Quiz — Formation de développements limités, I.pdf

Question 1 sur 10 20:00

[{"id":17659,"question":"Quel est l'ordre du développement limité de la fonction f(x) = sin(x) au voisinage de 0 ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement limité de sin(x) au voisinage de 0 est x - x³\/6 + ..., donc l'ordre minimal est 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\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":17660,"question":"La formule de Taylor-Young permet d'écrire un développement limité d'ordre n pour toute fonction dérivable n fois.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule de Taylor-Young s'applique si la fonction est de classe Cⁿ, c'est-à-dire n fois dérivable avec une dérivée n-ième continue.","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":17661,"question":"Quel est le développement limité de eˣ au voisinage de 0 à l'ordre 3 ?","option_a":"1 + x + x²\/2 + x³\/6","option_b":"1 + x + x²\/2","option_c":"x + x²\/2 + x³\/6","option_d":"1 + x²\/2 + x³\/6","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement limité de eˣ à l'ordre 3 est 1 + x + x²\/2 + x³\/6 + o(x³).","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 + x + x²\/2 + x³\/6\", \"b\": \"1 + x + x²\/2\", \"c\": \"x + x²\/2 + x","_debug_options_count":4},{"id":17662,"question":"Si f(x) = ln(1+x), quel est son développement limité à l'ordre 2 au voisinage de 0 ?","option_a":"x - x²\/2","option_b":"x + x²\/2","option_c":"x - x²","option_d":"x + x²","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement limité de ln(1+x) à l'ordre 2 est x - x²\/2 + o(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\": \"x - x²\/2\", \"b\": \"x + x²\/2\", \"c\": \"x - x²\", \"d\": \"x + x²\"}}","_debug_options_count":4},{"id":17663,"question":"Un développement limité d'ordre n est toujours unique.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un développement limité d'ordre n est unique si la fonction est de classe Cⁿ.","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":17664,"question":"Quel est le développement limité de (1+x)^α au voisinage de 0 à l'ordre 2 ?","option_a":"1 + αx + α(α-1)x²\/2","option_b":"1 + αx + αx²\/2","option_c":"1 + x + x²\/2","option_d":"1 + αx + α(α+1)x²\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement limité de (1+x)^α à l'ordre 2 est 1 + αx + α(α-1)x²\/2 + o(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\": \"1 + αx + α(α-1)x²\/2\", \"b\": \"1 + αx + αx²\/2\", \"c\": \"1 + x +","_debug_options_count":4},{"id":17665,"question":"Peut-on utiliser un développement limité pour calculer la limite de f(x) = (sin(x) - x)\/x³ quand x tend vers 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, en utilisant le développement limité de sin(x) = x - x³\/6 + o(x³), on obtient une forme indéterminée 0\/0 que l'on peut lever.","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":17666,"question":"Quel est l'ordre minimal nécessaire pour un développement limité de f(x) = 1\/(1+x) au voisinage de 0 afin d'étudier son comportement en +∞ ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un développement limité d'ordre 2 suffit pour étudier le comportement asymptotique : 1 - x + x² + o(x²).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":17667,"question":"La formule de Maclaurin est un cas particulier de la formule de Taylor.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule de Maclaurin est un développement limité au voisinage de 0, donc un cas particulier de la formule de Taylor.","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":17668,"question":"Quel est le développement limité de cos(x) au voisinage de 0 à l'ordre 4 ?","option_a":"1 - x²\/2 + x⁴\/24","option_b":"1 - x²\/2 + x⁴\/12","option_c":"1 + x²\/2 + x⁴\/24","option_d":"1 - x² + x⁴\/24","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le développement limité de cos(x) à l'ordre 4 est 1 - x²\/2 + x⁴\/24 + o(x⁴).","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 - x²\/2 + x⁴\/24\", \"b\": \"1 - x²\/2 + x⁴\/12\", \"c\": \"1 + x²\/2","_debug_options_count":4}]