Quiz — Série n°2 - Etude de fonctions.pdf

Question 1 sur 5 10:00

[{"id":26339,"question":"Quelle est la dérivée de la fonction f(x) = x² * ln(x) ?","option_a":"2x * ln(x)","option_b":"2x * ln(x) + x","option_c":"2x * ln(x) + 1","option_d":"x * ln(x) + 2x","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la formule de dérivation d'un produit : (uv)' = u'v + uv'. Ici, u = x² et v = ln(x), donc u' = 2x et v' = 1\/x. Ainsi, f'(x) = 2x * ln(x) + x² * (1\/x) = 2x * ln(x) + x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2x * ln(x)\", \"b\": \"2x * ln(x) + x\", \"c\": \"2x * ln(x) + 1\", \"d\": \"","_debug_options_count":4},{"id":26340,"question":"Quelle est la limite de la fonction f(x) = (3x² + 2x - 1)\/(2x² - x + 4) quand x tend vers +∞ ?","option_a":"0","option_b":"3\/2","option_c":"1","option_d":"+∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une fonction rationnelle, on compare les degrés du numérateur et du dénominateur. Ici, les deux sont de degré 2. La limite est donc le rapport des coefficients dominants : 3\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"3\/2\", \"c\": \"1\", \"d\": \"+∞\"}}","_debug_options_count":4},{"id":26341,"question":"Quelle est l'équation de l'asymptote horizontale de la fonction f(x) = (2x + 1)\/(x - 3) ?","option_a":"y = 0","option_b":"y = 1","option_c":"y = 2","option_d":"y = -3","option_e":"","option_f":"","bonne_reponse":"c","explication":"Pour une fonction rationnelle où le numérateur et le dénominateur ont le même degré, l'asymptote horizontale est donnée par le rapport des coefficients dominants : 2\/1 = 2. Donc y = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"y = 0\", \"b\": \"y = 1\", \"c\": \"y = 2\", \"d\": \"y = -3\"}}","_debug_options_count":4},{"id":26342,"question":"Soit f(x) = e^(3x - 1). Quelle est la valeur de f'(0) ?","option_a":"0","option_b":"1","option_c":"3","option_d":"e^(-1)","option_e":"","option_f":"","bonne_reponse":"d","explication":"La dérivée de f(x) = e^(3x - 1) est f'(x) = 3 * e^(3x - 1). Donc f'(0) = 3 * e^(-1) = 3\/e.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"3\", \"d\": \"e^(-1)\"}}","_debug_options_count":4},{"id":26343,"question":"Quelle est la solution de l'équation ln(x) = 2 ?","option_a":"x = 0","option_b":"x = 1","option_c":"x = e²","option_d":"x = 2","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'équation ln(x) = 2 équivaut à x = e², car ln(e²) = 2 par définition du logarithme népérien.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"x = 0\", \"b\": \"x = 1\", \"c\": \"x = e²\", \"d\": \"x = 2\"}}","_debug_options_count":4}]