Quiz interactif : Testez vos connaissances sur les fonctions dérivées !

Question 1 sur 10 10:00

[{"id":23928,"question":"Quel est le nombre dérivé de la fonction f(x) = x² en x = 3 ?","option_a":"0","option_b":"3","option_c":"6","option_d":"9","option_e":"","option_f":"","bonne_reponse":"C","explication":"Le nombre dérivé de f(x) = x² est f'(x) = 2x. En x = 3, f'(3) = 2 × 3 = 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23929,"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 à cet endroit.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23930,"question":"Quelle est la dérivée de la fonction f(x) = e^(3x) ?","option_a":"e^(3x)","option_b":"3e^(3x)","option_c":"e^x","option_d":"3e^x","option_e":"","option_f":"","bonne_reponse":"B","explication":"En utilisant la règle de la chaîne, f'(x) = e^(3x) × 3 = 3e^(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23931,"question":"Si f'(x) \u003E 0 sur un intervalle, alors la fonction f est :","option_a":"Décroissante","option_b":"Croissante","option_c":"Constante","option_d":"Non monotone","option_e":"","option_f":"","bonne_reponse":"B","explication":"Si la dérivée est positive sur un intervalle, la fonction est strictement croissante sur cet intervalle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23932,"question":"La dérivée de la fonction f(x) = ln(x² + 1) est :","option_a":"2x\/(x² + 1)","option_b":"1\/(x² + 1)","option_c":"2x","option_d":"x\/(x² + 1)","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la règle de la chaîne, f'(x) = (1\/(x² + 1)) × 2x = 2x\/(x² + 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23933,"question":"Une fonction peut-elle être dérivable sans être continue ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Si une fonction est dérivable en un point, elle est nécessairement continue en ce point. La réciproque n'est pas vraie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23934,"question":"Quelle est la dérivée de la fonction f(x) = sin(2x) ?","option_a":"cos(2x)","option_b":"2cos(2x)","option_c":"cos(x)","option_d":"2sin(x)","option_e":"","option_f":"","bonne_reponse":"B","explication":"En utilisant la règle de la chaîne, f'(x) = cos(2x) × 2 = 2cos(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23935,"question":"Si f'(a) = 0, alors la fonction f admet un extremum en x = a.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Une condition nécessaire pour un extremum est f'(a) = 0, mais ce n'est pas suffisant (il faut aussi étudier le signe de la dérivée autour de a).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23936,"question":"La dérivée de la fonction f(x) = x³ - 3x² + 2x est :","option_a":"3x² - 6x + 2","option_b":"3x² - 6x","option_c":"x² - 3x + 2","option_d":"3x² - 3x + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"En dérivant terme à terme, f'(x) = 3x² - 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":23937,"question":"La fonction f(x) = 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 1\/x n'est pas définie en x = 0, donc elle ne peut pas y être dérivable.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]