Quiz — 4._exponentielle_2-maths_Corrections..pdf

Question 1 sur 10 20:00

[{"id":68886,"question":"Quelle est la valeur de la dérivée de la fonction f(x) = e^(3x) ?","option_a":"e^(3x)","option_b":"3e^(3x)","option_c":"e^x","option_d":"3x e^(3x)","option_e":"","option_f":"","bonne_reponse":"b","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\": \"b\", \"options\": {\"a\": \"e^(3x)\", \"b\": \"3e^(3x)\", \"c\": \"e^x\", \"d\": \"3x e^(3x)\"}}","_debug_options_count":4},{"id":68887,"question":"L'équation e^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":"La fonction exponentielle est toujours strictement positive (e^x \u003E 0 pour tout x réel). L'équation e^x = -2 n'a donc pas de solution réelle.","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":68888,"question":"Quelle est la limite de e^(-x) lorsque x tend vers +∞ ?","option_a":"0","option_b":"+∞","option_c":"1","option_d":"e","option_e":"","option_f":"","bonne_reponse":"a","explication":"Lorsque x tend vers +∞, -x tend vers -∞. Or, 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\": \"a\", \"options\": {\"a\": \"0\", \"b\": \"+∞\", \"c\": \"1\", \"d\": \"e\"}}","_debug_options_count":4},{"id":68889,"question":"La fonction exponentielle est-elle strictement croissante sur ℝ ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction exponentielle est toujours positive (e^x \u003E 0), ce qui signifie qu'elle est strictement croissante sur tout son domaine de définition.","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":68890,"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 = ln(5)","option_e":"","option_f":"","bonne_reponse":"a","explication":"En égalant les exposants (car la fonction exponentielle 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 = ln(5)\"}}","_debug_options_count":4},{"id":68891,"question":"La fonction f(x) = e^x est-elle paire ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une fonction paire vérifie f(-x) = f(x). Or, e^(-x) ≠ e^x sauf si x=0. La fonction exponentielle n'est donc pas paire.","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":68892,"question":"Quelle est la valeur de ∫ e^(2x) dx ?","option_a":"e^(2x)\/2 + C","option_b":"2e^(2x) + C","option_c":"e^(2x) + C","option_d":"(e^(2x))\/x + C","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intégrale de e^(u(x)) est (1\/u'(x))e^(u(x)) + C. Ici, u(x)=2x donc u'(x)=2. Ainsi, ∫ e^(2x) dx = e^(2x)\/2 + C.","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^(2x)\/2 + C\", \"b\": \"2e^(2x) + C\", \"c\": \"e^(2x) + C\", \"d\": \"(e^(2","_debug_options_count":4},{"id":68893,"question":"La fonction exponentielle vérifie-t-elle e^(a+b) = e^a + e^b ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La bonne propriété est e^(a+b) = e^a * e^b. L'addition des exposants ne se traduit pas par une addition des résultats.","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":68894,"question":"Quelle est la solution de l'inéquation e^x \u003E 5 ?","option_a":"x \u003E ln(5)","option_b":"x \u003C ln(5)","option_c":"x \u003E 5","option_d":"x \u003C 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle est strictement croissante. L'inéquation e^x \u003E 5 équivaut à x \u003E ln(5).","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 ln(5)\", \"b\": \"x \u003C ln(5)\", \"c\": \"x \u003E 5\", \"d\": \"x \u003C 5\"}}","_debug_options_count":4},{"id":68895,"question":"La fonction exponentielle est-elle définie pour tout x réel ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle e^x est définie pour tout nombre réel x. Son domaine de définition est 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}]