Quiz — درس3.pdf

Question 1 sur 10 20:00

[{"id":77796,"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":"3x² + 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : (3x²)' = 6x, (2x)' = 2, (-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\": \"6x + 2\", \"b\": \"3x + 2\", \"c\": \"6x² + 2\", \"d\": \"3x² + 2x\"}}","_debug_options_count":4},{"id":77797,"question":"La dérivée d'une fonction constante est toujours nulle.","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 tangente est horizontale 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":77798,"question":"Quelle est la dérivée de f(x) = sin(2x) ?","option_a":"2cos(2x)","option_b":"cos(2x)","option_c":"2sin(x)","option_d":"cos(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de la chaîne : (sin(2x))' = cos(2x) × 2 = 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)\", \"d\": \"cos(x)\"}}","_debug_options_count":4},{"id":77799,"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 est croissante. 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":77800,"question":"Quelle est la dérivée de f(x) = (x² + 1)\/(x - 3) ?","option_a":"(2x(x-3) - (x²+1))\/(x-3)²","option_b":"(2x(x-3) + (x²+1))\/(x-3)²","option_c":"(2x(x-3))\/(x-3)²","option_d":"(x²+1)\/(x-3)²","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 = x² + 1, v = x - 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\": \"(2x(x-3) - (x²+1))\/(x-3)²\", \"b\": \"(2x(x-3) + (x²+1))\/(x-3)²\",","_debug_options_count":4},{"id":77801,"question":"La dérivée de f(x) = e^x est e^x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle est l'une des rares fonctions égales à sa dérivée : (e^x)' = e^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":77802,"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":"3\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de la chaîne : (ln(u))' = u'\/u. Ici u = 3x + 1, donc u' = 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\": \"3\/(3x + 1)\", \"b\": \"1\/(3x + 1)\", \"c\": \"3x\/(3x + 1)\", \"d\": \"3\/(x + ","_debug_options_count":4},{"id":77803,"question":"Si f'(a) = 0 et f''(a) \u003E 0, alors la fonction f admet un minimum local en x = a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f'(a) = 0 et f''(a) \u003E 0, la fonction est concave vers le haut en a, donc elle admet un minimum local en ce 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":77804,"question":"Quelle est la dérivée de f(x) = √(x² + 4) ?","option_a":"x\/√(x² + 4)","option_b":"2x\/√(x² + 4)","option_c":"(x² + 4)\/√(x² + 4)","option_d":"1\/√(x² + 4)","option_e":"","option_f":"","bonne_reponse":"a","explication":"On utilise la règle de la chaîne : (√u)' = u'\/(2√u). Ici u = x² + 4, donc u' = 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\": \"x\/√(x² + 4)\", \"b\": \"2x\/√(x² + 4)\", \"c\": \"(x² + 4)\/√(x² ","_debug_options_count":4},{"id":77805,"question":"La dérivée d'une fonction paire est toujours une fonction 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), donc f' est impaire.","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}]