Les dérivées et leurs applications — Terminale Mathématiques

Question 1 sur 10 10:00

[{"id":21538,"question":"Quelle est la dérivée de la fonction f(x) = 3x² - 5x + 2 ?","option_a":"6x - 5","option_b":"3x - 5","option_c":"6x² - 5x","option_d":"3x² - 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'un polynôme se calcule terme par terme : (3x²)' = 6x, (-5x)' = -5, et 2' = 0. Donc f'(x) = 6x - 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21539,"question":"La fonction f(x) = x³ admet un point d'inflexion en x = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Un point d'inflexion existe si la dérivée seconde s'annule et change de signe. Ici, f''(x) = 6x, qui s'annule en x=0 mais ne change pas de signe (toujours croissante).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21540,"question":"Quelle est la dérivée de la fonction g(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2xe^(2x-1)","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. D'où g'(x)=2e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21541,"question":"La fonction h(x) = ln(x² + 1) est définie pour tout x réel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le logarithme est défini lorsque son argument est strictement positif. x² + 1 \u003E 0 pour tout x réel, donc h est bien définie sur ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21542,"question":"Quelle est la solution de l'équation f'(x) = 0 pour f(x) = x³ - 3x² ?","option_a":"x = 0 et x = 2","option_b":"x = 0 seulement","option_c":"x = 1 seulement","option_d":"x = 2 seulement","option_e":"","option_f":"","bonne_reponse":"A","explication":"f'(x) = 3x² - 6x = 3x(x-2). L'équation f'(x)=0 donne x=0 ou x=2, qui sont les points critiques de f.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21543,"question":"La dérivée de la fonction k(x) = sin(3x) est k'(x) = 3cos(3x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","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. D'où k'(x)=3cos(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21544,"question":"Quelle est la dérivée de la fonction m(x) = (2x + 1)\/(x - 3) ?","option_a":"(2x - 7)\/(x - 3)²","option_b":"(2x + 7)\/(x - 3)²","option_c":"2\/(x - 3)²","option_d":"(2x - 5)\/(x - 3)²","option_e":"","option_f":"","bonne_reponse":"A","explication":"On utilise la formule (u\/v)' = (u'v - uv')\/v². Ici u=2x+1, v=x-3, donc u'=2, v'=1. D'où m'(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},{"id":21545,"question":"La fonction n(x) = x²e^(-x) admet un maximum en x = 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"n'(x) = 2xe^(-x) - x²e^(-x) = e^(-x)(2x - x²). n'(x)=0 donne x=0 ou x=2. Le signe de n' montre que n admet un maximum en x=2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21546,"question":"Quelle est la dérivée de la fonction p(x) = ln(5x + 1) ?","option_a":"5\/(5x + 1)","option_b":"1\/(5x + 1)","option_c":"5\/(x + 1)","option_d":"1\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici u(x)=5x+1 donc u'(x)=5. D'où p'(x)=5\/(5x+1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":21547,"question":"La dérivée de la fonction q(x) = √(x² + 4) est q'(x) = x\/√(x² + 4).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"On utilise la formule (√u)' = u'\/(2√u). Ici u=x²+4 donc u'=2x. D'où q'(x)=2x\/(2√(x²+4)) = x\/√(x²+4).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]