Quiz — Études de fonctions numériques, IV.pdf

Question 1 sur 10 20:00

[{"id":77116,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 5)\/(x² - 1) 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 - 5\/x²)\/(1 - 1\/x²) = 2\/1 = 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":77117,"question":"La fonction f(x) = x³ - 3x² + 2x est dérivable sur ℝ.","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 est f'(x) = 3x² - 6x + 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":77118,"question":"Quelle est la dérivée de la fonction f(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^(2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la formule (e^u)' = u'e^u avec u = 2x+1, on obtient f'(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)\", \"d\": \"","_debug_options_count":4},{"id":77119,"question":"La courbe représentative de f(x) = 1\/x admet une asymptote verticale en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction f(x) = 1\/x n'est pas définie en x = 0 et lim (x→0) f(x) = ±∞, donc la droite x=0 est une asymptote verticale.","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":77120,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x² + 1) ?","option_a":"A. 6x\/(3x² + 1)","option_b":"B. 3x\/(3x² + 1)","option_c":"C. 6x","option_d":"D. ln(6x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule (ln u)' = u'\/u avec u = 3x² + 1, on obtient f'(x) = 6x\/(3x² + 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. 6x\/(3x² + 1)\", \"b\": \"B. 3x\/(3x² + 1)\", \"c\": \"C. 6x\", \"d\": \"D","_debug_options_count":4},{"id":77121,"question":"La fonction f(x) = x² - 4x + 3 est croissante sur l'intervalle [2 ; +∞[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 2x - 4 est positive pour x ≥ 2, donc la fonction est croissante sur [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":77122,"question":"Quelle est la limite de la fonction f(x) = (sin x)\/x quand x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"On sait que lim (x→0) (sin x)\/x = 1 (limite fondamentale en analyse).","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":77123,"question":"La fonction f(x) = √(x² + 1) est paire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction est paire si f(-x) = f(x). Ici, f(-x) = √((-x)² + 1) = √(x² + 1) = f(x), donc elle est paire.","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":77124,"question":"Quelle est la dérivée de la fonction f(x) = (x + 1)\/(x - 2) ?","option_a":"A. 1\/(x - 2)²","option_b":"B. -3\/(x - 2)²","option_c":"C. 3\/(x - 2)","option_d":"D. (x - 2)\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la formule (u\/v)' = (u'v - uv')\/v² avec u = x+1 et v = x-2, on obtient f'(x) = -3\/(x - 2)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\/(x - 2)²\", \"b\": \"B. -3\/(x - 2)²\", \"c\": \"C. 3\/(x - 2)\", \"d\"","_debug_options_count":4},{"id":77125,"question":"La fonction f(x) = x³ - 3x + 1 admet un point d'inflexion en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée seconde f''(x) = 6x s'annule en x = 0, et change de signe autour de ce point, donc il y a un point d'inflexion.","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}]