Quiz interactif : Les dérivées et leurs applications en Terminale Math

Question 1 sur 10 10:00

[{"id":4104,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"A. 6x + 2","option_b":"B. 3x + 2","option_c":"C. 6x² + 2x","option_d":"D. 6x + 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'un polynôme s'obtient en appliquant la formule (ax^n)' = n·a·x^(n-1). Ici, f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":4105,"question":"La fonction f(x) = x³ - 3x est décroissante sur l'intervalle [-1, 1].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée f'(x) = 3x² - 3 est négative sur [-1, 1] (car 3x² ≤ 3), donc la fonction est décroissante sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":4106,"question":"Quelle est la dérivée de la fonction g(x) = e^(2x) ?","option_a":"A. 2e^x","option_b":"B. e^(2x)","option_c":"C. 2e^(2x)","option_d":"D. e^(x)","option_e":"","option_f":"","bonne_reponse":"C","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},{"id":4107,"question":"La fonction h(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) n'est définie que pour x \u003E 0. Elle n'est pas définie pour x ≤ 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":4108,"question":"Quelle est la dérivée de la fonction k(x) = (2x + 1)\/(x - 3) ?","option_a":"A. 2\/(x - 3)","option_b":"B. -7\/(x - 3)²","option_c":"C. (2x - 7)\/(x - 3)²","option_d":"D. 7\/(x - 3)","option_e":"","option_f":"","bonne_reponse":"B","explication":"En utilisant la formule de dérivation d'un quotient (u\/v)' = (u'v - uv')\/v², on obtient k'(x) = -7\/(x - 3)².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":4109,"question":"La dérivée d'une fonction constante est toujours égale à 0.","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 (où c est une constante) est bien f'(x) = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":4110,"question":"Quelle est la dérivée de la fonction m(x) = sin(3x) ?","option_a":"A. 3cos(3x)","option_b":"B. cos(3x)","option_c":"C. -3cos(3x)","option_d":"D. sin(3x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de sin(u(x)) est u'(x)·cos(u(x)). Ici, u(x) = 3x, donc u'(x) = 3. Ainsi, m'(x) = 3cos(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":4111,"question":"La fonction n(x) = x² + 4x + 4 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 n'(x) = 2x + 4 s'annule en x = -2. Comme n''(x) = 2 \u003E 0, la fonction admet bien un minimum en x = -2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":4112,"question":"Quelle est la dérivée de la fonction p(x) = √(2x + 1) ?","option_a":"A. 1\/√(2x + 1)","option_b":"B. 1\/√(2x)","option_c":"C. 1\/(2√(2x + 1))","option_d":"D. 2\/√(2x + 1)","option_e":"","option_f":"","bonne_reponse":"C","explication":"En utilisant la formule de dérivation d'une racine (√u)' = u'\/(2√u), on obtient p'(x) = 1\/(2√(2x + 1)).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":4113,"question":"La dérivée d'une fonction paire est toujours impaire.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Si f est paire (f(-x) = f(x)), alors f'(-x) = -f'(x), donc f' est impaire. Cette propriété est toujours vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]