Quiz — سلسلة تمارين حول الدوال العددية.pdf

Question 1 sur 10 20:00

[{"id":59204,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x + 2","option_c":"6x² + 2","option_d":"x² + 2x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la formule (ax^n)' = n*ax^(n-1). Ici, f'(x) = 6x + 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\": \"6x + 2\", \"b\": \"3x + 2\", \"c\": \"6x² + 2\", \"d\": \"x² + 2x - 5\"}}","_debug_options_count":4},{"id":59205,"question":"La fonction f(x) = x³ - 3x est-elle croissante sur l'intervalle [0 ; 2] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 3 est négative sur [0 ; 1] et positive sur [1 ; 2], donc la fonction n'est pas croissante sur tout l'intervalle.","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":59206,"question":"Quelle est la limite de la fonction f(x) = (2x + 1)\/(x - 3) lorsque x tend vers +∞ ?","option_a":"0","option_b":"2","option_c":"1\/3","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par x, on obtient f(x) = (2 + 1\/x)\/(1 - 3\/x), donc la limite est 2\/1 = 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\": \"2\", \"c\": \"1\/3\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":59207,"question":"La fonction f(x) = sin(x) est-elle continue sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction sinus est continue sur tout son domaine de définition, qui est ℝ, car elle est dérivable partout.","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":59208,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"2x\/(x² + 1)","option_b":"1\/(x² + 1)","option_c":"2x","option_d":"ln(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées, f'(x) = (2x)\/(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2x\/(x² + 1)\", \"b\": \"1\/(x² + 1)\", \"c\": \"2x\", \"d\": \"ln(2x)\"}}","_debug_options_count":4},{"id":59209,"question":"La fonction f(x) = x² - 4x + 3 admet-elle un minimum ? Si oui, en quel point ?","option_a":"Oui, en x = 1","option_b":"Oui, en x = 2","option_c":"Non","option_d":"Oui, en x = 0","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 2x - 4 s'annule en x = 2. Comme f''(x) = 2 \u003E 0, il s'agit d'un minimum.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Oui, en x = 1\", \"b\": \"Oui, en x = 2\", \"c\": \"Non\", \"d\": \"Oui, en x","_debug_options_count":4},{"id":59210,"question":"Quelle est l'équation de l'asymptote horizontale de la fonction f(x) = (3x² + 2)\/(x² - 1) lorsque x tend vers ±∞ ?","option_a":"y = 0","option_b":"y = 3","option_c":"y = 2","option_d":"y = x","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par x², on obtient f(x) = (3 + 2\/x²)\/(1 - 1\/x²), donc l'asymptote horizontale est y = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"y = 0\", \"b\": \"y = 3\", \"c\": \"y = 2\", \"d\": \"y = x\"}}","_debug_options_count":4},{"id":59211,"question":"La fonction f(x) = e^x est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = e^x est toujours positive sur ℝ, donc la fonction 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":59212,"question":"Quelle est la dérivée de la fonction f(x) = cos(3x + 2) ?","option_a":"-3sin(3x + 2)","option_b":"sin(3x + 2)","option_c":"-sin(3x + 2)","option_d":"3cos(3x + 2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions composées, f'(x) = -3sin(3x + 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\": \"-3sin(3x + 2)\", \"b\": \"sin(3x + 2)\", \"c\": \"-sin(3x + 2)\", \"d\": \"3c","_debug_options_count":4},{"id":59213,"question":"La fonction f(x) = 1\/x est-elle définie en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction f(x) = 1\/x n'est pas définie en x = 0 car le dénominateur s'annule, ce qui entraîne une discontinuité.","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}]