Quiz d'analyse : Fonctions et Dérivées — Terminale Math

Question 1 sur 10 10:00

[{"id":27288,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2xe^(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":27289,"question":"La fonction f(x) = x^3 - 3x est strictement croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée f'(x)=3x^2-3 s'annule en x=±1. La fonction n'est pas strictement croissante sur tout ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27290,"question":"Quelle est la limite de (sin(x))\/x quand x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"n'existe pas","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":27291,"question":"La fonction f(x) = ln(x) est définie pour :","option_a":"x \u003E 0","option_b":"x ≥ 0","option_c":"x ∈ ℝ","option_d":"x \u003C 0","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction logarithme népérien est définie uniquement pour x \u003E 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27292,"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},{"id":27293,"question":"Quelle est l'intégrale de f(x) = 3x^2 entre 0 et 2 ?","option_a":"4","option_b":"8","option_c":"12","option_d":"16","option_e":"","option_f":"","bonne_reponse":"C","explication":"L'intégrale de 3x^2 est x^3. Évaluée entre 0 et 2 : 2^3 - 0^3 = 8.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27294,"question":"La suite u_n = (n+1)\/n converge vers :","option_a":"0","option_b":"1","option_c":"∞","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"B","explication":"lim(n→∞) (n+1)\/n = lim(n→∞) 1 + 1\/n = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27295,"question":"La fonction f(x) = |x| est 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 0 (dérivée à gauche ≠ dérivée à droite).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27296,"question":"Quelle est la dérivée de f(x) = ln(3x) ?","option_a":"1\/x","option_b":"3\/x","option_c":"1\/(3x)","option_d":"3\/(3x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x)=3x donc u'(x)=3 : 3\/(3x)=1\/x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":27297,"question":"Le théorème des valeurs intermédiaires s'applique à toute fonction continue sur un intervalle fermé.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le TVI stipule que si f est continue sur [a,b] et k entre f(a) et f(b), alors il existe c ∈ [a,b] tel que f(c)=k.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]