Quiz — درس التفاضليات من رتب عليا دساتير تايلور.pdf

Question 1 sur 10 20:00

[{"id":98850,"question":"Quelle est la formule de Taylor-Young pour une fonction f au voisinage de a ?","option_a":"f(x) = f(a) + f'(a)(x-a) + (f''(a)\/2!)(x-a)^2 + ... + (f^(n)(a)\/n!)(x-a)^n + o((x-a)^n)","option_b":"f(x) = f(a) + f'(a)(x-a) + (f''(a)\/2!)(x-a)^2 + ... + (f^(n)(a)\/n!)(x-a)^n","option_c":"f(x) = f(a) + f'(a)(x-a) + (f''(a)\/2!)(x-a)^2","option_d":"f(x) = f(a) + f'(a)(x-a) + o(x-a)","option_e":"","option_f":"","bonne_reponse":"a","explication":"La formule de Taylor-Young inclut le terme reste o((x-a)^n) pour indiquer l'erreur d'approximation.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"f(x) = f(a) + f'(a)(x-a) + (f''(a)\/2!)(x-a)^2 + ... + (f^(n)(a)\/n","_debug_options_count":4},{"id":98851,"question":"Le développement limité d'une fonction polynôme est-il exact ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un polynôme est déjà sous forme développée, donc son développement limité est exact (sans reste).","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":98852,"question":"Quelle est la formule de Maclaurin pour f(x) = e^x ?","option_a":"e^x = 1 + x + x^2\/2! + x^3\/3! + ... + x^n\/n! + o(x^n)","option_b":"e^x = 1 + x + x^2\/2 + x^3\/6 + ...","option_c":"e^x = 1 + x + x^2 + x^3 + ...","option_d":"e^x = x + x^2\/2! + x^3\/3! + ...","option_e":"","option_f":"","bonne_reponse":"a","explication":"La série de Maclaurin pour e^x est la somme des termes x^n\/n! pour n de 0 à l'infini.","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 + x + x^2\/2! + x^3\/3! + ... + x^n\/n! + o(x^n)\", \"b\": \"e^x","_debug_options_count":4},{"id":98853,"question":"Le développement limité d'ordre 2 de f(x) = ln(1+x) au voisinage de 0 est :","option_a":"ln(1+x) ≈ x - x^2\/2","option_b":"ln(1+x) ≈ x + x^2\/2","option_c":"ln(1+x) ≈ 1 + x - x^2\/2","option_d":"ln(1+x) ≈ x - x^2","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée première de ln(1+x) est 1\/(1+x), et la dérivée seconde est -1\/(1+x)^2. En x=0, f(0)=0, f'(0)=1, f''(0)=-1, d'où le développement x - x^2\/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\": \"ln(1+x) ≈ x - x^2\/2\", \"b\": \"ln(1+x) ≈ x + x^2\/2\", \"c\": \"ln(1+","_debug_options_count":4},{"id":98854,"question":"La série de Taylor converge toujours vers la fonction d'origine.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La convergence dépend de la fonction et du point autour duquel on développe. Certaines fonctions n'ont pas de série de Taylor convergente.","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":98855,"question":"Quel est le développement limité d'ordre 3 de f(x) = sin(x) au voisinage de 0 ?","option_a":"sin(x) ≈ x - x^3\/6","option_b":"sin(x) ≈ x + x^3\/6","option_c":"sin(x) ≈ x - x^2\/2 + x^3\/6","option_d":"sin(x) ≈ 1 - x^2\/2","option_e":"","option_f":"","bonne_reponse":"a","explication":"Les dérivées successives de sin(x) en 0 donnent : f(0)=0, f'(0)=1, f''(0)=0, f'''(0)=-1. Le développement est donc x - x^3\/6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"sin(x) ≈ x - x^3\/6\", \"b\": \"sin(x) ≈ x + x^3\/6\", \"c\": \"sin(x) ","_debug_options_count":4},{"id":98856,"question":"Pour approximer f(x) = cos(x) près de 0 avec une erreur inférieure à 0.01, quel ordre de développement limité faut-il choisir ?","option_a":"Ordre 2","option_b":"Ordre 3","option_c":"Ordre 4","option_d":"Ordre 1","option_e":"","option_f":"","bonne_reponse":"b","explication":"Le développement limité d'ordre 3 de cos(x) est 1 - x^2\/2 + x^4\/24. L'erreur est de l'ordre de x^4\/24, donc pour |x| \u003C 0.2, l'erreur est \u003C 0.01.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Ordre 2\", \"b\": \"Ordre 3\", \"c\": \"Ordre 4\", \"d\": \"Ordre 1\"}}","_debug_options_count":4},{"id":98857,"question":"Le développement limité d'une fonction paire ne contient que des puissances paires de x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une fonction paire vérifie f(-x) = f(x). Son développement limité ne contient que des termes en x^(2k), donc des puissances paires.","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":98858,"question":"Quelle est la dérivée troisième de f(x) = x^4 - 3x^2 + 2 ?","option_a":"24x","option_b":"12x^2 - 6","option_c":"24x - 6","option_d":"12x^2","option_e":"","option_f":"","bonne_reponse":"a","explication":"f'(x) = 4x^3 - 6x, f''(x) = 12x^2 - 6, f'''(x) = 24x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"24x\", \"b\": \"12x^2 - 6\", \"c\": \"24x - 6\", \"d\": \"12x^2\"}}","_debug_options_count":4},{"id":98859,"question":"Le développement limité d'ordre 1 de f(x) = 1\/(1-x) au voisinage de 0 est :","option_a":"1 + x","option_b":"1 - x","option_c":"1 + x + x^2","option_d":"1 - x + x^2","option_e":"","option_f":"","bonne_reponse":"b","explication":"f(x) = 1\/(1-x) a pour dérivée f'(x) = 1\/(1-x)^2. En x=0, f(0)=1 et f'(0)=1. Le développement limité d'ordre 1 est donc 1 - x.","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 + x\", \"b\": \"1 - x\", \"c\": \"1 + x + x^2\", \"d\": \"1 - x + x^2\"}}","_debug_options_count":4}]