Maîtrisez la dérivation en 1ère Science Expérimentale — Quiz interactif

Question 1 sur 10 10:00

[{"id":52031,"question":"Quel est le nombre dérivé de la fonction f définie par f(x) = 3x² - 2x + 1 en x = 2 ?","option_a":"10","option_b":"8","option_c":"12","option_d":"6","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de f est f'(x) = 6x - 2. En x=2, f'(2) = 6*2 - 2 = 10.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":52032,"question":"La dérivée de la fonction g(x) = 1\/x est g'(x) = -1\/x².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"En appliquant la règle de dérivation d'une fonction inverse (1\/u)' = -u'\/u², on obtient bien g'(x) = -1\/x².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":52033,"question":"Quelle est l'équation de la tangente à la courbe de f(x) = x³ - 3x + 2 au point d'abscisse x = 1 ?","option_a":"y = 3x - 1","option_b":"y = 0","option_c":"y = -3x + 5","option_d":"y = x + 1","option_e":"","option_f":"","bonne_reponse":"C","explication":"f(1) = 0, f'(x) = 3x² - 3 donc f'(1) = 0. L'équation de la tangente est y = f'(1)(x-1) + f(1) = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":52034,"question":"La fonction h(x) = x⁴ - 4x³ admet-elle un extremum local en x = 3 ?","option_a":"Oui, c'est un maximum local","option_b":"Oui, c'est un minimum local","option_c":"Non, car h'(3) = 0 mais h''(3) \u003E 0","option_d":"Non, car h'(3) ≠ 0","option_e":"","option_f":"","bonne_reponse":"B","explication":"h'(x) = 4x³ - 12x², h'(3) = 0. h''(x) = 12x² - 24x, h''(3) = 36 \u003E 0 donc minimum local en x=3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":52035,"question":"La dérivée de la fonction k(x) = (2x + 1)\/(x - 3) est k'(x) = 2\/(x - 3)².","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"En appliquant la règle du quotient : k'(x) = [2(x-3) - (2x+1)*1]\/(x-3)² = (-7)\/(x-3)² ≠ 2\/(x-3)².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":52036,"question":"Quel est le sens de variation de la fonction f(x) = -x³ + 3x² sur l'intervalle [-1, 2] ?","option_a":"Croissante puis décroissante","option_b":"Décroissante puis croissante","option_c":"Toujours croissante","option_d":"Toujours décroissante","option_e":"","option_f":"","bonne_reponse":"A","explication":"f'(x) = -3x² + 6x = -3x(x-2). f'(x) \u003E 0 pour x ∈ ]0,2[ et f'(x) \u003C 0 pour x ∈ ]-∞,0[ ∪ ]2,+∞[. Donc f est croissante sur [0,2] et décroissante sur [-1,0].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":52037,"question":"La fonction m(x) = x² - 4x + 5 admet-elle un extremum global ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"m'(x) = 2x - 4. m'(x) = 0 ⇒ x = 2. m''(x) = 2 \u003E 0 donc minimum global en x=2 avec m(2)=1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":52038,"question":"Quelle est la dérivée de la fonction n(x) = (x² + 1)(3x - 2) ?","option_a":"n'(x) = 9x² - 4x + 3","option_b":"n'(x) = 3x² - 4x + 3","option_c":"n'(x) = 6x² - 4x + 3","option_d":"n'(x) = 9x² - 4x - 3","option_e":"","option_f":"","bonne_reponse":"A","explication":"En utilisant la règle du produit : n'(x) = (2x)(3x-2) + (x²+1)(3) = 6x² - 4x + 3x² + 3 = 9x² - 4x + 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":52039,"question":"La dérivée d'une fonction constante est toujours égale à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par définition, le taux d'accroissement d'une fonction constante est nul pour tout x, donc sa limite (nombre dérivé) est 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":52040,"question":"Pour la fonction p(x) = x⁴ - 4x², quel est l'ensemble des points où la tangente est horizontale ?","option_a":"{0, 2}","option_b":"{-√2, 0, √2}","option_c":"{0}","option_d":"{-2, 0, 2}","option_e":"","option_f":"","bonne_reponse":"B","explication":"p'(x) = 4x³ - 8x = 4x(x² - 2). p'(x) = 0 ⇒ x = 0 ou x = ±√2. Les tangentes sont horizontales en ces trois points.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]