Quiz — chapitre 02.pdf

Question 1 sur 10 20:00

[{"id":68931,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"f'(x) = 6x + 2","option_b":"f'(x) = 3x + 2","option_c":"f'(x) = 6x² + 2x","option_d":"f'(x) = 6x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme se calcule terme par terme : (3x²)' = 6x, (2x)' = 2, et (-5)' = 0. Ainsi, 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\": \"f'(x) = 6x + 2\", \"b\": \"f'(x) = 3x + 2\", \"c\": \"f'(x) = 6x² + 2x\",","_debug_options_count":4},{"id":68932,"question":"La dérivée de la fonction f(x) = e^x est toujours positive.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle e^x a une dérivée égale à elle-même, donc e^x \u003E 0 pour tout x réel. Sa dérivée est donc toujours positive.","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":68933,"question":"Quelle est la dérivée de f(x) = ln(2x + 1) ?","option_a":"f'(x) = 1\/(2x + 1)","option_b":"f'(x) = 2\/(2x + 1)","option_c":"f'(x) = 1\/x","option_d":"f'(x) = 2x\/(2x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la règle de la chaîne, la dérivée de ln(u) est u'\/u. Ici, u = 2x + 1, donc u' = 2. Ainsi, f'(x) = 2\/(2x + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"f'(x) = 1\/(2x + 1)\", \"b\": \"f'(x) = 2\/(2x + 1)\", \"c\": \"f'(x) = 1\/x","_debug_options_count":4},{"id":68934,"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":68935,"question":"Quelle est la dérivée de f(x) = sin(3x) ?","option_a":"f'(x) = 3cos(3x)","option_b":"f'(x) = cos(3x)","option_c":"f'(x) = 3sin(3x)","option_d":"f'(x) = -3cos(3x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant la règle de la chaîne, la dérivée de sin(u) est u'cos(u). Ici, u = 3x, donc u' = 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\": \"a\", \"options\": {\"a\": \"f'(x) = 3cos(3x)\", \"b\": \"f'(x) = cos(3x)\", \"c\": \"f'(x) = 3sin(3x)","_debug_options_count":4},{"id":68936,"question":"La dérivée d'une constante est toujours égale à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée d'une constante est toujours égale à 0, car une constante ne varie pas.","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":68937,"question":"Quelle est la dérivée de f(x) = (x² + 1)\/(x - 1) ?","option_a":"f'(x) = (2x(x - 1) - (x² + 1))\/(x - 1)²","option_b":"f'(x) = (2x(x - 1) + (x² + 1))\/(x - 1)²","option_c":"f'(x) = (x² + 1)\/(x - 1)²","option_d":"f'(x) = 2x\/(x - 1)","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 et v = x - 1, donc u' = 2x et v' = 1. Ainsi, f'(x) = (2x(x - 1) - (x² + 1))\/(x - 1)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\": \"f'(x) = (2x(x - 1) - (x² + 1))\/(x - 1)²\", \"b\": \"f'(x) = (2x(x -","_debug_options_count":4},{"id":68938,"question":"Si f'(a) = 0, alors la fonction f admet un extremum en x = a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un extremum peut exister en x = a si f'(a) = 0, mais ce n'est pas une condition suffisante. Il faut aussi étudier le signe de f' autour de a ou utiliser la dérivée seconde.","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":68939,"question":"Quelle est la dérivée de f(x) = √(x² + 4) ?","option_a":"f'(x) = x\/√(x² + 4)","option_b":"f'(x) = 2x\/√(x² + 4)","option_c":"f'(x) = 1\/√(x² + 4)","option_d":"f'(x) = x\/2√(x² + 4)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de la chaîne, la dérivée de √u est u'\/(2√u). Ici, u = x² + 4, donc u' = 2x. Ainsi, f'(x) = 2x\/(2√(x² + 4)) = x\/√(x² + 4).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f'(x) = x\/√(x² + 4)\", \"b\": \"f'(x) = 2x\/√(x² + 4)\", \"c\": \"f'","_debug_options_count":4},{"id":68940,"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 sa dérivée f' vérifie f'(-x) = -f'(x), ce qui signifie que 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}]