Quiz : Fonctions numériques — Dérivées et Limites

Question 1 sur 10 10:00

[{"id":32398,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"3x² - 4x + 5","option_b":"3x² - 4x + 5x","option_c":"x² - 2x + 5","option_d":"3x² - 4x + 1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle de dérivation terme à terme : (x^n)' = n*x^(n-1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":32399,"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":32400,"question":"Quelle est la limite de la fonction f(x) = (2x² - 3x + 1)\/(x² + 1) lorsque x tend vers +∞ ?","option_a":"2","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour les limites à l'infini d'un rapport de polynômes, on compare les termes de plus haut degré : (2x²)\/(x²) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":32401,"question":"La fonction f(x) = x³ - 3x + 2 admet-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"En x = 1, f'(x) = 0 et f''(x) = 6x \u003E 0, donc la fonction admet un minimum local en ce point.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":32402,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = e^x en x = 0 ?","option_a":"y = x","option_b":"y = e*x","option_c":"y = 1","option_d":"y = x + 1","option_e":"","option_f":"","bonne_reponse":"D","explication":"L'équation de la tangente en a est y = f'(a)(x - a) + f(a). Ici, f(0) = 1 et f'(0) = 1, donc y = x + 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":32403,"question":"La fonction f(x) = ln(x) est-elle définie pour x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction logarithme népérien ln(x) est définie uniquement pour x \u003E 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":32404,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"cos(2x)","option_b":"2cos(2x)","option_c":"cos(x)","option_d":"2sin(2x)","option_e":"","option_f":"","bonne_reponse":"B","explication":"En utilisant la règle de dérivation des fonctions composées : (sin(u))' = u'*cos(u). Ici, u = 2x, donc u' = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":32405,"question":"La fonction f(x) = x^4 - 4x³ admet-elle un point d'inflexion en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Un point d'inflexion correspond à un changement de concavité, vérifié par f''(x) = 12x² - 24x. En x = 1, f''(1) = -12 ≠ 0, donc pas d'inflexion.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":32406,"question":"Quelle est la limite de la fonction f(x) = (1 - cos(x))\/x² lorsque x tend vers 0 ?","option_a":"0","option_b":"1\/2","option_c":"∞","option_d":"1","option_e":"","option_f":"","bonne_reponse":"B","explication":"En utilisant le développement limité de cos(x) au voisinage de 0 : cos(x) ≈ 1 - x²\/2, donc (1 - cos(x))\/x² ≈ 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":32407,"question":"La fonction f(x) = x² - 4x + 3 est-elle 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 \u003E 2, donc la fonction est croissante sur [2, +∞[.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]