Quiz — 1-ère-année-Exercices-DanalyseHichem-MORTAD.pdf

Question 1 sur 10 20:00

[{"id":86893,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(x² + 1) lorsque x tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 3","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La limite d'un rapport de polynômes de même degré est égale au rapport des coefficients dominants. Ici, 3x²\/x² = 3.","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. 3\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":86894,"question":"Une fonction continue sur un intervalle fermé [a, b] est nécessairement dérivable sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction continue n'est pas forcément dérivable (exemple : f(x) = |x| en x=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":86895,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 7 ?","option_a":"A. 3x² - 4x + 5","option_b":"B. 3x² - 4x + 5x","option_c":"C. x³ - 4x + 5","option_d":"D. 3x² - 2x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle de dérivation terme à terme : (x^n)' = n*x^(n-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\": \"A. 3x² - 4x + 5\", \"b\": \"B. 3x² - 4x + 5x\", \"c\": \"C. x³ - 4x + ","_debug_options_count":4},{"id":86896,"question":"La fonction f(x) = 1\/x est-elle continue sur son ensemble de définition ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = 1\/x n'est pas définie en x=0, donc elle n'est pas continue sur ℝ. Elle est continue sur ℝ*.","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":86897,"question":"Quelle est la primitive de la fonction f(x) = 4x³ - 2x + 1 ?","option_a":"A. x⁴ - x² + x + C","option_b":"B. x⁴ - x² + C","option_c":"C. 12x² - 2 + C","option_d":"D. 4x⁴ - 2x² + x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"La primitive s'obtient en intégrant terme à terme : ∫4x³ dx = x⁴, ∫-2x dx = -x², ∫1 dx = 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\": \"A. x⁴ - x² + x + C\", \"b\": \"B. x⁴ - x² + C\", \"c\": \"C. 12x² ","_debug_options_count":4},{"id":86898,"question":"Si une suite (uₙ) est croissante et majorée, alors elle converge.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est 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},{"id":86899,"question":"Quelle est la valeur de l'intégrale ∫₀¹ (2x + 1) dx ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"c","explication":"Calcul de l'intégrale : [x² + x]₀¹ = (1 + 1) - (0 + 0) = 2. L'intégrale vaut donc 2.","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":86900,"question":"La fonction f(x) = x² - 4x + 3 est-elle dérivable en x=2 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction polynôme est dérivable sur ℝ. Sa dérivée f'(x) = 2x - 4 existe donc en x=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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":86901,"question":"Quelle est la limite de la suite uₙ = (n² + 1)\/(2n² - 3) lorsque n tend vers l'infini ?","option_a":"A. 0","option_b":"B. 1\/2","option_c":"C. 1","option_d":"D. +∞","option_e":"","option_f":"","bonne_reponse":"c","explication":"La limite d'une suite rationnelle de même degré au numérateur et dénominateur est le rapport des coefficients dominants : n²\/2n² = 1\/2. Ici, le numérateur est n²+1 et le dénominateur 2n²-3, donc la limite est 1\/2.","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\/2\", \"c\": \"C. 1\", \"d\": \"D. +∞\"}}","_debug_options_count":4},{"id":86902,"question":"Si f est une fonction dérivable sur ℝ, alors f est nécessairement continue sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction dérivable est continue. La dérivabilité implique la continuité, mais l'inverse n'est pas vrai.","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}]