Maîtrisez les fonctions numériques : dérivées, limites et continuité

Question 1 sur 10 10:00

[{"id":48571,"question":"Quelle est la dérivée de la fonction f(x) = x³ - 2x² + 5x - 1 ?","option_a":"3x² - 4x + 5","option_b":"3x² - 4x + 5x","option_c":"x³ - 4x + 5","option_d":"3x² - 2x + 5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de x³ est 3x², celle de -2x² est -4x, celle de 5x est 5, et la constante -1 disparaît.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":48572,"question":"Si f'(x) = 0 pour tout x dans un intervalle, alors f est constante sur cet intervalle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Si la dérivée est nulle sur un intervalle, la fonction est constante sur cet intervalle (théorème de Rolle étendu).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":48573,"question":"Quelle est la limite de f(x) = (2x² + 3)\/(x² - 1) quand x tend vers l'infini ?","option_a":"2","option_b":"0","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"A","explication":"Divisez numérateur et dénominateur par x² : (2 + 3\/x²)\/(1 - 1\/x²) → 2\/1 = 2 quand x → ∞.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":48574,"question":"La fonction f(x) = x² + 3x - 4 est-elle continue sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les fonctions polynomiales sont continues sur ℝ car dérivables en tout point.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":48575,"question":"Quelle est la dérivée de f(x) = e^(3x) ?","option_a":"3e^(3x)","option_b":"e^(3x)","option_c":"3xe^(3x)","option_d":"e^(x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de e^u est u'·e^u, ici u = 3x donc u' = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":48576,"question":"Si f est continue sur [a, b] et f(a) \u003C 0 \u003C f(b), alors il existe c dans ]a, b[ tel que f(c) = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"C'est le théorème des valeurs intermédiaires (TVI).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":48577,"question":"Quelle est la limite de f(x) = (sin x)\/x quand x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"n'existe pas","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":48578,"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 0 car la dérivée à gauche (-1) ≠ dérivée à droite (1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":48579,"question":"Quelle est la dérivée de f(x) = ln(2x + 1) ?","option_a":"1\/(2x + 1)","option_b":"2\/(2x + 1)","option_c":"1\/x","option_d":"2\/(x + 1)","option_e":"","option_f":"","bonne_reponse":"B","explication":"La dérivée de ln(u) est u'\/u, ici u = 2x + 1 donc u' = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":48580,"question":"Si f est dérivable en a et f'(a) = 0, alors a est un extremum local de f.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"C'est une condition nécessaire mais pas suffisante (exemple : f(x) = x³ en x = 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]