Les fonctions dérivées et leurs applications — Terminale Mathématiques

Question 1 sur 10 10:00

[{"id":61097,"question":"Soit f(x) = x³ - 3x² + 2. Quelle est la dérivée de f en x = 2 ?","option_a":"3","option_b":"0","option_c":"6","option_d":"12","option_e":"","option_f":"","bonne_reponse":"C","explication":"La dérivée de f(x) = x³ - 3x² + 2 est f'(x) = 3x² - 6x. En x = 2, f'(2) = 3*(4) - 6*(2) = 12 - 12 = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61098,"question":"Le théorème de Rolle s'applique-t-il à f(x) = x² sur [-1, 1] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le théorème de Rolle nécessite f(a) = f(b). Ici, f(-1) = 1 et f(1) = 1, donc f(-1) = f(1). De plus, f est continue et dérivable sur [-1, 1]. Le théorème s'applique.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61099,"question":"Quelle est la limite de f(x) = (x² - 1)\/(x - 1) lorsque x tend vers 1 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"B","explication":"Factorisez le numérateur : (x² - 1) = (x - 1)(x + 1). Ainsi, f(x) = (x + 1) pour x ≠ 1. La limite lorsque x → 1 est donc 1 + 1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61100,"question":"Soit f(x) = e^x. Quelle est la dérivée de f en x = 0 ?","option_a":"0","option_b":"1","option_c":"e","option_d":"e^0","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée de f(x) = e^x est f'(x) = e^x. En x = 0, f'(0) = e^0 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61101,"question":"Une fonction peut-elle être continue en un point sans y être dérivable ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, par exemple f(x) = |x| est continue en x = 0 mais n'y est pas dérivable (dérivée à gauche ≠ dérivée à droite).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61102,"question":"Quelle est la dérivée de f(x) = ln(x) pour x \u003E 0 ?","option_a":"1\/x","option_b":"x","option_c":"e^x","option_d":"1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de la fonction logarithme népérien est f'(x) = 1\/x pour x \u003E 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61103,"question":"Soit f(x) = x^4 - 4x³ + 6x². Quel est le nombre de points critiques de f ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"C","explication":"Les points critiques sont les solutions de f'(x) = 0. f'(x) = 4x³ - 12x² + 12x = 4x(x² - 3x + 3). Le discriminant de x² - 3x + 3 est négatif, donc seule solution réelle est x = 0. Cependant, f'(x) = 4x(x-1)(x-3) donne trois points critiques : x = 0, x = 1, x = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61104,"question":"Le théorème des accroissements finis garantit-il l'existence d'un point où la dérivée est égale à la pente de la corde ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, le théorème des accroissements finis stipule qu'il existe un point c dans ]a,b[ où f'(c) = [f(b) - f(a)] \/ (b - a).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61105,"question":"Quelle est la dérivée de f(x) = sin(x)cos(x) ?","option_a":"cos(2x)","option_b":"sin(2x)","option_c":"cos²(x) - sin²(x)","option_d":"2sin(x)cos(x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"Utilisez l'identité sin(2x) = 2sin(x)cos(x). Ainsi, f(x) = (1\/2)sin(2x), et f'(x) = cos(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61106,"question":"Soit f(x) = x² - 2x + 1. Quel est le minimum de f sur ℝ ?","option_a":"-1","option_b":"0","option_c":"1","option_d":"2","option_e":"","option_f":"","bonne_reponse":"B","explication":"f'(x) = 2x - 2. Le point critique est x = 1. f(1) = 1 - 2 + 1 = 0. Comme f''(x) = 2 \u003E 0, c'est un minimum global.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]