Quiz — cours_eucRn.pdf

Question 1 sur 10 20:00

[{"id":62438,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x + 2x - 5","option_d":"6x² + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x² est 2x, donc 3x² donne 6x. La dérivée de 2x est 2, et la constante -5 disparaît.","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\": \"3x² + 2\", \"c\": \"6x + 2x - 5\", \"d\": \"6x² + 2\"}}","_debug_options_count":4},{"id":62439,"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":"La dérivée de e^u est u' × e^u. Ici, u = 2x, donc u' = 2. D'où 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\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":62440,"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) est u'\/u. Ici, u = 3x + 1, donc u' = 3. D'où 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":62441,"question":"Si f'(x) \u003E 0 sur un intervalle, alors 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. 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":62442,"question":"Quelle est la dérivée de f(x) = (2x + 1)\/(x - 3) ?","option_a":"(2x - 7)\/(x - 3)²","option_b":"2\/(x - 3)","option_c":"(2x + 1)\/(x - 3)","option_d":"2\/(x - 3)²","option_e":"","option_f":"","bonne_reponse":"a","explication":"Utilisez la formule (u\/v)' = (u'v - uv')\/v². Ici, u = 2x + 1 (u' = 2) et v = x - 3 (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\": \"2\/(x - 3)\", \"c\": \"(2x + 1)\/(x - 3)\", \"","_debug_options_count":4},{"id":62443,"question":"La dérivée de f(x) = x³ est toujours positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² est toujours positive sauf en x = 0 où elle est nulle. Elle n'est jamais strictement positive partout.","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":62444,"question":"Quelle est la dérivée de f(x) = sin(5x) ?","option_a":"5cos(5x)","option_b":"cos(5x)","option_c":"sin(5x)","option_d":"5sin(5x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin(u) est u' × cos(u). Ici, u = 5x, donc u' = 5. D'où f'(x) = 5cos(5x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"5cos(5x)\", \"b\": \"cos(5x)\", \"c\": \"sin(5x)\", \"d\": \"5sin(5x)\"}}","_debug_options_count":4},{"id":62445,"question":"Si f'(a) = 0, alors la tangente à la courbe de f en x = a est horizontale.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f'(a) = 0, la pente de la tangente est nulle, donc elle 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":62446,"question":"Quelle est la dérivée de f(x) = √(x² + 1) ?","option_a":"x\/√(x² + 1)","option_b":"2x\/√(x² + 1)","option_c":"1\/√(x² + 1)","option_d":"(x² + 1)\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de √u est u'\/(2√u). Ici, u = x² + 1, donc u' = 2x. D'où 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\": \"2x\/√(x² + 1)\", \"c\": \"1\/√(x² + 1)\", \"","_debug_options_count":4},{"id":62447,"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":"La dérivée d'une constante k est 0, car sa pente est nulle partout.","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}]