Quiz — تجميعة تمارين في الدوال الأسية مع حلولها للنهائي شعب علمية (باك 2019).pdf

Question 1 sur 10 20:00

[{"id":14404,"question":"Quelle est la dérivée de la fonction f(x) = e^(3x) ?","option_a":"3e^(3x)","option_b":"e^(3x)","option_c":"3x e^(3x)","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) = 3x, donc u'(x) = 3. Ainsi, f'(x) = 3e^(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\": \"3e^(3x)\", \"b\": \"e^(3x)\", \"c\": \"3x e^(3x)\", \"d\": \"e^(x)\"}}","_debug_options_count":4},{"id":14405,"question":"L’équation e^x = 0 admet-elle une solution réelle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La fonction exponentielle e^x est toujours strictement positive pour tout x réel. Elle ne s’annule jamais.","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":14406,"question":"Quelle est la limite de e^(-x) lorsque x tend vers +∞ ?","option_a":"+∞","option_b":"0","option_c":"1","option_d":"-∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Lorsque x tend vers +∞, -x tend vers -∞, et e^(-∞) = 0. La limite est donc 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"+∞\", \"b\": \"0\", \"c\": \"1\", \"d\": \"-∞\"}}","_debug_options_count":4},{"id":14407,"question":"Si f(x) = e^(x²), alors f'(x) = ?","option_a":"2x e^(x²)","option_b":"x e^(x²)","option_c":"e^(x²)","option_d":"2x e^(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la dérivée de e^(u(x)), ici u(x) = x², donc u'(x) = 2x. Ainsi, f'(x) = 2x e^(x²).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"2x e^(x²)\", \"b\": \"x e^(x²)\", \"c\": \"e^(x²)\", \"d\": \"2x e^(x)\"}}","_debug_options_count":4},{"id":14408,"question":"La fonction exponentielle est-elle croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle e^x est strictement croissante sur ℝ car sa dérivée e^x est toujours positive.","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":14409,"question":"Quelle est la solution de l’équation e^(2x) = e^5 ?","option_a":"x = 5\/2","option_b":"x = 2\/5","option_c":"x = 5","option_d":"x = -5\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Si e^(2x) = e^5, alors 2x = 5 (car la fonction exponentielle est injective). Ainsi, 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 = -5\/2\"}}","_debug_options_count":4},{"id":14410,"question":"La limite de e^x lorsque x tend vers -∞ est égale à :","option_a":"+∞","option_b":"0","option_c":"1","option_d":"-∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Lorsque x tend vers -∞, e^x tend vers 0. C’est une propriété fondamentale de la fonction exponentielle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"+∞\", \"b\": \"0\", \"c\": \"1\", \"d\": \"-∞\"}}","_debug_options_count":4},{"id":14411,"question":"Si f(x) = e^(x) + x, alors f'(x) = ?","option_a":"e^(x) + 1","option_b":"e^(x)","option_c":"e^(x) + x","option_d":"x e^(x)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de e^(x) est e^(x), et la dérivée de x est 1. Ainsi, f'(x) = e^(x) + 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"e^(x) + 1\", \"b\": \"e^(x)\", \"c\": \"e^(x) + x\", \"d\": \"x e^(x)\"}}","_debug_options_count":4},{"id":14412,"question":"L’inéquation e^x \u003E 1 est équivalente à :","option_a":"x \u003E 0","option_b":"x \u003C 0","option_c":"x = 0","option_d":"x ≥ 0","option_e":"","option_f":"","bonne_reponse":"a","explication":"e^x \u003E 1 équivaut à x \u003E 0, car e^0 = 1 et la fonction exponentielle est croissante.","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 \u003E 0\", \"b\": \"x \u003C 0\", \"c\": \"x = 0\", \"d\": \"x ≥ 0\"}}","_debug_options_count":4},{"id":14413,"question":"La fonction g(x) = e^(-x²) admet-elle un maximum ? Si oui, en quel point ?","option_a":"Oui, en x = 0","option_b":"Oui, en x = 1","option_c":"Non, elle n’a pas de maximum","option_d":"Oui, en x = -1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction g(x) = e^(-x²) admet un maximum en x = 0, car -x² est maximal en 0, et e^0 = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Oui, en x = 0\", \"b\": \"Oui, en x = 1\", \"c\": \"Non, elle n’a pas d","_debug_options_count":4}]