Quiz — ct24m0708 LPA.pdf

Question 1 sur 10 20:00

[{"id":8858,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"A) 6x + 2","option_b":"B) 3x + 2","option_c":"C) 6x² + 2x","option_d":"D) 6x + 2x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : (3x²)' = 6x, (2x)' = 2 et (-5)' = 0. Donc 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\": \"A) 6x + 2\", \"b\": \"B) 3x + 2\", \"c\": \"C) 6x² + 2x\", \"d\": \"D) 6x + ","_debug_options_count":4},{"id":8859,"question":"La dérivée d'une fonction constante est toujours égale à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction constante f(x) = k a pour dérivée f'(x) = 0, car sa pente est nulle en tout point.","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":8860,"question":"Quelle est la dérivée de f(x) = e^(2x) ?","option_a":"A) e^(2x)","option_b":"B) 2e^(2x)","option_c":"C) e^(x)","option_d":"D) 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\": \"A) e^(2x)\", \"b\": \"B) 2e^(2x)\", \"c\": \"C) e^(x)\", \"d\": \"D) 2e^x\"}}","_debug_options_count":4},{"id":8861,"question":"Si f'(x) \u003E 0 sur un intervalle, alors la fonction f est décroissante sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Si f'(x) \u003E 0, la fonction f est croissante sur l'intervalle. Si f'(x) \u003C 0, elle est décroissante.","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":8862,"question":"Quelle est la dérivée de f(x) = ln(x² + 1) ?","option_a":"A) 2x\/(x² + 1)","option_b":"B) 1\/(x² + 1)","option_c":"C) x\/(x² + 1)","option_d":"D) 2\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici u(x) = x² + 1, donc u'(x) = 2x. Ainsi 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\": \"A) 2x\/(x² + 1)\", \"b\": \"B) 1\/(x² + 1)\", \"c\": \"C) x\/(x² + 1)\", \"","_debug_options_count":4},{"id":8863,"question":"La fonction f(x) = x³ - 3x a-t-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 3x² - 3. En x = 1, f'(1) = 0. De plus, f''(x) = 6x, donc f''(1) = 6 \u003E 0 : la fonction admet un minimum local en x = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":8864,"question":"Quelle est la dérivée de f(x) = (x + 1)\/(x - 2) ?","option_a":"A) 1","option_b":"B) -3\/(x - 2)²","option_c":"C) 3\/(x - 2)²","option_d":"D) (x - 2)\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v². Ici u = x + 1, v = x - 2, donc u' = 1, v' = 1. Ainsi f'(x) = (1*(x - 2) - (x + 1)*1)\/(x - 2)² = -3\/(x - 2)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) 1\", \"b\": \"B) -3\/(x - 2)²\", \"c\": \"C) 3\/(x - 2)²\", \"d\": \"D) (x","_debug_options_count":4},{"id":8865,"question":"Si f'(x) = 0 en un point x = a, alors f admet nécessairement un extremum en x = a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une condition nécessaire pour un extremum est f'(a) = 0, mais ce n'est pas suffisant. Il faut aussi étudier le signe de f' autour de a ou calculer f''(a).","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":8866,"question":"Quelle est la dérivée de f(x) = sin(3x) ?","option_a":"A) cos(3x)","option_b":"B) 3cos(3x)","option_c":"C) -3cos(3x)","option_d":"D) sin(3x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de sin(u(x)) est u'(x)cos(u(x)). Ici u(x) = 3x, donc u'(x) = 3. Ainsi f'(x) = 3cos(3x).","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) cos(3x)\", \"b\": \"B) 3cos(3x)\", \"c\": \"C) -3cos(3x)\", \"d\": \"D) si","_debug_options_count":4},{"id":8867,"question":"La dérivée d'une fonction paire est toujours impaire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f est paire (f(-x) = f(x)), alors f'(-x) = -f'(x) : la dérivée est donc impaire. C'est un résultat général en analyse.","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}]