Quiz — scéance 12( énoncé)-DS1.pdf

Question 1 sur 10 20:00

[{"id":84185,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 5x - 2 ?","option_a":"A: 6x + 5","option_b":"B: 3x + 5","option_c":"C: 6x² + 5","option_d":"D: 6x + 5x - 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme par terme : f'(x) = 6x + 5.","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: 6x + 5\", \"b\": \"B: 3x + 5\", \"c\": \"C: 6x² + 5\", \"d\": \"D: 6x + 5","_debug_options_count":4},{"id":84186,"question":"Une suite géométrique de raison q = 2 et de premier terme u₀ = 3 est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une suite géométrique converge uniquement si |q| \u003C 1. Ici, q = 2, donc elle diverge.","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":84187,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² + 5) quand x tend vers +∞ ?","option_a":"A: 0","option_b":"B: 2","option_c":"C: +∞","option_d":"D: -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par x², on obtient lim (2 + 3\/x - 1\/x²)\/(1 + 5\/x²) = 2.","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: 2\", \"c\": \"C: +∞\", \"d\": \"D: -∞\"}}","_debug_options_count":4},{"id":84188,"question":"Si f'(x) \u003E 0 pour tout x ∈ [a, b], alors la fonction f est :","option_a":"A: décroissante sur [a, b]","option_b":"B: croissante sur [a, b]","option_c":"C: constante sur [a, b]","option_d":"D: non définie sur [a, b]","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si la dérivée est positive sur un intervalle, la fonction est strictement croissante sur cet intervalle.","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: décroissante sur [a, b]\", \"b\": \"B: croissante sur [a, b]\", \"c","_debug_options_count":4},{"id":84189,"question":"La suite définie par uₙ = (-1)ⁿ est-elle bornée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Tous les termes de la suite valent soit -1, soit 1, donc elle est bornée par -1 et 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":84190,"question":"Quelle est la dérivée de la fonction g(x) = e^(2x + 1) ?","option_a":"A: e^(2x + 1)","option_b":"B: 2e^(2x + 1)","option_c":"C: (2x + 1)e^(2x)","option_d":"D: e^(x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^u est u' * e^u. Ici, u = 2x + 1, donc u' = 2, d'où g'(x) = 2e^(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\": \"A: e^(2x + 1)\", \"b\": \"B: 2e^(2x + 1)\", \"c\": \"C: (2x + 1)e^(2x)\", ","_debug_options_count":4},{"id":84191,"question":"Une fonction f est continue en un point a si :","option_a":"A: f(a) existe et lim(x→a) f(x) existe","option_b":"B: lim(x→a) f(x) = f(a)","option_c":"C: f est dérivable en a","option_d":"D: f est définie sur un intervalle ouvert contenant a","option_e":"","option_f":"","bonne_reponse":"b","explication":"La continuité en a exige que la limite de f(x) quand x tend vers a soit égale à f(a).","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: f(a) existe et lim(x→a) f(x) existe\", \"b\": \"B: lim(x→a) f(","_debug_options_count":4},{"id":84192,"question":"La suite uₙ = n² - 5n + 6 est-elle croissante à partir d'un certain rang ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La suite est croissante si uₙ₊₁ - uₙ \u003E 0. Ici, uₙ₊₁ - uₙ = 2n - 4, qui devient positif pour n ≥ 3.","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":84193,"question":"Quelle est la dérivée de la fonction h(x) = ln(3x + 2) ?","option_a":"A: 1\/(3x + 2)","option_b":"B: 3\/(3x + 2)","option_c":"C: 3x\/(3x + 2)","option_d":"D: 1\/(x + 2)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x + 2, donc u' = 3, d'où h'(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\": \"b\", \"options\": {\"a\": \"A: 1\/(3x + 2)\", \"b\": \"B: 3\/(3x + 2)\", \"c\": \"C: 3x\/(3x + 2)\", \"d\":","_debug_options_count":4},{"id":84194,"question":"Si une suite est majorée et croissante, alors elle est convergente.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après le théorème de convergence monotone, une suite croissante et majorée converge vers sa borne supérieure.","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}]