Quiz — Études de fonctions numériques, I.pdf

Question 1 sur 10 20:00

[{"id":47268,"question":"Quelle est la dérivée de la fonction f(x) = x² + 3x - 5 ?","option_a":"f'(x) = 2x + 3","option_b":"f'(x) = x² + 3","option_c":"f'(x) = 2x + 5","option_d":"f'(x) = x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : la dérivée de x² est 2x, celle de 3x est 3, et celle de -5 est 0. Ainsi, f'(x) = 2x + 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f'(x) = 2x + 3\", \"b\": \"f'(x) = x² + 3\", \"c\": \"f'(x) = 2x + 5\", \"","_debug_options_count":4},{"id":47269,"question":"La fonction f(x) = x³ - 3x² + 2 est croissante sur l'intervalle [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 6x. Sur [-1, 1], f'(x) est négative (par exemple, f'(-0.5) = 3*(0.25) - 6*(-0.5) = 0.75 + 3 = 3.75 \u003E 0 ? Non, f'(0.5) = 3*(0.25) - 6*(0.5) = 0.75 - 3 = -2.25 \u003C 0). La fonction n'est donc pas croissante sur cet 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":47270,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 4) lorsque x tend vers l'infini ?","option_a":"2","option_b":"1","option_c":"0","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour x → ∞, on divise le numérateur et le dénominateur par x² : f(x) = (2 + 3\/x - 1\/x²)\/(1 - 4\/x²). Les termes en 1\/x et 1\/x² 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\": \"a\", \"options\": {\"a\": \"2\", \"b\": \"1\", \"c\": \"0\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":47271,"question":"La fonction f(x) = |x| est 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 à cet endroit. La dérivée à gauche (f'(x) = -1) et à droite (f'(x) = 1) ne sont pas égales.","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":47272,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"f'(x) = e^(2x)","option_b":"f'(x) = 2e^(2x)","option_c":"f'(x) = e^x","option_d":"f'(x) = 2e^x","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de e^(u(x)) est u'(x) * e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, f'(x) = 2e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"f'(x) = e^(2x)\", \"b\": \"f'(x) = 2e^(2x)\", \"c\": \"f'(x) = e^x\", \"d\":","_debug_options_count":4},{"id":47273,"question":"La fonction f(x) = ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction logarithme népérien ln(x) n'est définie que pour x \u003E 0. Elle n'est donc pas définie pour x ≤ 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":47274,"question":"Quelle est la limite de la fonction f(x) = (sin(x))\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"-1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une limite classique : lim(x→0) sin(x)\/x = 1. Cette limite est fondamentale en analyse et peut être démontrée par le théorème des gendarmes ou les développements limités.","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\": \"1\", \"c\": \"-1\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":47275,"question":"La fonction f(x) = x^4 - 4x³ + 6x² est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La convexité est déterminée par la dérivée seconde : f''(x) = 12x² - 24x + 12 = 12(x² - 2x + 1) = 12(x-1)² ≥ 0. Comme f''(x) ≥ 0 pour tout x, la fonction est convexe sur ℝ.","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":47276,"question":"Quelle est la dérivée de la fonction f(x) = √(x² + 1) ?","option_a":"f'(x) = x\/√(x² + 1)","option_b":"f'(x) = 1\/√(x² + 1)","option_c":"f'(x) = x\/(2√(x² + 1))","option_d":"f'(x) = 2x\/√(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la dérivée des fonctions composées, f'(x) = (1\/(2√(x² + 1))) * 2x = x\/√(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\": \"f'(x) = x\/√(x² + 1)\", \"b\": \"f'(x) = 1\/√(x² + 1)\", \"c\": \"f'(","_debug_options_count":4},{"id":47277,"question":"La fonction f(x) = 1\/x est continue sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction 1\/x n'est pas définie en x = 0, donc elle n'est pas continue sur ℝ. Elle est cependant continue sur 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]