Quiz — td_fcts.pdf

Question 1 sur 10 20:00

[{"id":36990,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 5x - 2 ?","option_a":"6x + 5","option_b":"3x² + 5","option_c":"6x + 5x - 2","option_d":"9x² + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme se calcule terme par terme : (3x²)' = 6x, (5x)' = 5 et (-2)' = 0. Donc f'(x) = 6x + 5.","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 + 5\", \"b\": \"3x² + 5\", \"c\": \"6x + 5x - 2\", \"d\": \"9x² + 5\"}}","_debug_options_count":4},{"id":36991,"question":"La dérivée de f(x) = e^(2x) est f'(x) = 2e^(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : la dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici u(x) = 2x donc u'(x) = 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":36992,"question":"Quelle est la dérivée de f(x) = ln(3x + 1) ?","option_a":"3\/(3x + 1)","option_b":"1\/(3x + 1)","option_c":"3x\/(3x + 1)","option_d":"1\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici u(x) = 3x + 1 donc u'(x) = 3. Ainsi f'(x) = 3\/(3x + 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\": \"3\/(3x + 1)\", \"b\": \"1\/(3x + 1)\", \"c\": \"3x\/(3x + 1)\", \"d\": \"1\/x\"}}","_debug_options_count":4},{"id":36993,"question":"La fonction f(x) = x³ - 3x² + 2 admet un extremum local en x = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : f'(x) = 3x² - 6x. f'(1) = 0 et f''(1) = 6 - 6 = 0 (changement de signe de f' autour de 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":36994,"question":"Quelle est la dérivée de f(x) = (2x + 1)\/(x - 3) ?","option_a":"(2)\/(x - 3)","option_b":"(2x - 7)\/(x - 3)²","option_c":"2\/(x - 3)²","option_d":"(x - 3)\/(2x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"On utilise la formule (u\/v)' = (u'v - uv')\/v². Ici u = 2x + 1, v = x - 3 donc u' = 2, v' = 1. Ainsi f'(x) = (2(x - 3) - (2x + 1)(1))\/(x - 3)² = (2x - 6 - 2x - 1)\/(x - 3)² = -7\/(x - 3)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\": \"(2)\/(x - 3)\", \"b\": \"(2x - 7)\/(x - 3)²\", \"c\": \"2\/(x - 3)²\", \"d\":","_debug_options_count":4},{"id":36995,"question":"La dérivée de f(x) = sin(2x) est f'(x) = 2cos(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : la dérivée de sin(u(x)) est u'(x)cos(u(x)). Ici u(x) = 2x donc u'(x) = 2. Ainsi f'(x) = 2cos(2x).","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":36996,"question":"Quelle est la dérivée de f(x) = √(x² + 1) ?","option_a":"x\/√(x² + 1)","option_b":"1\/√(x² + 1)","option_c":"2x\/√(x² + 1)","option_d":"(x² + 1)\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la dérivée d'une fonction composée : (√u)' = u'\/(2√u). Ici u = x² + 1 donc u' = 2x. Ainsi f'(x) = 2x\/(2√(x² + 1)) = 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\": \"x\/√(x² + 1)\", \"b\": \"1\/√(x² + 1)\", \"c\": \"2x\/√(x² + 1)\", \"","_debug_options_count":4},{"id":36997,"question":"La fonction f(x) = x⁴ - 4x³ admet un point d'inflexion en x = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : f''(x) = 12x² - 24x. f''(2) = 48 - 48 = 0 et f'' change de signe autour de x=2 (concavité).","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":36998,"question":"Quelle est la dérivée de f(x) = cos(3x + π\/2) ?","option_a":"-3sin(3x)","option_b":"3sin(3x + π\/2)","option_c":"-3sin(3x + π\/2)","option_d":"sin(3x + π\/2)","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée de cos(u(x)) est -u'(x)sin(u(x)). Ici u(x) = 3x + π\/2 donc u'(x) = 3. Ainsi f'(x) = -3sin(3x + π\/2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"-3sin(3x)\", \"b\": \"3sin(3x + π\/2)\", \"c\": \"-3sin(3x + π\/2)\", \"d\":","_debug_options_count":4},{"id":36999,"question":"La dérivée de f(x) = e^(-x²) est f'(x) = -2xe^(-x²).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai : la dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici u(x) = -x² donc u'(x) = -2x. Ainsi f'(x) = -2xe^(-x²).","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}]