{"id":409,"date":"2012-05-27T21:14:29","date_gmt":"2012-05-27T12:14:29","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=409"},"modified":"2021-10-23T03:47:57","modified_gmt":"2021-10-22T18:47:57","slug":"wsr201204","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201204\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52012\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\u95a2\u6570\r\n\\[\r\nf(x) = \\log \\left( 1+\\sqrt{1-x^2} \\right) -\\sqrt{1-x^2} -\\log x \\quad ( 0 \\lt x \\lt 1 )\r\n\\]\r\n\u306b\u3064\u3044\u3066, \u3064\u304e\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(f'(x)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e\u6982\u5f62\u3092\u63cf\u3051.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u66f2\u7dda \\(y = f(x)\\) \u4e0a\u3092\u52d5\u304f\u70b9\u3092 P \u3068\u3059\u308b. \u70b9 Q \u306f, \u66f2\u7dda \\(y = f(x)\\) \u306e P \u306b\u304a\u3051\u308b\u63a5\u7dda\u4e0a\u306b\u3042\u308a, P \u3068\u306e\u8ddd\u96e2\u304c \\(1\\) \u3067, \u305d\u306e \\(x\\) \u5ea7\u6a19\u304c P \u306e \\(x\\) \u5ea7\u6a19\u3088\u308a\u5c0f\u3055\u3044\u3082\u306e\u3068\u3059\u308b. Q \u306e\u8ecc\u8de1\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\nf'(x) & = \\dfrac{1}{1+\\sqrt{1-x^2}} \\cdot \\dfrac{-2x}{2 \\sqrt{1-x^2}} -\\dfrac{-2x}{2 \\sqrt{1-x^2}} -\\dfrac{1}{x} \\\\\r\n& = \\dfrac{x}{\\sqrt{1-x^2}} \\left( 1 -\\dfrac{1}{1 +\\sqrt{1-x^2}} \\right) -\\dfrac{1}{x} \\\\\r\n& = \\dfrac{x}{1 +\\sqrt{1-x^2}} -\\dfrac{1}{x} \\\\\r\n& = \\dfrac{x \\left( 1 -\\sqrt{1-x^2} \\right)}{1 -(1-x^2)} -\\dfrac{1}{x} \\\\\r\n& = \\dfrac{1 -\\sqrt{1-x^2}}{x} -\\dfrac{1}{x} \\\\\r\n& = \\underline{-\\dfrac{\\sqrt{1-x^2}}{x}}\r\n\\end{align}\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\[\r\n\\displaystyle\\lim _ {x \\rightarrow 0} f(x) = \\infty , \\quad \\displaystyle\\lim _ {x \\rightarrow 1} f(x) = 0\r\n\\]\r\n\u307e\u305f, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nf'(x) \\lt 0 , \\quad \\displaystyle\\lim _ {x \\rightarrow 0} f'(x) = -\\infty , \\quad \\displaystyle\\lim _ {x \\rightarrow 1} f'(x) = 0\r\n\\]\r\n\u3088\u3063\u3066, \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n\"waseda_r_2012_04_01\"\r\n

    (3)<\/strong><\/p>\r\n

    P \\(( t , f(t))\\) \u3068\u304a\u304f.
    \r\nP \u306b\u304a\u3051\u308b\u63a5\u7dda\u306e\u65b9\u5411\u30d9\u30af\u30c8\u30eb\u3092 \\(\\overrightarrow{v}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\overrightarrow{v} & = \\left( \\begin{array}{c} 1 \\\\ f'(t) \\end{array} \\right) , \\\\\r\n\\left| \\overrightarrow{v} \\right| & = \\sqrt{1+\\left\\{ f'(t) \\right\\}^2} = \\dfrac{1}{t}\r\n\\end{align}\\]\r\n\\(\\text{PQ} = 1\\) \u304b\u3064 Q \u306e \\(x\\) \u5ea7\u6a19\u306f P \u3088\u308a\u5c0f\u3055\u3044\u306e\u3067\r\n\\[\r\n\\overrightarrow{\\text{PQ}} = -\\dfrac{\\overrightarrow{v}}{\\left| \\overrightarrow{v} \\right|} = \\left( \\begin{array}{c} -t \\\\ -t f'(t) \\end{array} \\right)\r\n\\]\r\n\u3088\u3063\u3066, Q \\(( X , Y )\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\nX & = t -t =0 , \\\\\r\nY & = f(t) -tf'(t) \\\\\r\n& = \\log \\dfrac{1+\\sqrt{1-t^2}}{t} \\\\\r\n& = \\log \\left( \\dfrac{1}{t} +\\sqrt{\\dfrac{1}{t^2}-1} \\right) \\\\\r\n& \\gt \\log \\left( 1 +\\sqrt{1-1} \\right) = 0\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u8ecc\u8de1\u306f\r\n\\[\r\n\\underline{\\text{\u534a\u76f4\u7dda} : \\ x = 0 \\quad ( y \\gt 0 )}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\[ f(x) = \\log \\left( 1+\\sqrt{1-x^2} \\right) -\\sqrt{1-x^2} -\\log x \\quad ( 0 \\lt x \\lt 1 ) \\] \u306b\u3064\u3044\u3066, \u3064\u304e\u306e\u554f\u306b\u7b54\u3048 … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[64],"tags":[147,68],"class_list":["post-409","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2012","tag-waseda_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/409","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=409"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/409\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=409"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=409"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=409"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}