Quiz — حل تمارين الدالة الأسية السلسلة رقم 02 (Réparé).docx

Question 1 sur 10 20:00

[{"id":47198,"question":"Quelle est la valeur de exp(0) ?","option_a":"0","option_b":"1","option_c":"e","option_d":"ln(1)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction exponentielle vérifie exp(0) = 1, car c'est une propriété fondamentale de cette fonction.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"e\", \"d\": \"ln(1)\"}}","_debug_options_count":4},{"id":47199,"question":"La fonction exponentielle est-elle toujours dérivable ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car exp(x) est dérivable sur ℝ et sa dérivée est exp(x) elle-même.","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":47200,"question":"Quelle est la dérivée de la fonction f(x) = exp(3x) ?","option_a":"3exp(3x)","option_b":"exp(3x)","option_c":"3xexp(3x-1)","option_d":"exp(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de exp(u(x)) est u'(x)×exp(u(x)). Ici, u(x)=3x, donc u'(x)=3. Ainsi, f'(x)=3exp(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3exp(3x)\", \"b\": \"exp(3x)\", \"c\": \"3xexp(3x-1)\", \"d\": \"exp(x)\"}}","_debug_options_count":4},{"id":47201,"question":"L'équation exp(x) = -2 admet-elle une solution réelle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux, car exp(x) est toujours strictement positif pour tout réel x. L'équation n'a donc pas de solution.","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":47202,"question":"Quelle propriété algébrique vérifie exp(a+b) ?","option_a":"exp(a) + exp(b)","option_b":"exp(a) × exp(b)","option_c":"exp(a) - exp(b)","option_d":"exp(a) \/ exp(b)","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction exponentielle vérifie exp(a+b) = exp(a) × exp(b), une propriété clé pour simplifier les calculs.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"exp(a) + exp(b)\", \"b\": \"exp(a) × exp(b)\", \"c\": \"exp(a) - exp(b)\"","_debug_options_count":4},{"id":47203,"question":"La fonction exp(x) est-elle périodique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux, car exp(x) est strictement croissante et ne se répète jamais. Elle n'est donc pas périodique.","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":47204,"question":"Résoudre l'équation exp(2x) = exp(5).","option_a":"x = 5\/2","option_b":"x = 2\/5","option_c":"x = 5","option_d":"x = 2","option_e":"","option_f":"","bonne_reponse":"a","explication":"En égalant les exposants (car exp est injective), on obtient 2x = 5, donc x = 5\/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\": \"x = 5\/2\", \"b\": \"x = 2\/5\", \"c\": \"x = 5\", \"d\": \"x = 2\"}}","_debug_options_count":4},{"id":47205,"question":"La fonction exp(x) est-elle convexe sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car sa dérivée seconde exp(x) est toujours positive, ce qui signifie que la fonction est convexe.","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":47206,"question":"Quelle est la limite de exp(x) lorsque x tend vers -∞ ?","option_a":"0","option_b":"+∞","option_c":"1","option_d":"-∞","option_e":"","option_f":"","bonne_reponse":"a","explication":"Lorsque x tend vers -∞, exp(x) tend vers 0, car la fonction exponentielle décroît rapidement 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\": \"0\", \"b\": \"+∞\", \"c\": \"1\", \"d\": \"-∞\"}}","_debug_options_count":4},{"id":47207,"question":"La fonction exp(x) est-elle égale à sa dérivée ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car la dérivée de exp(x) est exp(x) elle-même, une propriété unique de cette fonction.","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}]