Quiz — 07-suites-series-fonctions.pdf

Question 1 sur 10 20:00

[{"id":41885,"question":"Soit la suite définie par u_{n+1} = 2u_n + 1 avec u_0 = 0. Quelle est sa limite si elle existe ?","option_a":"A. 0","option_b":"B. -1","option_c":"C. +∞","option_d":"D. La suite diverge","option_e":"","option_f":"","bonne_reponse":"c","explication":"La suite est arithmético-géométrique. En résolvant l'équation l = 2l + 1, on trouve l = -1. Cependant, comme u_0 = 0, la suite diverge vers +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. -1\", \"c\": \"C. +∞\", \"d\": \"D. La suite diverge\"}}","_debug_options_count":4},{"id":41886,"question":"La série de terme général u_n = 1\/n^2 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série de Riemann Σ 1\/n^α converge si et seulement si α \u003E 1. Ici, α = 2 \u003E 1, donc la série converge.","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":41887,"question":"Soit f(x) = x^3 - 3x + 2. Combien de racines réelles distinctes possède cette fonction ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"En étudiant les variations de f (f'(x) = 3x^2 - 3), on trouve deux extrema locaux. En évaluant f en ces points et aux bornes, on déduit qu'il y a 3 racines réelles distinctes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 1\", \"b\": \"B. 2\", \"c\": \"C. 3\", \"d\": \"D. 4\"}}","_debug_options_count":4},{"id":41888,"question":"La fonction f(x) = |x| est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction valeur absolue n'est pas dérivable en 0 car la dérivée à gauche (-1) ≠ dérivée à droite (+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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":41889,"question":"Soit la série Σ (-1)^n \/ n. Cette série est-elle absolument convergente ?","option_a":"A. Oui","option_b":"B. Non","option_c":"C. Parfois","option_d":"D. On ne peut pas savoir","option_e":"","option_f":"","bonne_reponse":"b","explication":"La série des valeurs absolues est Σ 1\/n, qui est la série harmonique divergente. La série initiale est semi-convergente (critère de Leibniz).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. Oui\", \"b\": \"B. Non\", \"c\": \"C. Parfois\", \"d\": \"D. On ne peut pa","_debug_options_count":4},{"id":41890,"question":"Le théorème de Rolle s'applique-t-il à f(x) = x^2 sur l'intervalle [-1, 1] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f est continue sur [-1, 1], dérivable sur ]-1, 1[, et f(-1) = f(1) = 1. Le théorème de Rolle garantit l'existence d'un c ∈ ]-1, 1[ tel que f'(c) = 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":41891,"question":"Soit la suite u_n = n \/ (n + 1). Quelle est sa limite ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par n, on obtient u_n = 1 \/ (1 + 1\/n). Quand n → +∞, u_n → 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\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":41892,"question":"La fonction f(x) = sin(1\/x) est-elle continue en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction n'est pas définie en x = 0, donc elle ne peut pas être continue en ce point. Même si on la prolonge par f(0) = 0, la limite 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":41893,"question":"Soit la série Σ n \/ 2^n. Cette série est-elle convergente ?","option_a":"A. Oui","option_b":"B. Non","option_c":"C. Parfois","option_d":"D. On ne peut pas savoir","option_e":"","option_f":"","bonne_reponse":"a","explication":"On peut appliquer le critère de d'Alembert : lim |u_{n+1}\/u_n| = lim (n+1)\/2n = 1\/2 \u003C 1. La série converge donc absolument.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. Oui\", \"b\": \"B. Non\", \"c\": \"C. Parfois\", \"d\": \"D. On ne peut pa","_debug_options_count":4},{"id":41894,"question":"Le théorème des accroissements finis s'applique-t-il à f(x) = x^3 sur [0, 2] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f est continue sur [0, 2], dérivable sur ]0, 2[, donc le théorème s'applique. Il existe c ∈ ]0, 2[ tel que f'(c) = (f(2) - f(0))\/(2 - 0) = 4.","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}]