Quiz — Corrigé__magazine 27 (fct.ln).pdf

Question 1 sur 10 20:00

[{"id":37890,"question":"Quelle est la dérivée de la fonction f(x) = ln(x) pour x \u003E 0 ?","option_a":"1\/x","option_b":"1","option_c":"x","option_d":"e^x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction logarithme népérien est 1\/x pour tout x \u003E 0, d'après le cours de Terminale.","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\/x\", \"b\": \"1\", \"c\": \"x\", \"d\": \"e^x\"}}","_debug_options_count":4},{"id":37891,"question":"L'équation différentielle y' = y admet pour solution générale y = Ce^x où C est une constante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est exact : la solution générale de y' = y est bien y = Ce^x, avec C ∈ ℝ.","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":37892,"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)\/x tend vers 0 lorsque 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":37893,"question":"La fonction exponentielle est-elle la réciproque de la fonction logarithme népérien ?","option_a":"Oui, pour tout x ∈ ℝ","option_b":"Non, seulement pour x \u003E 0","option_c":"Oui, mais uniquement pour x = 1","option_d":"Non, elles n'ont aucun lien","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction exponentielle est bien la réciproque de ln(x), mais uniquement pour x \u003E 0, car ln(x) n'est défini 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\": \"Oui, pour tout x ∈ ℝ\", \"b\": \"Non, seulement pour x \u003E 0\", \"c\":","_debug_options_count":4},{"id":37894,"question":"Quelle est la primitive de la fonction f(x) = e^(2x) ?","option_a":"(1\/2)e^(2x) + C","option_b":"e^(2x) + C","option_c":"2e^(2x) + C","option_d":"(1\/2x)e^(2x) + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive de e^(2x) est (1\/2)e^(2x) + C, car la dérivée de (1\/2)e^(2x) est e^(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\": \"(1\/2)e^(2x) + C\", \"b\": \"e^(2x) + C\", \"c\": \"2e^(2x) + C\", \"d\": \"(1","_debug_options_count":4},{"id":37895,"question":"L'équation ln(x) = 0 admet-elle une solution ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car ln(1) = 0, donc x = 1 est solution de l'équation ln(x) = 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":37896,"question":"Quelle est la valeur de l'intégrale ∫(de 1 à e) (1\/x) dx ?","option_a":"0","option_b":"1","option_c":"e","option_d":"e - 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'intégrale de 1\/x entre 1 et e vaut ln(e) - ln(1) = 1 - 0 = 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\": \"0\", \"b\": \"1\", \"c\": \"e\", \"d\": \"e - 1\"}}","_debug_options_count":4},{"id":37897,"question":"La fonction f(x) = e^(-x²) est-elle toujours positive ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car e^(-x²) \u003E 0 pour tout x ∈ ℝ, car l'exponentielle est toujours positive.","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":37898,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 avec y(0) = 3 ?","option_a":"y = 3e^(2x)","option_b":"y = 3e^(-2x)","option_c":"y = -3e^(2x)","option_d":"y = 3e^(x\/2)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La solution générale de y' + 2y = 0 est y = Ce^(-2x). Avec y(0) = 3, on trouve C = 3, donc y = 3e^(-2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"y = 3e^(2x)\", \"b\": \"y = 3e^(-2x)\", \"c\": \"y = -3e^(2x)\", \"d\": \"y =","_debug_options_count":4},{"id":37899,"question":"La fonction f(x) = ln(x² + 1) est-elle définie pour tout x ∈ ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, car x² + 1 \u003E 0 pour tout x ∈ ℝ, donc ln(x² + 1) est défini sur ℝ.","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}]