Quiz — 4aM-ds1-kef2009.pdf

Question 1 sur 10 20:00

[{"id":21749,"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":"6x + 2x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynomiale se calcule terme par terme. Pour 3x², la dérivée est 6x. Pour 2x, c'est 2. La constante -5 a une dérivée nulle.","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\": \"6x + 2x - 5\"}}","_debug_options_count":4},{"id":21750,"question":"La dérivée de f(x) = e^x est-elle toujours positive ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle e^x a une dérivée égale à elle-même, donc toujours positive pour tout x réel.","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":21751,"question":"Quelle est la dérivée de f(x) = (2x + 1)\/(x - 3) ?","option_a":"(2x - 7)\/(x - 3)²","option_b":"(2x + 1)\/(x - 3)²","option_c":"2\/(x - 3)","option_d":"(2x - 7)\/(x - 3)","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v². Ici, u = 2x + 1 et v = x - 3, donc u' = 2 et v' = 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 - 7)\/(x - 3)²\", \"b\": \"(2x + 1)\/(x - 3)²\", \"c\": \"2\/(x - 3)\",","_debug_options_count":4},{"id":21752,"question":"Si f'(x) = 0, que peut-on dire de la tangente à la courbe en ce point ?","option_a":"Elle est horizontale","option_b":"Elle est verticale","option_c":"Elle est parallèle à l'axe des ordonnées","option_d":"Elle n'existe pas","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée f'(x) est nulle en un point, la tangente à la courbe en ce point est horizontale.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Elle est horizontale\", \"b\": \"Elle est verticale\", \"c\": \"Elle est ","_debug_options_count":4},{"id":21753,"question":"La dérivée de f(x) = ln(x) est-elle 1\/x ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction logarithme népérien ln(x) est bien 1\/x pour x \u003E 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":21754,"question":"Quelle est la dérivée de f(x) = sin(2x) ?","option_a":"cos(2x)","option_b":"2cos(2x)","option_c":"sin(2x)","option_d":"2sin(2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la règle de la chaîne : la dérivée de sin(u) est u'cos(u). Ici, u = 2x, donc u' = 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\": \"cos(2x)\", \"b\": \"2cos(2x)\", \"c\": \"sin(2x)\", \"d\": \"2sin(2x)\"}}","_debug_options_count":4},{"id":21755,"question":"Si f(x) = x³ - 3x² + 2, quel est le nombre de points critiques de f ?","option_a":"1","option_b":"2","option_c":"3","option_d":"0","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les points critiques sont les solutions de f'(x) = 0. f'(x) = 3x² - 6x = 3x(x - 2). Les solutions sont x = 0 et x = 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\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"0\"}}","_debug_options_count":4},{"id":21756,"question":"La dérivée de f(x) = √x est-elle 1\/(2√x) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de √x (ou x^(1\/2)) est (1\/2)x^(-1\/2) = 1\/(2√x).","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":21757,"question":"Quelle est la dérivée de f(x) = cos(3x) ?","option_a":"-sin(3x)","option_b":"-3sin(3x)","option_c":"sin(3x)","option_d":"3cos(3x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la règle de la chaîne : la dérivée de cos(u) est -u'sin(u). Ici, u = 3x, donc u' = 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\": \"-sin(3x)\", \"b\": \"-3sin(3x)\", \"c\": \"sin(3x)\", \"d\": \"3cos(3x)\"}}","_debug_options_count":4},{"id":21758,"question":"Si f'(x) \u003E 0 sur un intervalle, que peut-on dire de la fonction f sur cet intervalle ?","option_a":"Elle est décroissante","option_b":"Elle est croissante","option_c":"Elle est constante","option_d":"Elle est concave","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si la dérivée f'(x) est positive sur un intervalle, la fonction f est strictement 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\": \"Elle est décroissante\", \"b\": \"Elle est croissante\", \"c\": \"Elle e","_debug_options_count":4}]