Quiz : Maîtrisez les fonctions dérivées et les limites !

Question 1 sur 10 10:00

[{"id":71972,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x² + 2x","option_d":"3x + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynôme se calcule en appliquant la formule (ax^n)' = n*ax^(n-1). Ici, f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71973,"question":"La fonction f(x) = (x² + 1)\/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 n'est pas définie en x = 0 (division par zéro), donc elle n'est pas dérivable en ce point.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71974,"question":"Quelle est la limite de f(x) = (x² - 4)\/(x - 2) quand x tend vers 2 ?","option_a":"0","option_b":"4","option_c":"2","option_d":"Indéterminée","option_e":"","option_f":"","bonne_reponse":"C","explication":"En factorisant le numérateur, on obtient (x-2)(x+2)\/(x-2) = x+2 pour x ≠ 2. La limite est donc 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71975,"question":"La fonction f(x) = x³ - 3x est-elle continue sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les fonctions polynômes sont continues sur tout leur domaine de définition, ici ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71976,"question":"Quelle est la dérivée de f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2x*e^(2x)","option_d":"e^x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de e^(u(x)) est u'(x)*e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71977,"question":"La limite de f(x) = (sin x)\/x quand x tend vers 0 est égale à :","option_a":"0","option_b":"1","option_c":"∞","option_d":"Indéterminée","option_e":"","option_f":"","bonne_reponse":"B","explication":"C'est une limite fondamentale en analyse : lim(x→0) (sin x)\/x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71978,"question":"Une fonction dérivable en un point est-elle nécessairement continue en ce point ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Toute fonction dérivable en un point est continue en ce point (théorème fondamental).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71979,"question":"Quelle est la dérivée de f(x) = ln(x² + 1) ?","option_a":"2x\/(x² + 1)","option_b":"1\/(x² + 1)","option_c":"2x*ln(x² + 1)","option_d":"(x² + 1)\/2x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x) = x² + 1, donc u'(x) = 2x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71980,"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 x = 0 car elle présente un point anguleux (dérivée à gauche ≠ dérivée à droite).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":71981,"question":"Quelle est la limite de f(x) = (1 - cos x)\/x² quand x tend vers 0 ?","option_a":"0","option_b":"1\/2","option_c":"1","option_d":"Indéterminée","option_e":"","option_f":"","bonne_reponse":"B","explication":"En utilisant le développement limité de cos x, on obtient lim(x→0) (1 - (1 - x²\/2 + o(x²)))\/x² = 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]