Quiz — série28-Fonction exponentielle corrigé.pdf

Question 1 sur 10 20:00

[{"id":31300,"question":"Quelle est la valeur de exp(1) ?","option_a":"e ≈ 2.718","option_b":"1","option_c":"0","option_d":"π ≈ 3.14","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle exp(x) est définie de telle sorte que exp(1)=e, où e est la constante d'Euler (≈2.718).","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 ≈ 2.718\", \"b\": \"1\", \"c\": \"0\", \"d\": \"π ≈ 3.14\"}}","_debug_options_count":4},{"id":31301,"question":"La dérivée de la fonction f(x)=exp(2x) est f'(x)=2exp(2x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant la règle de dérivation des fonctions composées, f'(x)=2*exp(2x), ce qui est correct.","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":31302,"question":"Quelle est la solution de l'équation exp(x)=5 ?","option_a":"x=ln(5)","option_b":"x=5","option_c":"x=exp(5)","option_d":"x=0","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation exp(x)=k admet pour solution x=ln(k), donc ici x=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=ln(5)\", \"b\": \"x=5\", \"c\": \"x=exp(5)\", \"d\": \"x=0\"}}","_debug_options_count":4},{"id":31303,"question":"La fonction exponentielle est toujours croissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La fonction exponentielle exp(x) est strictement croissante sur ℝ car sa dérivée exp(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":31304,"question":"Quelle est la limite de exp(x)\/x quand x tend vers +∞ ?","option_a":"+∞","option_b":"0","option_c":"1","option_d":"e","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par croissance comparée, exp(x) domine x quand x→+∞, donc exp(x)\/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\": \"+∞\", \"b\": \"0\", \"c\": \"1\", \"d\": \"e\"}}","_debug_options_count":4},{"id":31305,"question":"La fonction f(x)=exp(-x) est décroissante sur ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de f(x)=exp(-x) est f'(x)=-exp(-x), toujours négative, donc la fonction est décroissante.","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":31306,"question":"Quelle est la solution générale de l'équation différentielle y'=y ?","option_a":"y(x)=C*exp(x)","option_b":"y(x)=C*x","option_c":"y(x)=C","option_d":"y(x)=ln(x)+C","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation différentielle y'=y a pour solution générale y(x)=C*exp(x), 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\": \"y(x)=C*exp(x)\", \"b\": \"y(x)=C*x\", \"c\": \"y(x)=C\", \"d\": \"y(x)=ln(x)+","_debug_options_count":4},{"id":31307,"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":"Oui, la fonction exponentielle exp(x) est définie et continue pour tout nombre réel 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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":31308,"question":"Quelle est la valeur de exp(ln(3)) ?","option_a":"3","option_b":"ln(3)","option_c":"e^3","option_d":"0","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, exp(ln(x))=x pour tout x\u003E0, donc exp(ln(3))=3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3\", \"b\": \"ln(3)\", \"c\": \"e^3\", \"d\": \"0\"}}","_debug_options_count":4},{"id":31309,"question":"La fonction f(x)=exp(x²) est-elle paire ou impaire ?","option_a":"Ni paire ni impaire","option_b":"Paire","option_c":"Impaire","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"f(-x)=exp((-x)²)=exp(x²)=f(x), donc la fonction est paire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Ni paire ni impaire\", \"b\": \"Paire\", \"c\": \"Impaire\", \"d\": \"\"}}","_debug_options_count":4}]