Quiz — dzexams-docs-2as-903985.pdf

Question 1 sur 10 20:00

[{"id":95459,"question":"Quelle est la dérivée de la fonction f(x) = 3x^4 - 2x^2 + 5 ?","option_a":"12x^3 - 4x","option_b":"12x^3 - 4x + 5","option_c":"9x^3 - 4x","option_d":"12x^3 - 4x^2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle f'(x) = n*ax^(n-1) à chaque terme. Ici, 3x^4 donne 12x^3, -2x^2 donne -4x, et 5 donne 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\": \"12x^3 - 4x\", \"b\": \"12x^3 - 4x + 5\", \"c\": \"9x^3 - 4x\", \"d\": \"12x^3","_debug_options_count":4},{"id":95460,"question":"Si f'(x) = 0 pour tout x dans un intervalle, alors la fonction f est constante sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si la dérivée est nulle sur tout un intervalle, cela signifie que la fonction ne varie pas, donc elle est constante sur cet intervalle.","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":95461,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2xe^(2x-1)","option_d":"e^(x)","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, 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\": \"2e^(2x)\", \"b\": \"e^(2x)\", \"c\": \"2xe^(2x-1)\", \"d\": \"e^(x)\"}}","_debug_options_count":4},{"id":95462,"question":"La tangente à la courbe d'une fonction f en un point d'abscisse a a pour équation y = f'(a)(x - a) + f(a).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est la formule de l'équation de la tangente à une courbe en un point, où f'(a) est le coefficient directeur et f(a) l'ordonnée à l'origine.","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":95463,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x + 1) ?","option_a":"3\/(3x + 1)","option_b":"1\/(3x + 1)","option_c":"3x\/(3x + 1)","option_d":"(3x + 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, 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\": \"(3x + 1","_debug_options_count":4},{"id":95464,"question":"Si une fonction est croissante sur un intervalle, alors sa dérivée est positive sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction est croissante sur un intervalle si et seulement si sa dérivée est positive sur cet intervalle.","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":95465,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"2cos(2x)","option_b":"cos(2x)","option_c":"2sin(2x)","option_d":"-2cos(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, d'où 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(2x)\", \"d\": \"-2cos(2x)\"}}","_debug_options_count":4},{"id":95466,"question":"La dérivée d'une fonction 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 fonction constante est toujours égale à 0, car la pente de la tangente est nulle.","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":95467,"question":"Quelle est la dérivée de la fonction f(x) = 1\/x ?","option_a":"-1\/x^2","option_b":"1\/x^2","option_c":"-x^2","option_d":"x^2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de 1\/x = x^(-1) est -1*x^(-2) = -1\/x^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\": \"-1\/x^2\", \"b\": \"1\/x^2\", \"c\": \"-x^2\", \"d\": \"x^2\"}}","_debug_options_count":4},{"id":95468,"question":"Si f'(x) \u003E 0 pour tout x dans 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. Une dérivée positive indique une croissance.","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}]