Quiz : Fonctions ln, exponentielle et intégrales — Terminale Math

Question 1 sur 10 10:00

[{"id":26028,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x + 2) ?","option_a":"A) 3\/(3x + 2)","option_b":"B) 1\/(3x + 2)","option_c":"C) 3x\/(3x + 2)","option_d":"D) 1\/(x + 2)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u) est u'\/u. Ici, u = 3x + 2, donc u' = 3. Ainsi, f'(x) = 3\/(3x + 2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":26029,"question":"L'équation e^x = -2 admet une solution réelle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction exponentielle est toujours strictement positive. e^x \u003E 0 pour tout x réel, donc pas de solution.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":26030,"question":"Quelle est la primitive de f(x) = e^(2x) ?","option_a":"A) e^(2x) + C","option_b":"B) (1\/2)e^(2x) + C","option_c":"C) 2e^(2x) + C","option_d":"D) e^(2x)\/x + C","option_e":"","option_f":"","bonne_reponse":"B","explication":"La primitive de e^(u) est (1\/u')e^(u) + C. Ici, u = 2x, donc u' = 2. Ainsi, ∫e^(2x) dx = (1\/2)e^(2x) + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":26031,"question":"La fonction ln(x) est définie pour x \u003E 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le logarithme népérien est défini uniquement pour les nombres strictement positifs (x \u003E 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":26032,"question":"Calculer ∫(x e^x) dx :","option_a":"A) x e^x - e^x + C","option_b":"B) e^x (x - 1) + C","option_c":"C) e^x (x + 1) + C","option_d":"D) x e^x + C","option_e":"","option_f":"","bonne_reponse":"B","explication":"Utiliser l'intégration par parties : u = x, dv = e^x dx → du = dx, v = e^x. ∫x e^x dx = x e^x - ∫e^x dx = x e^x - e^x + C = e^x (x - 1) + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":26033,"question":"Quelle est la limite de (ln(x))^2 \/ x quand x tend vers +∞ ?","option_a":"A) 0","option_b":"B) +∞","option_c":"C) 1","option_d":"D) -∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par croissance comparée, ln(x) croît plus lentement que toute puissance de x. Ainsi, (ln(x))^2 \/ x → 0 quand x → +∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":26034,"question":"La fonction f(x) = e^(x^2) est une solution de f'(x) = 2x f(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Calculons f'(x) : f'(x) = 2x e^(x^2) = 2x f(x). L'affirmation est donc vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":26035,"question":"Quelle est la valeur de ∫(1\/(1 + x^2)) dx ?","option_a":"A) ln(1 + x^2) + C","option_b":"B) arctan(x) + C","option_c":"C) x\/(1 + x^2) + C","option_d":"D) 2x\/(1 + x^2)^2 + C","option_e":"","option_f":"","bonne_reponse":"B","explication":"La primitive de 1\/(1 + x^2) est arctan(x) + C, une formule fondamentale à connaître.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":26036,"question":"La fonction ln(1\/x) est égale à -ln(x) pour x \u003E 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par propriété des logarithmes : ln(1\/x) = ln(x^(-1)) = -ln(x). L'affirmation est donc vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":26037,"question":"Résoudre l'inéquation ln(x) \u003E 2 :","option_a":"A) x \u003E 2","option_b":"B) x \u003E e^2","option_c":"C) x \u003C e^2","option_d":"D) x \u003E 0","option_e":"","option_f":"","bonne_reponse":"B","explication":"ln(x) \u003E 2 ⇔ x \u003E e^2 (car la fonction ln est strictement croissante).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]