de ln(x) quand x tend vers 0+ est :","option_a":"0","option_b":"+∞","option_c":"-∞","option_d":"1","option_e":"","option_f":"","bonne_reponse":"c","explication":"lim_{x→0+} ln(x) = -∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"0\", \"b\": \"+∞\", \"c\": \"-∞\", \"d\": \"1\"}}","_debug_options_count":4},{"id":13791,"question":"Si f(x) = ln(x^2), alors f'(x) = :","option_a":"2\/x","option_b":"1\/x","option_c":"2x","option_d":"x\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"f(x) = 2ln(x) ⇒ f'(x) = 2\/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\": \"2\/x\", \"b\": \"1\/x\", \"c\": \"2x\", \"d\": \"x\/2\"}}","_debug_options_count":4},{"id":13792,"question":"L'équation ln(x) = x - 1 admet-elle une solution ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, x = 1 est une solution évidente (ln(1) = 0 et 1 - 1 = 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":13793,"question":"Quelle est la valeur de ln(√e) ?","option_a":"1\/2","option_b":"2","option_c":"√e","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"ln(√e) = ln(e^{1\/2}) = 1\/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\": \"1\/2\", \"b\": \"2\", \"c\": \"√e\", \"d\": \"1\"}}","_debug_options_count":4}]