Quiz : Maîtrisez les fonctions numériques et la dérivation

Question 1 sur 10 10:00

[{"id":65382,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"3x² - 4x + 5","option_b":"3x² - 4x + 1","option_c":"x² - 2x + 5","option_d":"x³ - 2x + 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle (x^n)' = n*x^(n-1) à chaque terme.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65383,"question":"La fonction f(x) = |x| est 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 0 car elle présente un point anguleux.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65384,"question":"Quelle est la limite de la fonction f(x) = (2x² + 3x - 1)\/(x² - 5) quand x tend vers l'infini ?","option_a":"2","option_b":"1","option_c":"0","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"Pour les fonctions rationnelles, la limite à l'infini est égale au rapport des coefficients dominants (2\/1 = 2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65385,"question":"Si f'(x) \u003E 0 sur un intervalle I, alors la fonction f est décroissante sur I.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Si f'(x) \u003E 0, la fonction f est croissante sur I. Une dérivée positive indique une croissance.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65386,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2xe^(2x)","option_d":"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":65387,"question":"La fonction f(x) = x^4 - 4x³ + 6x² est convexe sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La convexité se détermine avec la dérivée seconde : f''(x) = 12x² - 24x + 12 = 12(x-1)² ≥ 0, donc f est convexe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65388,"question":"Quelle est la valeur de la limite lim(x→0) (sin(x))\/x ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"Indéterminé","option_e":"","option_f":"","bonne_reponse":"B","explication":"C'est une limite fondamentale en analyse : lim(x→0) (sin(x))\/x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65389,"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(x→a) f(x) = f(a).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65390,"question":"Quelle est la dérivée de la fonction f(x) = ln(3x + 2) ?","option_a":"3\/(3x + 2)","option_b":"1\/(3x + 2)","option_c":"3x\/(3x + 2)","option_d":"1\/x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de ln(u(x)) est u'(x)\/u(x). Ici, u(x) = 3x + 2, donc u'(x) = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":65391,"question":"La fonction f(x) = x^3 - 3x² + 2x admet un extremum local en x = 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"f'(x) = 3x² - 6x + 2. f'(1) = -1 ≠ 0, donc x=1 n'est pas un point critique. Pas d'extremum local en x=1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]