Quiz — Cours Math Etude de fonction (1).pdf

Question 1 sur 10 20:00

[{"id":31220,"question":"Quelle est la limite de la fonction f(x) = (2x + 1)\/(x - 3) lorsque x tend vers +∞ ?","option_a":"A. 0","option_b":"B. 2","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant le numérateur et le dénominateur par x, on obtient f(x) = (2 + 1\/x)\/(1 - 3\/x). Lorsque x tend vers +∞, 1\/x et 3\/x tendent vers 0, 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\": \"A. 0\", \"b\": \"B. 2\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":31221,"question":"La fonction f(x) = x^3 - 3x^2 + 2 est continue sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les fonctions polynomiales sont continues sur ℝ, donc f(x) = x^3 - 3x^2 + 2 est bien continue sur tout son ensemble de définition.","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":31222,"question":"Quelle est la dérivée de la fonction f(x) = 4x^5 - 2x^3 + x ?","option_a":"A. 20x^4 - 6x^2 + 1","option_b":"B. 20x^4 - 6x^2","option_c":"C. 4x^4 - 2x^2 + 1","option_d":"D. 20x^5 - 6x^3 + x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée se calcule terme à terme : (4x^5)' = 20x^4, (-2x^3)' = -6x^2, et (x)' = 1. Donc f'(x) = 20x^4 - 6x^2 + 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\": \"A. 20x^4 - 6x^2 + 1\", \"b\": \"B. 20x^4 - 6x^2\", \"c\": \"C. 4x^4 - 2x^","_debug_options_count":4},{"id":31223,"question":"Si f'(x) \u003E 0 sur un intervalle I, alors la fonction f est :","option_a":"A. Décroissante sur I","option_b":"B. Croissante sur I","option_c":"C. Constante sur I","option_d":"D. Nulle sur I","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le signe de la dérivée f'(x) indique le sens de variation de la fonction : si f'(x) \u003E 0, alors f est croissante sur I.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. Décroissante sur I\", \"b\": \"B. Croissante sur I\", \"c\": \"C. Con","_debug_options_count":4},{"id":31224,"question":"La fonction f(x) = x^2 - 4x + 3 admet un minimum en x = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 2x - 4 s'annule en x = 2. Comme f''(x) = 2 \u003E 0, il s'agit bien d'un minimum local.","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":31225,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x^2 en x = 1 ?","option_a":"A. y = 2x - 1","option_b":"B. y = x + 1","option_c":"C. y = 2x + 1","option_d":"D. y = x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente en x = a a pour équation y = f'(a)(x - a) + f(a). Ici, f(1) = 1, f'(1) = 2, donc l'équation est y = 2(x - 1) + 1 = 2x - 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\": \"A. y = 2x - 1\", \"b\": \"B. y = x + 1\", \"c\": \"C. y = 2x + 1\", \"d\": \"","_debug_options_count":4},{"id":31226,"question":"Si une fonction f est dérivable en un point a, alors elle est nécessairement continue en a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute fonction dérivable en un point est continue en ce point. La réciproque n'est pas vraie.","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":31227,"question":"Quelle est la limite de f(x) = (sin x)\/x lorsque x tend vers 0 ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. +∞","option_d":"D. -∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une limite classique : lim(x→0) (sin x)\/x = 1, souvent démontrée par encadrement ou règle de l'Hôpital.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 1\", \"c\": \"C. +∞\", \"d\": \"D. -∞\"}}","_debug_options_count":4},{"id":31228,"question":"La fonction f(x) = 1\/x est dérivable sur ℝ.","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, donc elle n'est pas dérivable sur ℝ. Elle est dérivable sur ℝ* (ℝ privé de 0).","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":31229,"question":"Quel est le maximum de la fonction f(x) = -x^2 + 4x sur l'intervalle [0, 4] ?","option_a":"A. 0","option_b":"B. 4","option_c":"C. 2","option_d":"D. -4","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée f'(x) = -2x + 4 s'annule en x = 2. f(2) = -4 + 8 = 4. C'est le maximum sur l'intervalle [0, 4].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A. 0\", \"b\": \"B. 4\", \"c\": \"C. 2\", \"d\": \"D. -4\"}}","_debug_options_count":4}]