Quiz : Maîtrisez l'étude des fonctions en Terminale Math

Question 1 sur 10 10:00

[{"id":26398,"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},{"id":26399,"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},{"id":26400,"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},{"id":26401,"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},{"id":26402,"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},{"id":26403,"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},{"id":26404,"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},{"id":26405,"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},{"id":26406,"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},{"id":26407,"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}]