Quiz — série n°2 Fonct num.docx

Question 1 sur 10 20:00

[{"id":24814,"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 - 2x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle : (x^n)' = n*x^(n-1). Ainsi, f'(x) = 3x² - 4x + 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3x² - 4x + 5\", \"b\": \"3x² - 4x + 5x\", \"c\": \"x² - 2x + 5\", \"d\": ","_debug_options_count":4},{"id":24815,"question":"La fonction f(x) = x² - 4x + 3 admet-elle un minimum ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction polynôme du second degré admet toujours un extremum (minimum ou maximum). Ici, comme le coefficient de x² est positif, c'est un minimum.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":24816,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 5) quand x tend vers l'infini ?","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 des fonctions rationnelles, on compare les degrés des polynômes. Ici, les degrés sont égaux (2), donc la limite est le rapport des coefficients dominants : 2\/1 = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2\", \"b\": \"0\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":24817,"question":"La fonction f(x) = |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 valeur absolue n'est pas dérivable en x = 0 car sa courbe présente un point anguleux (dérivée à gauche ≠ dérivée à droite).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":24818,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x² - 3x + 2 au point d'abscisse x = 1 ?","option_a":"y = -x + 1","option_b":"y = x - 1","option_c":"y = -x - 1","option_d":"y = 2x - 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation de la tangente en x = a est y = f'(a)(x - a) + f(a). Ici, f(1) = 0 et f'(x) = 2x - 3, donc f'(1) = -1. D'où y = -1(x - 1) + 0 = -x + 1. (Correction : réponse correcte est y = -x + 1, donc correct = 0)","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y = -x + 1\", \"b\": \"y = x - 1\", \"c\": \"y = -x - 1\", \"d\": \"y = 2x - ","_debug_options_count":4},{"id":24819,"question":"La fonction f(x) = x³ + 2x est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction est strictement croissante si sa dérivée est strictement positive. Ici, f'(x) = 3x² + 2 \u003E 0 pour tout x ∈ ℝ, donc f est strictement croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":24820,"question":"Quelle est la dérivée de la fonction f(x) = √(x² + 1) ?","option_a":"(x)\/√(x² + 1)","option_b":"x\/√(x² + 1)","option_c":"(2x)\/√(x² + 1)","option_d":"1\/√(x² + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la formule de dérivation des fonctions composées : (√u)' = u'\/(2√u). Ici, u = x² + 1, donc u' = 2x. Ainsi, f'(x) = (2x)\/(2√(x² + 1)) = x\/√(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"(x)\/√(x² + 1)\", \"b\": \"x\/√(x² + 1)\", \"c\": \"(2x)\/√(x² + 1)","_debug_options_count":4},{"id":24821,"question":"La fonction f(x) = sin(x) est-elle périodique ? Si oui, quelle est sa période ?","option_a":"Non","option_b":"Oui, période 2π","option_c":"Oui, période π","option_d":"Oui, période π\/2","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction sinus est périodique de période 2π, car sin(x + 2π) = sin(x) pour tout x ∈ ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Non\", \"b\": \"Oui, période 2π\", \"c\": \"Oui, période π\", \"d\": \"Ou","_debug_options_count":4},{"id":24822,"question":"La fonction f(x) = e^x est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction est convexe si sa dérivée seconde est positive. Ici, f''(x) = e^x \u003E 0 pour tout x ∈ ℝ, donc f est convexe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":24823,"question":"Quelle est la valeur de l'intégrale ∫(de 0 à 1) (3x² + 2x) dx ?","option_a":"2","option_b":"1","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale d'un polynôme s'obtient en intégrant terme à terme : ∫(3x² + 2x) dx = x³ + x² + C. En évaluant entre 0 et 1, on obtient (1 + 1) - (0 + 0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2\", \"b\": \"1\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4}]