Maîtrisez les fonctions et dérivées — Quiz interactif

Question 1 sur 10 10:00

[{"id":24428,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x² + 2","option_d":"3x + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction polynôme s'obtient en multipliant chaque terme par son exposant et en réduisant l'exposant de 1. Ainsi, f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":24429,"question":"L'équation 2x² - 5x + 2 = 0 admet-elle deux solutions réelles distinctes ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le discriminant Δ = b² - 4ac = 25 - 16 = 9 \u003E 0, donc l'équation admet bien deux solutions réelles distinctes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":24430,"question":"Quelle est la limite de la fonction f(x) = (x² - 1)\/(x - 1) lorsque x tend vers 1 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"En factorisant le numérateur, on obtient f(x) = (x - 1)(x + 1)\/(x - 1) = x + 1 pour x ≠ 1. La limite est donc 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":24431,"question":"Une fonction dérivable en un point est-elle nécessairement continue en ce point ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, si une fonction est dérivable en un point, elle est nécessairement continue en ce point (théorème fondamental).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":24432,"question":"Quelle est la dérivée de la fonction g(x) = e^(3x) ?","option_a":"3e^(3x)","option_b":"e^(3x)","option_c":"3x e^(3x)","option_d":"e^(x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'une fonction exponentielle e^(u(x)) est u'(x) * e^(u(x)). Ici, u(x) = 3x, donc g'(x) = 3e^(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":24433,"question":"La fonction h(x) = x³ - 3x + 2 admet-elle un extremum local en x = 1 ?","option_a":"Oui","option_b":"Non","option_c":"On ne peut pas savoir","option_d":"Seulement un maximum","option_e":"","option_f":"","bonne_reponse":"A","explication":"h'(x) = 3x² - 3. En x = 1, h'(1) = 0 et h''(1) = 6 \u003E 0, donc la fonction admet un minimum local en x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":24434,"question":"Quelle est la solution de l'inéquation x² - 4x + 3 \u003E 0 ?","option_a":"x \u003C 1 ou x \u003E 3","option_b":"1 \u003C x \u003C 3","option_c":"x \u003E 1","option_d":"x \u003C 3","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les racines de l'équation x² - 4x + 3 = 0 sont x = 1 et x = 3. Le polynôme est positif à l'extérieur des racines.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":24435,"question":"La fonction k(x) = ln(x) est-elle définie pour x = 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Non, la fonction logarithme népérien ln(x) est définie uniquement pour x \u003E 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":24436,"question":"Quelle est la dérivée de la fonction m(x) = sin(2x) ?","option_a":"2cos(2x)","option_b":"cos(2x)","option_c":"2sin(2x)","option_d":"-2cos(2x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de sin(u(x)) est u'(x) * cos(u(x)). Ici, u(x) = 2x, donc m'(x) = 2cos(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":24437,"question":"La fonction n(x) = x\/(x² + 1) admet-elle une asymptote horizontale en ±∞ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, car lim(x→±∞) n(x) = 0, donc la droite y = 0 est une asymptote horizontale.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]