Quiz — سلسلة الدوال للثانية علمي.doc

Question 1 sur 10 20:00

[{"id":18009,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"6x² + 2","option_c":"3x + 2","option_d":"6x - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de 3x² est 6x, celle de 2x est 2, et celle de -5 est 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\": \"6x² + 2\", \"c\": \"3x + 2\", \"d\": \"6x - 5\"}}","_debug_options_count":4},{"id":18010,"question":"La dérivée d'une fonction constante est toujours égale à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction constante (ex: f(x) = 5) est bien 0, car sa pente est nulle 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":18011,"question":"Quelle est la dérivée de la fonction f(x) = (2x + 1)(x - 3) ?","option_a":"4x - 5","option_b":"2x - 5","option_c":"3x - 5","option_d":"4x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule (uv)' = u'v + uv', on obtient f'(x) = 2(x - 3) + (2x + 1)(1) = 4x - 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\": \"4x - 5\", \"b\": \"2x - 5\", \"c\": \"3x - 5\", \"d\": \"4x + 5\"}}","_debug_options_count":4},{"id":18012,"question":"Si f'(x) \u003E 0 sur un intervalle, alors la fonction f est croissante sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale des dérivées : si f'(x) \u003E 0, f est strictement croissante sur l'intervalle considéré.","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":18013,"question":"Quelle est la dérivée de la fonction f(x) = 1\/x ?","option_a":"-1\/x²","option_b":"1\/x²","option_c":"-x","option_d":"1\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de 1\/x est -1\/x², car (1\/x)' = -1\/x² (dérivée 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\": \"-1\/x²\", \"b\": \"1\/x²\", \"c\": \"-x\", \"d\": \"1\/x\"}}","_debug_options_count":4},{"id":18014,"question":"La dérivée de la fonction f(x) = sin(x) est cos(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est exact : la dérivée de sin(x) est bien cos(x), une propriété fondamentale des fonctions trigonométriques.","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":18015,"question":"Quelle est la dérivée de la fonction f(x) = e^(3x) ?","option_a":"3e^(3x)","option_b":"e^(3x)","option_c":"3x e^(3x)","option_d":"e^(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la formule de dérivation des fonctions exponentielles (e^u)' = u'e^u, on obtient f'(x) = 3e^(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\": \"3e^(3x)\", \"b\": \"e^(3x)\", \"c\": \"3x e^(3x)\", \"d\": \"e^(x)\"}}","_debug_options_count":4},{"id":18016,"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":"Faux. f'(a) = 0 est une condition nécessaire mais pas suffisante : il faut aussi étudier le signe de f' autour de a pour conclure.","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":18017,"question":"Quelle est la dérivée de la fonction f(x) = ln(x) ?","option_a":"1\/x","option_b":"x","option_c":"-1\/x","option_d":"ln(x)\/x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de ln(x) est 1\/x, une propriété essentielle des fonctions logarithmiques.","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\/x\", \"b\": \"x\", \"c\": \"-1\/x\", \"d\": \"ln(x)\/x\"}}","_debug_options_count":4},{"id":18018,"question":"La dérivée de la fonction f(x) = x³ + 2x² - x + 4 est f'(x) = 3x² + 4x - 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est exact : la dérivée de x³ est 3x², celle de 2x² est 4x, celle de -x est -1, et celle de 4 est 0.","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}]