Quiz express : Dérivées et limites — Niveau 4ème année secondaire

Question 1 sur 10 10:00

[{"id":75382,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x + 2","option_c":"6x² + 2","option_d":"3x² + 2x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle de puissance : f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75383,"question":"La limite de (sin x)\/x quand x tend vers 0 est égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est un résultat fondamental en analyse : lim (x→0) (sin x)\/x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75384,"question":"Quelle est la dérivée de la fonction 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":"En utilisant la règle de dérivation des fonctions exponentielles : (e^u)' = u' e^u, ici u = 2x donc u' = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75385,"question":"La fonction f(x) = 1\/x a une limite finie quand x tend vers 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La limite de 1\/x quand x tend vers 0 est +∞ ou -∞ selon le sens d'approche, donc elle n'est pas finie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75386,"question":"Quelle est la dérivée de la fonction f(x) = ln(x) ?","option_a":"1\/x","option_b":"x","option_c":"1\/ln(x)","option_d":"e^x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(x) est 1\/x, une formule fondamentale en analyse.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75387,"question":"Si f'(x) = 0 pour tout x dans un intervalle, alors f est constante sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est le théorème de Lagrange : si la dérivée est nulle sur un intervalle, la fonction est constante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75388,"question":"Quelle est la limite de (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 simplifiant l'expression : (x² - 4)\/(x - 2) = (x - 2)(x + 2)\/(x - 2) = x + 2 pour x ≠ 2, donc la limite est 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75389,"question":"La dérivée de la fonction f(x) = cos(x) est f'(x) = -sin(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est une formule de dérivation de base : (cos x)' = -sin x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75390,"question":"Quelle est la dérivée de la fonction f(x) = √x ?","option_a":"1\/(2√x)","option_b":"√x","option_c":"1\/√x","option_d":"x^(-1\/2)","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la règle de dérivation des racines : (√x)' = 1\/(2√x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":75391,"question":"Si f est dérivable en a, alors f est continue en a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est un théorème fondamental : la dérivabilité implique la continuité.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]