Quiz — 65d77b4f03e21_corrigé_ex 3 série 12.pdf

Question 1 sur 10 20:00

[{"id":92052,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"3x² - 4x + 5","option_b":"3x² - 4x + 5x","option_c":"x³ - 4x + 5","option_d":"3x² - 2x + 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de x³ est 3x², celle de -2x² est -4x, celle de 5x est 5, et la dérivée de -1 est 0. Ainsi, f'(x) = 3x² - 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\": \"3x² - 4x + 5\", \"b\": \"3x² - 4x + 5x\", \"c\": \"x³ - 4x + 5\", \"d\": ","_debug_options_count":4},{"id":92053,"question":"La fonction f(x) = x² - 4x + 3 admet un minimum en x = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = 2x - 4 s'annule en x = 2. Comme f''(x) = 2 \u003E 0, il s'agit bien d'un minimum.","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":92054,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x² au point d'abscisse x = 1 ?","option_a":"y = 2x - 1","option_b":"y = x + 1","option_c":"y = 2x + 1","option_d":"y = x - 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La tangente a pour équation y = f'(a)(x - a) + f(a). Ici, f(1) = 1 et f'(1) = 2, donc y = 2(x - 1) + 1 = 2x - 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\": \"y = 2x - 1\", \"b\": \"y = x + 1\", \"c\": \"y = 2x + 1\", \"d\": \"y = x - 1","_debug_options_count":4},{"id":92055,"question":"La fonction f(x) = (x² + 1)\/(x - 1) est dérivable en x = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction n'est pas définie en x = 1 (dénominateur nul), donc elle n'est pas dérivable en ce point.","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":92056,"question":"Quelle est la limite de f(x) = (x² - 4)\/(x - 2) lorsque x tend vers 2 ?","option_a":"0","option_b":"4","option_c":"2","option_d":"La limite n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"En factorisant, f(x) = (x - 2)(x + 2)\/(x - 2) = x + 2 pour x ≠ 2. Ainsi, lim(x→2) f(x) = 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"4\", \"c\": \"2\", \"d\": \"La limite n'existe pas\"}}","_debug_options_count":4},{"id":92057,"question":"La dérivée de la fonction exponentielle 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(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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":92058,"question":"Quelle est la solution de l'équation différentielle y' + 2y = 0 ?","option_a":"y = Ce^(-2x)","option_b":"y = Cx + 2","option_c":"y = Ce^(2x)","option_d":"y = 2x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une équation différentielle linéaire du premier ordre. La solution générale est y = Ce^(-2x), où C est une constante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"y = Ce^(-2x)\", \"b\": \"y = Cx + 2\", \"c\": \"y = Ce^(2x)\", \"d\": \"y = 2","_debug_options_count":4},{"id":92059,"question":"La fonction f(x) = ln(x) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction ln(x) est définie uniquement pour x \u003E 0. Elle n'est pas définie pour 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":92060,"question":"Quelle est la dérivée de la fonction f(x) = sin(3x) ?","option_a":"cos(3x)","option_b":"3cos(3x)","option_c":"sin(3x)","option_d":"-3cos(3x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée de sin(u(x)) est u'(x)cos(u(x)). Ici, u(x) = 3x, donc u'(x) = 3, d'où 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\": \"b\", \"options\": {\"a\": \"cos(3x)\", \"b\": \"3cos(3x)\", \"c\": \"sin(3x)\", \"d\": \"-3cos(3x)\"}}","_debug_options_count":4},{"id":92061,"question":"La fonction f(x) = x\/(x² + 1) admet un maximum en x = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée f'(x) = (1 - x²)\/(x² + 1)² s'annule en x = 1. Comme f''(1) \u003C 0, il s'agit bien d'un maximum.","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}]