Les dérivées en Terminale Sciences : concepts et applications

Question 1 sur 10 10:00

[{"id":57061,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x + 2","option_c":"6x² + 2x","option_d":"3x² + 2","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'un polynôme se calcule terme à terme : f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":57062,"question":"Si f'(x) = 0 pour une valeur x = a, 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":"Un extremum nécessite que f'(a) = 0 ET que f' change de signe en a (ou que f''(a) ≠ 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":57063,"question":"Quelle est la dérivée de la fonction g(x) = (2x + 1)\/(x - 3) ?","option_a":"(2x - 7)\/(x - 3)²","option_b":"2\/(x - 3)²","option_c":"(x - 3)\/(2x + 1)","option_d":"2","option_e":"","option_f":"","bonne_reponse":"A","explication":"Utilisez la formule de dérivation d'un quotient : (u\/v)' = (u'v - uv')\/v².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":57064,"question":"La fonction h(x) = x³ - 3x² + 2 est croissante sur l'intervalle [-1, 3].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"h'(x) = 3x² - 6x = 3x(x - 2). h'(x) \u003E 0 sur [-1, 0[ et ]2, 3], donc h est croissante sur ces intervalles.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":57065,"question":"Quelle est la dérivée de la fonction k(x) = e^(2x + 1) ?","option_a":"e^(2x + 1)","option_b":"2e^(2x + 1)","option_c":"(2x + 1)e^(2x)","option_d":"2x e^(2x + 1)","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée de e^u est u' e^u, ici u = 2x + 1 donc u' = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":57066,"question":"Une fonction dont la dérivée est toujours positive est strictement croissante.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée doit être strictement positive (pas nulle) pour garantir une croissance stricte.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":57067,"question":"Quelle est la dérivée de la fonction m(x) = ln(3x² + 1) ?","option_a":"6x\/(3x² + 1)","option_b":"3x\/(3x² + 1)","option_c":"6x²\/(3x² + 1)","option_d":"3\/(3x² + 1)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u) est u'\/u, ici u = 3x² + 1 donc u' = 6x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":57068,"question":"La tangente à la courbe de f en x = 2 a pour équation y = 5x - 3. Que vaut f(2) ?","option_a":"7","option_b":"5","option_c":"3","option_d":"1","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'équation de la tangente est y = f'(2)(x - 2) + f(2). En x = 2, y = f(2) = 5*2 - 3 = 7.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":57069,"question":"Si f'(x) = x² - 4, alors f est décroissante sur l'intervalle ]-2, 2[.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"f'(x) \u003C 0 sur ]-2, 2[ car x² - 4 \u003C 0 dans cet intervalle, donc f est décroissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":57070,"question":"Quelle est la dérivée de la fonction n(x) = √(x² + 4) ?","option_a":"x\/√(x² + 4)","option_b":"2x\/√(x² + 4)","option_c":"1\/√(x² + 4)","option_d":"x\/√(x² - 4)","option_e":"","option_f":"","bonne_reponse":"A","explication":"Utilisez la formule de dérivation de √u : (√u)' = u'\/(2√u), ici u = x² + 4 donc u' = 2x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]