Quiz : Maîtrisez les familles de fonctions en Terminale Math

Question 1 sur 10 10:00

[{"id":5395,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"3x² - 4x + 5","option_b":"x² - 2x + 5","option_c":"3x² - 4x + 5x","option_d":"x³ - 4x + 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'un polynôme se calcule en appliquant la formule (x^n)' = n*x^(n-1) à chaque terme.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5396,"question":"La fonction f(x) = (x² + 1)\/(x - 1) admet-elle une asymptote verticale ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une asymptote verticale existe si le dénominateur s'annule sans que le numérateur ne s'annule au même point. Ici, x - 1 = 0 pour x = 1, et le numérateur vaut 2 en x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5397,"question":"Quelle est la limite de la fonction f(x) = (x² - 4)\/(x - 2) lorsque x tend vers 2 ?","option_a":"0","option_b":"4","option_c":"2","option_d":"Indéterminée","option_e":"","option_f":"","bonne_reponse":"D","explication":"La fonction est indéterminée en x = 2 (forme 0\/0). En factorisant, on obtient f(x) = (x + 2), donc la limite est 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5398,"question":"La fonction f(x) = e^(2x) est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de f(x) = e^(2x) est f'(x) = 2e^(2x), qui est toujours positive sur ℝ. La fonction est donc strictement croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5399,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = ln(x) au point d'abscisse x = 1 ?","option_a":"y = x - 1","option_b":"y = x","option_c":"y = 1","option_d":"y = 2x - 1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La tangente en x = 1 a pour équation y = f(1) + f'(1)(x - 1). Ici, f(1) = 0 et f'(1) = 1, donc y = x - 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5400,"question":"La fonction f(x) = x^4 - 4x³ + 6 admet-elle un extremum local en x = 3 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"En calculant f'(x) = 4x³ - 12x² et f''(x) = 12x² - 24x, on trouve f'(3) = 0 et f''(3) = 36 \u003E 0. Il s'agit donc d'un minimum local.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5401,"question":"Quelle est l'asymptote oblique de la fonction f(x) = (x² + 1)\/x ?","option_a":"y = x","option_b":"y = x + 1","option_c":"y = 1","option_d":"Aucune asymptote oblique","option_e":"","option_f":"","bonne_reponse":"A","explication":"En effectuant la division euclidienne, on obtient f(x) = x + 1\/x. La partie entière x est l'asymptote oblique.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5402,"question":"La fonction f(x) = sin(x) est-elle périodique ? Si oui, quelle est sa période ?","option_a":"Non","option_b":"Oui, π","option_c":"Oui, 2π","option_d":"Oui, π\/2","option_e":"","option_f":"","bonne_reponse":"C","explication":"La fonction sinus est périodique de période 2π, car sin(x + 2π) = sin(x) pour tout x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5403,"question":"Quelle est la dérivée de la fonction f(x) = √(x² + 1) ?","option_a":"x\/√(x² + 1)","option_b":"(x² + 1)\/√(x² + 1)","option_c":"√(x² + 1)\/x","option_d":"1\/(2√(x² + 1))","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la dérivée des fonctions composées, f'(x) = (1\/2)(x² + 1)^(-1\/2) * 2x = x\/√(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":5404,"question":"La fonction f(x) = x^3 - 3x + 2 admet-elle un point d'inflexion ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée seconde f''(x) = 6x s'annule en x = 0. Comme f''(x) change de signe en x = 0, il s'agit d'un point d'inflexion.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]