Quiz interactif : Maîtrisez les fonctions dérivées en Terminale Math

Question 1 sur 10 10:00

[{"id":61207,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x + 2x - 5","option_d":"3x + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynôme se calcule terme à terme : (3x²)' = 6x, (2x)' = 2, et (-5)' = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61208,"question":"La fonction f(x) = x³ est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction x³ est convexe sur ℝ car sa dérivée seconde f''(x) = 6x est positive pour x \u003E 0 et négative pour x \u003C 0, mais pas partout positive.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61209,"question":"Quel est le coefficient directeur de la tangente à la courbe de f(x) = 4x² - 3x + 1 au point d'abscisse x = 2 ?","option_a":"13","option_b":"10","option_c":"11","option_d":"14","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le coefficient directeur est f'(2). f'(x) = 8x - 3, donc f'(2) = 16 - 3 = 13.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61210,"question":"La fonction f(x) = e^x a une dérivée égale à elle-même.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de e^x est bien e^x, c'est une propriété fondamentale de la fonction exponentielle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61211,"question":"Quelle est la dérivée de f(x) = ln(x) ?","option_a":"1\/x","option_b":"x","option_c":"e^x","option_d":"1","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(x) est 1\/x, une formule essentielle à connaître en analyse.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61212,"question":"La fonction f(x) = x^4 - 4x³ + 6x² est croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée f'(x) = 4x³ - 12x² + 12x = 4x(x² - 3x + 3) n'est pas toujours positive (exemple : x = 1 donne f'(1) = 4 - 12 + 12 = 4 \u003E 0, mais x = 0 donne f'(0) = 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61213,"question":"Quel est le point critique de la fonction f(x) = x³ - 3x² + 2 ?","option_a":"x = 0 et x = 2","option_b":"x = 1 et x = 2","option_c":"x = 0 et x = 1","option_d":"x = -1 et x = 2","option_e":"","option_f":"","bonne_reponse":"B","explication":"Les points critiques sont les solutions de f'(x) = 0. f'(x) = 3x² - 6x = 3x(x - 2), donc x = 0 ou x = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61214,"question":"La fonction f(x) = sin(x) a une dérivée égale à cos(x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de sin(x) est bien cos(x), une formule fondamentale en trigonométrie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61215,"question":"Quelle est la dérivée de f(x) = (2x + 1)\/(x - 3) ?","option_a":"(2x - 7)\/(x - 3)²","option_b":"2\/(x - 3)²","option_c":"(2x + 7)\/(x - 3)²","option_d":"(2x - 7)\/(x - 3)","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v², avec u = 2x + 1 et v = x - 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":61216,"question":"La fonction f(x) = x² - 4x + 5 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, et la dérivée seconde f''(x) = 2 \u003E 0 confirme qu'il s'agit d'un minimum.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]