Quiz — 3-Derivée ( n ).pdf

Question 1 sur 10 20:00

[{"id":23119,"question":"Quelle est la dérivée de la fonction f(x) = 3x^4 - 2x^2 + 5 ?","option_a":"A. 12x^3 - 4x","option_b":"B. 12x^3 - 4x + 5","option_c":"C. 3x^3 - 2x","option_d":"D. 12x^3 - 4x^2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en multipliant chaque coefficient par l'exposant et en diminuant l'exposant de 1. Ainsi, (3x^4)' = 12x^3 et (-2x^2)' = -4x. La constante 5 a une dérivée nulle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. 12x^3 - 4x\", \"b\": \"B. 12x^3 - 4x + 5\", \"c\": \"C. 3x^3 - 2x\", \"d","_debug_options_count":4},{"id":23120,"question":"La dérivée de la fonction 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":"C'est vrai. La dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, 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":23121,"question":"Quelle est la dérivée de la fonction f(x) = ln(x) ?","option_a":"A. 1\/x","option_b":"B. -1\/x","option_c":"C. x","option_d":"D. 1\/x^2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction logarithme népérien ln(x) est 1\/x. Cette propriété est fondamentale en calcul différentiel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. 1\/x\", \"b\": \"B. -1\/x\", \"c\": \"C. x\", \"d\": \"D. 1\/x^2\"}}","_debug_options_count":4},{"id":23122,"question":"La fonction f(x) = x^3 - 3x^2 + 2x admet-elle un extremum local en x = 1 ?","option_a":"A. Oui, un maximum","option_b":"B. Oui, un minimum","option_c":"C. Non","option_d":"D. Oui, mais on ne peut pas savoir","option_e":"","option_f":"","bonne_reponse":"b","explication":"Calculons f'(x) = 3x^2 - 6x + 2. En x = 1, f'(1) = 3 - 6 + 2 = -1 ≠ 0, donc pas d'extremum en x = 1. L'extremum se trouve là où f'(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\": \"A. Oui, un maximum\", \"b\": \"B. Oui, un minimum\", \"c\": \"C. Non\", \"d","_debug_options_count":4},{"id":23123,"question":"La dérivée de la fonction f(x) = sin(2x) est f'(x) = 2cos(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai. La dérivée de sin(u(x)) est u'(x)cos(u(x)). Ici, u(x) = 2x, donc u'(x) = 2. Ainsi, 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":23124,"question":"Quelle est la dérivée de la fonction f(x) = 1\/x ?","option_a":"A. -1\/x^2","option_b":"B. 1\/x^2","option_c":"C. x^2","option_d":"D. -x^2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de 1\/x est -1\/x^2. On peut aussi l'écrire comme -x^(-2) et appliquer la règle de dérivation des puissances.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A. -1\/x^2\", \"b\": \"B. 1\/x^2\", \"c\": \"C. x^2\", \"d\": \"D. -x^2\"}}","_debug_options_count":4},{"id":23125,"question":"La fonction f(x) = x^2 - 4x + 3 est-elle croissante sur l'intervalle [2, +∞[ ?","option_a":"A. Oui, car sa dérivée est positive","option_b":"B. Non, car sa dérivée est négative","option_c":"C. On ne peut pas savoir","option_d":"D. Oui, mais seulement si x \u003E 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"Calculons f'(x) = 2x - 4. Sur [2, +∞[, f'(x) ≥ 0 (car 2x - 4 ≥ 0 pour x ≥ 2), donc la fonction est croissante 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\": \"A. Oui, car sa dérivée est positive\", \"b\": \"B. Non, car sa dér","_debug_options_count":4},{"id":23126,"question":"La dérivée de la fonction f(x) = √x est f'(x) = 1\/(2√x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai. La dérivée de √x = x^(1\/2) est (1\/2)x^(-1\/2) = 1\/(2√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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":23127,"question":"Quelle est la dérivée de la fonction f(x) = (3x^2 + 2)\/(x - 1) ?","option_a":"A. (6x(x - 1) - (3x^2 + 2))\/(x - 1)^2","option_b":"B. (6x)\/(x - 1)","option_c":"C. (3x^2 + 2)\/(x - 1)^2","option_d":"D. (6x - 1)\/(x - 1)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Il faut utiliser la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v^2. Ici, u = 3x^2 + 2 et v = x - 1, donc u' = 6x et v' = 1. Ainsi, f'(x) = (6x(x - 1) - (3x^2 + 2))\/(x - 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\": \"A. (6x(x - 1) - (3x^2 + 2))\/(x - 1)^2\", \"b\": \"B. (6x)\/(x - 1)\", \"","_debug_options_count":4},{"id":23128,"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":"C'est vrai. Une fonction constante f(x) = k a une dérivée nulle, car son taux de variation est toujours égal à 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}]