Quiz — 270.._sujet-3-sc.pdf

Question 1 sur 10 20:00

[{"id":33170,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x + 2","option_c":"6x² + 2x","option_d":"3x² + 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle : (ax^n)' = n*ax^(n-1). Ici, 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 + 2x\", \"d\": \"3x² + 2\"}}","_debug_options_count":4},{"id":33171,"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. 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":33172,"question":"Quelle est la dérivée de la fonction g(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2xe^(2x)","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. Ainsi, g'(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)\", \"d\": \"e^x\"}}","_debug_options_count":4},{"id":33173,"question":"La tangente à la courbe de f au 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":"L'équation de la tangente à la courbe de f au point d'abscisse a est bien y = f'(a)(x - a) + f(a).","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":33174,"question":"Quelle est la dérivée de la fonction h(x) = ln(3x + 1) ?","option_a":"1\/(3x + 1)","option_b":"3\/(3x + 1)","option_c":"1\/(x + 1)","option_d":"3\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x) = 3x + 1, donc u'(x) = 3. Ainsi, h'(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\": \"b\", \"options\": {\"a\": \"1\/(3x + 1)\", \"b\": \"3\/(3x + 1)\", \"c\": \"1\/(x + 1)\", \"d\": \"3\/(x + 1)","_debug_options_count":4},{"id":33175,"question":"Si f'(x) = 0 en un point, alors ce point est nécessairement un extremum local.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un point où f'(x) = 0 peut être un extremum, mais ce n'est pas toujours le cas (exemple : f(x) = x³ en x = 0).","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":33176,"question":"Quelle est la dérivée de la fonction k(x) = (2x + 1)\/(x - 3) ?","option_a":"2\/(x - 3)²","option_b":"(2x - 7)\/(x - 3)²","option_c":"(2x + 1)\/(x - 3)²","option_d":"2\/(x - 3)","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour dériver un quotient u\/v, on utilise la formule (u'v - uv')\/v². Ici, u = 2x + 1, v = x - 3, donc u' = 2 et v' = 1. Ainsi, k'(x) = (2(x - 3) - (2x + 1)(1))\/(x - 3)² = (2x - 6 - 2x - 1)\/(x - 3)² = -7\/(x - 3)².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2\/(x - 3)²\", \"b\": \"(2x - 7)\/(x - 3)²\", \"c\": \"(2x + 1)\/(x - 3)²","_debug_options_count":4},{"id":33177,"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 fonction constante f(x) = c est f'(x) = 0, car la pente de la tangente 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":33178,"question":"Quelle est la dérivée de la fonction m(x) = sin(4x) ?","option_a":"4cos(4x)","option_b":"cos(4x)","option_c":"4sin(4x)","option_d":"sin(4x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de sin(u(x)) est u'(x)cos(u(x)). Ici, u(x) = 4x, donc u'(x) = 4. Ainsi, m'(x) = 4cos(4x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"4cos(4x)\", \"b\": \"cos(4x)\", \"c\": \"4sin(4x)\", \"d\": \"sin(4x)\"}}","_debug_options_count":4},{"id":33179,"question":"Si f'(x) change de signe autour d'un point critique, alors ce point est un extremum local.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si f'(x) change de signe autour d'un point critique (par exemple de positif à négatif), alors ce point est un maximum local. Si le changement est de négatif à positif, c'est un minimum local.","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}]