Quiz — 4. Sciences derivation.....pdf

Question 1 sur 10 20:00

[{"id":57607,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"6x - 5","option_b":"3x - 5","option_c":"6x² - 5x","option_d":"3x² - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme se calcule terme à terme : (3x²)' = 6x, (-5x)' = -5, et la dérivée d'une constante est 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\": \"3x - 5\", \"c\": \"6x² - 5x\", \"d\": \"3x² - 5\"}}","_debug_options_count":4},{"id":57608,"question":"La dérivée de f(x) = (2x + 1)\/(x - 3) est-elle égale à (2x - 7)\/(x - 3)² ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant la formule de dérivation d'un quotient (u\/v)' = (u'v - uv')\/v², on obtient bien (2(x-3) - (2x+1)(1))\/(x-3)² = (2x-6-2x-1)\/(x-3)² = -7\/(x-3)². La réponse proposée est donc fausse.","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":57609,"question":"Quelle est la dérivée de f(x) = √(x) ?","option_a":"1\/(2√x)","option_b":"1\/√x","option_c":"√x\/2","option_d":"2√x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de √x = 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\": \"1\/(2√x)\", \"b\": \"1\/√x\", \"c\": \"√x\/2\", \"d\": \"2√x\"}}","_debug_options_count":4},{"id":57610,"question":"La fonction f(x) = x³ - 3x² + 2x est-elle croissante sur l'intervalle [0, 2] ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Calculons f'(x) = 3x² - 6x + 2. Sur [0,2], f'(x) est négative (par exemple, f'(1) = -1), donc la fonction est décroissante 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":57611,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x² + 1 au point d'abscisse x = 1 ?","option_a":"y = 2x + 1","option_b":"y = 2x - 1","option_c":"y = x + 1","option_d":"y = 2x","option_e":"","option_f":"","bonne_reponse":"b","explication":"La tangente a pour équation y = f'(a)(x - a) + f(a). Ici, f(1) = 2 et f'(1) = 2, donc y = 2(x - 1) + 2 = 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\": \"y = 2x + 1\", \"b\": \"y = 2x - 1\", \"c\": \"y = x + 1\", \"d\": \"y = 2x\"}}","_debug_options_count":4},{"id":57612,"question":"La dérivée de f(x) = e^(2x) est-elle f'(x) = 2e^(2x) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2, et f'(x) = 2e^(2x). La réponse est donc 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":57613,"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 + 1\/3)","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, et 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":57614,"question":"La fonction f(x) = x^4 - 4x³ + 4x² admet-elle un extremum en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Calculons f'(x) = 4x³ - 12x² + 8x. f'(1) = 0, et f''(1) = 12 \u003E 0, donc x = 1 est un minimum local. La réponse est 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":57615,"question":"Quelle est la dérivée de f(x) = sin(2x) ?","option_a":"2cos(2x)","option_b":"cos(2x)","option_c":"2sin(x)cos(x)","option_d":"-2sin(2x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin(u(x)) est u'(x)cos(u(x)). Ici, u(x) = 2x, donc u'(x) = 2, et 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\": \"2cos(2x)\", \"b\": \"cos(2x)\", \"c\": \"2sin(x)cos(x)\", \"d\": \"-2sin(2x)\"","_debug_options_count":4},{"id":57616,"question":"La dérivée de f(x) = 1\/x est-elle f'(x) = -1\/x² ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de 1\/x = x^(-1) est -x^(-2) = -1\/x². La réponse est donc 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}]