Quiz — Fonctions usuelles.pdf

Question 1 sur 10 20:00

[{"id":64311,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x + 2","option_c":"6x² + 2","option_d":"3x² + 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'une fonction polynôme s'obtient en appliquant la règle de dérivation terme à terme : (3x²)' = 6x, (2x)' = 2, et (-5)' = 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\": \"6x + 2\", \"b\": \"3x + 2\", \"c\": \"6x² + 2\", \"d\": \"3x² + 2x\"}}","_debug_options_count":4},{"id":64312,"question":"La fonction f(x) = e^x est toujours croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle a une dérivée toujours positive (e^x \u003E 0), donc elle est strictement croissante sur ℝ.","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":64313,"question":"Quelle est la limite de la fonction f(x) = (2x + 1)\/(x - 3) quand x tend vers +∞ ?","option_a":"2","option_b":"0","option_c":"1\/2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour x → +∞, on divise numérateur et dénominateur par x : (2 + 1\/x)\/(1 - 3\/x) → 2\/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\": \"2\", \"b\": \"0\", \"c\": \"1\/2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":64314,"question":"La fonction f(x) = ln(x) est définie pour x ∈ ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction logarithme népérien est définie uniquement pour x \u003E 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":64315,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"cos(2x)","option_b":"2cos(2x)","option_c":"sin(2x)","option_d":"-2sin(2x)","option_e":"","option_f":"","bonne_reponse":"b","explication":"En utilisant la règle de dérivation des fonctions composées : (sin(u))' = u'cos(u), ici u = 2x donc u' = 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\": \"cos(2x)\", \"b\": \"2cos(2x)\", \"c\": \"sin(2x)\", \"d\": \"-2sin(2x)\"}}","_debug_options_count":4},{"id":64316,"question":"La fonction f(x) = x³ - 3x² + 2 admet-elle un extremum local en x = 1 ?","option_a":"Oui, un maximum","option_b":"Oui, un minimum","option_c":"Non","option_d":"Oui, un point d'inflexion","option_e":"","option_f":"","bonne_reponse":"b","explication":"La dérivée f'(x) = 3x² - 6x s'annule en x = 0 et x = 2. En x = 1, f'(1) = -3 \u003C 0, donc la fonction est décroissante : pas d'extremum en x = 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\": \"Oui, un maximum\", \"b\": \"Oui, un minimum\", \"c\": \"Non\", \"d\": \"Oui, ","_debug_options_count":4},{"id":64317,"question":"Quelle est l'asymptote horizontale de la fonction f(x) = (3x² + 1)\/(x² + 2) quand x tend vers ±∞ ?","option_a":"y = 0","option_b":"y = 3","option_c":"y = 1\/2","option_d":"Aucune asymptote","option_e":"","option_f":"","bonne_reponse":"b","explication":"En divisant numérateur et dénominateur par x², on obtient (3 + 1\/x²)\/(1 + 2\/x²) → 3\/1 = 3 quand x → ±∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"y = 0\", \"b\": \"y = 3\", \"c\": \"y = 1\/2\", \"d\": \"Aucune asymptote\"}}","_debug_options_count":4},{"id":64318,"question":"La fonction f(x) = 1\/x admet une asymptote verticale en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction 1\/x tend vers ±∞ quand x tend vers 0, donc elle admet une asymptote verticale en x = 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},{"id":64319,"question":"Quelle est la dérivée de la fonction f(x) = ln(5x) ?","option_a":"1\/x","option_b":"5\/x","option_c":"1\/(5x)","option_d":"5\/(5x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la règle de dérivation des fonctions composées : (ln(u))' = u'\/u, ici u = 5x donc u' = 5. Ainsi, f'(x) = 5\/(5x) = 1\/x.","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\": \"5\/x\", \"c\": \"1\/(5x)\", \"d\": \"5\/(5x)\"}}","_debug_options_count":4},{"id":64320,"question":"La fonction f(x) = x² - 4x + 3 est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée seconde f''(x) = 2 \u003E 0, donc la fonction est convexe sur ℝ.","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}]