Quiz d’analyse : Fonctions, suites et intégrales — Terminale SM

Question 1 sur 10 10:00

[{"id":29748,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x+1) ?","option_a":"A) e^(2x+1)","option_b":"B) 2e^(2x+1)","option_c":"C) e^(2x)","option_d":"D) 2e^(x+1)","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée de e^u est u' * e^u. Ici, u = 2x+1, donc u' = 2. Ainsi, f'(x) = 2e^(2x+1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29749,"question":"La suite définie par u_n = (-1)^n est-elle convergente ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La suite alterne entre -1 et 1, elle n’a pas de limite finie. Elle est donc divergente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29750,"question":"Quelle est la primitive de f(x) = 3x^2 + 2x - 1 ?","option_a":"A) x^3 + x^2 - x + C","option_b":"B) x^3 + x^2 - x","option_c":"C) 6x + 2 + C","option_d":"D) x^3 + x^2 + C","option_e":"","option_f":"","bonne_reponse":"A","explication":"La primitive de x^n est x^(n+1)\/(n+1). Ainsi, 3x^2 donne x^3, 2x donne x^2, et -1 donne -x. La constante C est ajoutée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29751,"question":"Si lim(x→∞) f(x) = 5, alors la droite y=5 est une asymptote horizontale à la courbe de f.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par définition, si la limite de f(x) en l’infini est L, alors y=L est une asymptote horizontale.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29752,"question":"Quelle est la valeur de l’intégrale ∫(0 à 1) (2x + 1) dx ?","option_a":"A) 1","option_b":"B) 2","option_c":"C) 3","option_d":"D) 4","option_e":"","option_f":"","bonne_reponse":"C","explication":"La primitive de 2x+1 est x^2 + x. Évaluée entre 0 et 1 : (1+1) - (0+0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29753,"question":"La fonction f(x) = x^3 - 3x + 2 est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Sa 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":29754,"question":"Quelle est la limite de la suite u_n = (n^2 + 1)\/(n^2 + n) quand n tend vers l’infini ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) n","option_d":"D) -1","option_e":"","option_f":"","bonne_reponse":"B","explication":"En divisant numérateur et dénominateur par n^2, on obtient (1 + 1\/n^2)\/(1 + 1\/n) → 1\/1 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29755,"question":"L’intégration par parties permet de calculer ∫ u dv = uv - ∫ v du.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La formule correcte est ∫ u dv = uv - ∫ v du. La formulation donnée est exacte.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29756,"question":"Quelle est la solution de l’équation différentielle y' = 2y avec y(0) = 3 ?","option_a":"A) y = 3e^(2x)","option_b":"B) y = 2e^(3x)","option_c":"C) y = 3e^(-2x)","option_d":"D) y = 2e^(-3x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La solution générale de y' = ky est y = Ce^(kx). Ici, k=2 et y(0)=3 donne C=3. Ainsi, y=3e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":29757,"question":"Si f est continue sur [a,b] et f(a) \u003C 0 \u003C f(b), alors f s’annule au moins une fois sur [a,b].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C’est le théorème des valeurs intermédiaires. f étant continue, elle prend toutes les valeurs entre f(a) et f(b), donc 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]