Quiz interactif : Fonctions dérivées et continuité en Terminale Sciences

Question 1 sur 10 10:00

[{"id":39555,"question":"Quelle est la dérivée de la fonction f(x) = x³ + 2x² - 5x + 1 ?","option_a":"f'(x) = 3x² + 4x - 5","option_b":"f'(x) = 3x² + 4x + 1","option_c":"f'(x) = 3x² + 2x - 5","option_d":"f'(x) = x³ + 4x - 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une somme est la somme des dérivées. f'(x) = 3x² + 4x - 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":39556,"question":"La fonction f(x) = (x² + 1)\/x est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction n'est pas définie en x = 0, donc elle n'est pas dérivable en ce point.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":39557,"question":"Quelle est la limite de f(x) = (2x² + 3x - 1)\/(x² - 4) lorsque x tend vers +∞ ?","option_a":"2","option_b":"1","option_c":"0","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"En divisant numérateur et dénominateur par x², on obtient lim (2 + 3\/x - 1\/x²)\/(1 - 4\/x²) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":39558,"question":"Si f est continue sur [a, b] et f(a) = -2, f(b) = 3, alors l'équation f(x) = 0 admet au moins une solution dans [a, b].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est le théorème des valeurs intermédiaires : si f est continue et change de signe, elle s'annule au moins une fois.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":39559,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"f'(x) = 2e^(2x)","option_b":"f'(x) = e^(2x)","option_c":"f'(x) = 2xe^(2x)","option_d":"f'(x) = 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.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":39560,"question":"La fonction f(x) = |x| est-elle dérivable en x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction valeur absolue n'est pas dérivable en x = 0 car elle présente un point anguleux (dérivée à gauche ≠ dérivée à droite).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":39561,"question":"Quelle est la limite de f(x) = (sin x)\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"-1","option_e":"","option_f":"","bonne_reponse":"B","explication":"C'est une limite fondamentale en analyse : lim (sin x)\/x = 1 lorsque x tend vers 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":39562,"question":"Si f est dérivable en a, alors f est continue en a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivabilité implique la continuité : si f est dérivable en a, alors lim (f(x) - f(a))\/(x - a) existe, donc lim f(x) = f(a).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":39563,"question":"Quelle est la dérivée de la fonction f(x) = ln(x² + 1) ?","option_a":"f'(x) = 2x\/(x² + 1)","option_b":"f'(x) = 1\/(x² + 1)","option_c":"f'(x) = 2x","option_d":"f'(x) = x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x) = x² + 1, donc u'(x) = 2x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":39564,"question":"La fonction f(x) = x³ - 3x + 1 est-elle continue sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les fonctions polynômes sont continues sur ℝ, donc f est continue sur tout son domaine de définition.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]