Maîtrisez la partie entière et ses variations

Question 1 sur 10 10:00

[{"id":58677,"question":"Quelle est la valeur de ⌊3.7⌋ ?","option_a":"3","option_b":"4","option_c":"3.7","option_d":"3.5","option_e":"","option_f":"","bonne_reponse":"A","explication":"La partie entière de 3.7 est le plus grand entier inférieur ou égal à 3.7, soit 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":58678,"question":"La fonction partie entière est-elle continue sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction partie entière est discontinue en tous les points entiers, car elle présente des sauts brusques.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":58679,"question":"Quelle est l'image de l'intervalle [2, 3[ par la fonction partie entière ?","option_a":"{2}","option_b":"{2, 3}","option_c":"{3}","option_d":"{2, 3, 4}","option_e":"","option_f":"","bonne_reponse":"A","explication":"Sur [2, 3[, la partie entière vaut toujours 2, donc l'image est {2}.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":58680,"question":"L'équation ⌊x⌋ = 5 admet-elle des solutions ?","option_a":"Oui, x = 5","option_b":"Oui, x ∈ [5, 6[","option_c":"Non, aucune solution","option_d":"Oui, x ∈ ]5, 6]","option_e":"","option_f":"","bonne_reponse":"B","explication":"Tous les réels x tels que 5 ≤ x \u003C 6 vérifient ⌊x⌋ = 5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":58681,"question":"La fonction partie entière est-elle croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Si x ≤ y, alors ⌊x⌋ ≤ ⌊y⌋, donc la fonction est croissante.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":58682,"question":"Quelle est la limite de ⌊x⌋ lorsque x tend vers 2 par la gauche ?","option_a":"1","option_b":"2","option_c":"−∞","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"Pour x ∈ [1, 2[, ⌊x⌋ = 1, mais pour x ∈ [2, 3[, ⌊x⌋ = 2. La limite à gauche de 2 est donc 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":58683,"question":"L'inéquation ⌊x⌋ ≤ 4 admet-elle des solutions ?","option_a":"Oui, x ≤ 4","option_b":"Oui, x \u003C 5","option_c":"Non, aucune solution","option_d":"Oui, x ∈ ]−∞, 5[","option_e":"","option_f":"","bonne_reponse":"B","explication":"Tous les réels x tels que x \u003C 5 vérifient ⌊x⌋ ≤ 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":58684,"question":"La fonction partie entière est-elle périodique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction partie entière est périodique de période 1, car ⌊x + 1⌋ = ⌊x⌋ + 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":58685,"question":"Quelle est la valeur de ⌊−2.3⌋ ?","option_a":"−3","option_b":"−2","option_c":"−2.3","option_d":"−1","option_e":"","option_f":"","bonne_reponse":"A","explication":"Le plus grand entier inférieur ou égal à −2.3 est −3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":58686,"question":"La fonction partie entière est-elle dérivable sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"La fonction partie entière n'est dérivable en aucun point, car elle est discontinue partout.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]