Quiz — ds1 2016, .pdf

Question 1 sur 10 20:00

[{"id":42975,"question":"Quelle est la dérivée de la fonction f(x) = 3x^4 - 2x^2 + 5 ?","option_a":"12x^3 - 4x","option_b":"12x^3 - 4x + 5","option_c":"3x^3 - 2x","option_d":"12x^3 - 2x","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée d'un polynôme s'obtient en appliquant la règle de puissance : d\/dx(x^n) = n*x^(n-1). Ici, f'(x) = 3*4x^3 - 2*2x + 0 = 12x^3 - 4x.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"12x^3 - 4x\", \"b\": \"12x^3 - 4x + 5\", \"c\": \"3x^3 - 2x\", \"d\": \"12x^3","_debug_options_count":4},{"id":42976,"question":"La limite de f(x) = (x^2 - 1)\/(x - 1) quand x tend vers 1 est :","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Factorisez le numérateur : (x^2 - 1) = (x - 1)(x + 1). La limite devient lim(x→1) (x + 1) = 2. Cependant, la simplification donne 2, mais la question semble incorrecte. La bonne réponse est 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"2\", \"d\": \"∞\"}}","_debug_options_count":4},{"id":42977,"question":"Vrai ou Faux : La dérivée de f(x) = e^x est toujours égale à e^x.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est une propriété fondamentale des fonctions exponentielles : d\/dx(e^x) = e^x pour tout x réel.","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":42978,"question":"Quelle est la dérivée de f(x) = ln(x) ?","option_a":"1\/x","option_b":"x","option_c":"e^x","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"La dérivée de la fonction logarithme népérien est f'(x) = 1\/x, définie pour x \u003E 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\": \"1\/x\", \"b\": \"x\", \"c\": \"e^x\", \"d\": \"1\"}}","_debug_options_count":4},{"id":42979,"question":"Vrai ou Faux : Si f'(x) = 0 en un point, alors ce point est toujours un extremum.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une dérivée nulle en un point peut indiquer un extremum, mais aussi un point d'inflexion ou une tangente horizontale sans extremum (exemple : f(x) = x^3 en x=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\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":42980,"question":"Quelle est la limite de f(x) = (sin x)\/x quand x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"∞","option_d":"n'existe pas","option_e":"","option_f":"","bonne_reponse":"b","explication":"C'est une limite fondamentale en analyse : lim(x→0) (sin x)\/x = 1. Cette limite est souvent utilisée en trigonométrie et en calcul différentiel.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"∞\", \"d\": \"n'existe pas\"}}","_debug_options_count":4},{"id":42981,"question":"La dérivée de f(x) = √x est :","option_a":"1\/(2√x)","option_b":"1\/√x","option_c":"2√x","option_d":"x^(-1\/2)","option_e":"","option_f":"","bonne_reponse":"a","explication":"Réécrivez √x comme x^(1\/2). En appliquant la règle de puissance, f'(x) = (1\/2)x^(-1\/2) = 1\/(2√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\": \"1\/(2√x)\", \"b\": \"1\/√x\", \"c\": \"2√x\", \"d\": \"x^(-1\/2)\"}}","_debug_options_count":4},{"id":42982,"question":"Vrai ou Faux : Une fonction dérivable en un point est toujours continue en ce point.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est un théorème fondamental : si une fonction est dérivable en un point, alors elle est nécessairement continue en ce point. La réciproque est fausse.","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":42983,"question":"Quelle est la dérivée de f(x) = x*ln(x) ?","option_a":"ln(x) + 1","option_b":"1\/x","option_c":"x + ln(x)","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Appliquez la règle du produit : f'(x) = 1*ln(x) + x*(1\/x) = ln(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\": \"ln(x) + 1\", \"b\": \"1\/x\", \"c\": \"x + ln(x)\", \"d\": \"1\"}}","_debug_options_count":4},{"id":42984,"question":"La limite de f(x) = (1 - cos x)\/x^2 quand x tend vers 0 est :","option_a":"0","option_b":"1\/2","option_c":"1","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"b","explication":"Utilisez le développement limité de cos x : 1 - cos x ≈ x^2\/2 pour x proche de 0. Ainsi, (1 - cos x)\/x^2 ≈ 1\/2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0\", \"b\": \"1\/2\", \"c\": \"1\", \"d\": \"∞\"}}","_debug_options_count":4}]