Quiz — 6361522921356_corrigé_Séance 31-10-2022(Dérivabilités).pdf

Question 1 sur 10 20:00

[{"id":36250,"question":"Si une fonction f est dérivable en a, alors :","option_a":"f est nécessairement continue en a","option_b":"f est nécessairement discontinue en a","option_c":"f peut être discontinue en a","option_d":"f n'est pas définie en a","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivabilité implique la continuité, mais pas l'inverse. Une fonction peut être continue sans être dérivable (ex: f(x)=|x| en 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f est nécessairement continue en a\", \"b\": \"f est nécessairement","_debug_options_count":4},{"id":36251,"question":"Le nombre dérivé de f en a est égal à :","option_a":"lim (h→0) [f(a+h) - f(a)] \/ h","option_b":"lim (x→0) [f(x) - f(a)] \/ (x - a)","option_c":"lim (h→0) [f(a+h) - f(a-h)] \/ (2h)","option_d":"lim (x→a) [f(x) - f(a)] \/ (x - a)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le nombre dérivé est défini par la limite du taux d'accroissement lorsque h tend vers 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"lim (h→0) [f(a+h) - f(a)] \/ h\", \"b\": \"lim (x→0) [f(x) - f(a)]","_debug_options_count":4},{"id":36252,"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 (dérivée à gauche ≠ dérivée à droite).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":36253,"question":"Soit f(x) = x² sin(1\/x) pour x ≠ 0 et f(0) = 0. La fonction f est dérivable en 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant le taux d'accroissement et le théorème des gendarmes, on montre que f'(0) = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":36254,"question":"Si f'(a) = 0, alors la fonction f admet un extremum local en a.","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. Il faut aussi vérifier le changement de signe de f' autour de a (ou utiliser la dérivée seconde).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":36255,"question":"La dérivée de la fonction f(x) = √x est :","option_a":"1\/(2√x)","option_b":"1\/√x","option_c":"2√x","option_d":"√x \/ 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la dérivée des fonctions composées ou la définition du nombre dérivé, on obtient f'(x) = 1\/(2√x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"1\/(2√x)\", \"b\": \"1\/√x\", \"c\": \"2√x\", \"d\": \"√x \/ 2\"}}","_debug_options_count":4},{"id":36256,"question":"Soit f une fonction dérivable sur ℝ telle que f'(x) = 2x + 1. Alors f(x) est égale à :","option_a":"x² + x + C","option_b":"x² + x","option_c":"2x² + x + C","option_d":"x² - x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"En intégrant f'(x), on obtient f(x) = x² + x + C, où C est une constante réelle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x² + x + C\", \"b\": \"x² + x\", \"c\": \"2x² + x + C\", \"d\": \"x² - x ","_debug_options_count":4},{"id":36257,"question":"La fonction f(x) = x³ est dérivable sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction cube est dérivable en tout point de ℝ, car son taux d'accroissement admet une limite finie pour tout a ∈ ℝ.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":36258,"question":"Si f est dérivable en a, alors f est continue en a. Cette affirmation est :","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème fondamental : la dérivabilité implique la continuité. La réciproque est fausse (ex: f(x)=|x| en 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":36259,"question":"Soit f(x) = e^x. La dérivée de f en x = ln(2) est égale à :","option_a":"2","option_b":"1\/2","option_c":"ln(2)","option_d":"e^ln(2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^x est e^x. En x = ln(2), on a e^ln(2) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2\", \"b\": \"1\/2\", \"c\": \"ln(2)\", \"d\": \"e^ln(2)\"}}","_debug_options_count":4}]