{"id":43854,"date":"2024-06-27T06:34:55","date_gmt":"2024-06-27T06:34:55","guid":{"rendered":"https:\/\/science-hub.click\/%E5%B9%B3%E6%96%B9%E6%A0%B9-%E5%AE%9A%E7%BE%A9\/"},"modified":"2024-06-27T06:34:55","modified_gmt":"2024-06-27T06:34:55","slug":"%E5%B9%B3%E6%96%B9%E6%A0%B9-%E5%AE%9A%E7%BE%A9","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=43854","title":{"rendered":"\u5e73\u65b9\u6839 – \u5b9a\u7fa9"},"content":{"rendered":"

\u6570\u5b66\u3067\u306f\u3001\u6570\u5024x<\/i>\u306e\u5e73\u65b9\u6839\u306f<\/strong>\u30012 \u4e57 (\u6570\u5024\u3092\u305d\u308c\u81ea\u8eab\u3067\u4e57\u7b97\u3057\u305f\u5024) \u304cx \u3068<\/i>\u306a\u308b\u6570\u5024\u3067\u3059\u3002\u3059\u3079\u3066\u306e\u6b63\u306e\u5b9f\u6570\u306b\u306f\u3001\u4e3b\u5e73\u65b9\u6839<\/i>\u3068\u547c\u3070\u308c\u308b\u56fa\u6709\u306e\u6b63\u306e\u5e73\u65b9\u6839\u304c\u3042\u308a\u3001\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002

$$ {\\sqrt{x}} $$<\/div> \u3002\u305f\u3068\u3048\u3070\u30019 \u306e\u4e3b\u5e73\u65b9\u6839\u306f 3 \u3067\u3042\u308a\u3001\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002
$$ {\\sqrt{9}=3} $$<\/div>\u306a\u305c\u306a\u3089
$$ {3^2 = 3 \\times 3 = 9} $$<\/div> \u3002 9 \u306e\u3082\u3046\u4e00\u65b9\u306e (\u4e00\u6b21\u4ee5\u5916\u306e) \u5e73\u65b9\u6839\u306f -3 \u3067\u3059\u3002<\/p>

\u4ee3\u6570\u5b66<\/a><\/span>\u306e\u57fa\u672c\u5b9a\u7406<\/span>\u306b\u3088\u308c\u3070\u3001\u4efb\u610f\u306e\u6570<\/a><\/span>\u306e\u5e73\u65b9\u6839\u3092\u5b9a\u7fa9\u3059\u308b\u65b9\u7a0b\u5f0f\u306b<\/a><\/span>\u306f\u5e38\u306b 2 \u3064\u306e\u89e3 (\u4e92\u3044\u306b\u7b49\u3057\u3044\u5834\u5408\u3082\u3042\u308a\u307e\u3059) \u304c\u5b58\u5728\u3057\u307e\u3059\u3002\u6b63\u306e\u5b9f\u6570<\/a><\/span>\u306e\u5834\u5408\u3001\u305d\u306e 2 \u3064\u306e\u5e73\u65b9\u6839\u306f\u4e3b\u5e73\u65b9\u6839\u3068\u8ca0\u306e\u5e73\u65b9\u6839 (

$$ {-\\sqrt{x}} $$<\/div> \uff09\u3002\u6570\u5024\u306e 2 \u3064\u306e\u6839\u306f\u4e00\u7dd2\u306b\u66f8\u304b\u308c\u307e\u3059
$$ {\\pm\\sqrt{x}} $$<\/div> \u3002\u865a\u6570\u3068\u8907\u7d20\u6570\u306e\u6982\u5ff5\u306f\u3001\u8ca0\u306e\u6570\u306e\u5e73\u65b9\u6839\u3092\u89e3\u304f\u305f\u3081\u306b\u958b\u767a\u3055\u308c\u307e\u3057\u305f\u3002<\/p>

\u5b8c\u5168\u306a\u5e73\u65b9<\/a><\/span>\u3067\u306f\u306a\u3044\u6574\u6570\u306e\u5e73\u65b9\u6839\u306f\u5e38\u306b\u7121\u7406\u6570<\/a><\/span>\u3067\u3059\u3002\u3064\u307e\u308a\u3001\u5206\u6570\u3068\u3057\u3066\u8868\u3059\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u4f8b\u3048\u3070\u3001

$$ {\\sqrt{2}} $$<\/div> m\/n<\/i> ( m<\/i>\u3068n<\/i>\u306f\u6574\u6570) \u306e\u5f62\u5f0f\u3067\u8a18\u8ff0\u3059\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u305f\u3060\u3057\u3001\u3053\u308c\u306f\u9577\u3055 1 \u306e\u6b63\u65b9\u5f62<\/a><\/span>\u306e\u5bfe\u89d2\u7dda<\/a><\/span>\u306e\u6b63\u78ba\u306a\u9577\u3055<\/a><\/span>\u3067\u3059\u3002<\/p>

\u30b5\u30a4\u30f3

$$ {\\sqrt{\\ }} $$<\/div>\u3092\u30e9\u30b8\u30ab\u30eb<\/i>\u3068\u3044\u3044\u307e\u3059\u3002\u305d\u306e\u8d77\u6e90\u306f\u3001\u30eb\u30fc\u30c8<\/i>\u3092\u610f\u5473\u3059\u308b\u30e9\u30c6\u30f3\u8a9e\u306eradix<\/i>\u306e\u982d\u6587\u5b57\u3067\u3042\u308b\u6587\u5b57\u300cr\u300d\u306e\u6b6a\u307f\u3067\u3059\u3002 1525 \u5e74\u306b\u6570\u5b66\u8005\u306e<\/a><\/span>\u30af\u30ea\u30b9\u30c8\u30d5 \u30eb\u30c9\u30eb\u30d5\u306b\u3088\u3063\u3066\u5c0e\u5165\u3055\u308c\u307e\u3057\u305f\u3002<\/p>

\u610f\u5473<\/a><\/span><\/span><\/h2>

\u30a2\u30d7\u30ea

$$ {x\\mapsto x^2} $$<\/div>\u5168\u5358\u5c04\u3067\u3059
$$ {\\mathbb{R}^+\\rightarrow \\mathbb{R}^+} $$<\/div>\u305d\u306e\u53cd\u5bfe\u304c\u6ce8\u76ee\u3055\u308c\u308b
$$ {x\\mapsto \\sqrt{x}} $$<\/div> \u3002\u3053\u306e\u95a2\u6570\u306f\u5e73\u65b9\u6839\u95a2\u6570<\/i>\u3068\u547c\u3070\u308c\u307e\u3059\u3002\u5e7e\u4f55\u5b66\u7684\u306b\u306f\u3001\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u5e73\u9762\u306e\u6b63\u65b9\u5f62\u306e\u9762\u7a4d\u306e\u5e73\u65b9\u6839\u304c\u305d\u306e\u8fba\u306e\u9577\u3055\u3067\u3042\u308b\u3068\u8a00\u3048\u307e\u3059\u3002<\/p>

\u6ce8\u610f: \u9762\u7a4d\u306f\u30e6\u30cb\u30d0\u30fc\u30b5\u30eb \u30b7\u30b9\u30c6\u30e0\u3067\u5e73\u65b9\u30e1\u30fc\u30c8\u30eb<\/a><\/span>\u3001\u9577\u3055\u306f\u30e1\u30fc\u30c8\u30eb<\/a><\/span>\u3067\u8868\u3055\u308c\u307e\u3059\u3002\u5e73\u65b9\u30e1\u30fc\u30c8\u30eb\u3067\u8868\u3055\u308c\u308b\u91cf<\/a><\/span>\u306e\u5e73\u65b9\u6839\u3092\u53d6\u308b\u3068\u3001\u30e1\u30fc\u30c8\u30eb\u3067\u8868\u3055\u308c\u308b\u91cf\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u7269\u7406\u5b66\u8005\u306f\u5358\u4f4d\u306e\u5206\u6790\u3092\u7279\u306b\u91cd\u8981\u8996\u3057\u307e\u3059\u3002\u3053\u306e\u5074\u9762\u306f\u6570\u5b66<\/a><\/span>\u3067\u306f\u6d88\u53bb\u3055\u308c\u307e\u3059\u3002\u5b9f\u6570\u306f\u5358\u4f4d\u306e\u306a\u3044\u5b9a\u6570\u3067\u3042\u308a\u3001\u6b63\u306e\u5b9f\u6570\u306e\u5e73\u65b9\u6839\u306f\u6b63\u306e\u5b9f\u6570\u3067\u3059\u3002<\/small><\/p>

\u5206\u6790<\/span><\/h3>

\u5e73\u65b9\u6839\u95a2\u6570\u306f\u3001\u3059\u3079\u3066\u306e\u6b63\u306e\u5b9f\u6570x<\/i>\u304a\u3088\u3073y<\/i>\u306b\u6709\u52b9\u306a\u6b21\u306e\u57fa\u672c\u30d7\u30ed\u30d1\u30c6\u30a3\u3092\u30c1\u30a7\u30c3\u30af\u3057\u307e\u3059\u3002 <\/p>

$$ {\\sqrt{x} = x^{\\frac{1}{2}}} $$<\/div><\/dd>
$$ {\\sqrt{x \\times y} = \\sqrt{x} \\times \\sqrt{y}} $$<\/div><\/dd>
$$ {\\sqrt{\\frac{x}{y}} = \\frac{\\sqrt{x}}{\\sqrt{y}}} $$<\/div> \uff08\u6761\u4ef6\u7684\u306b\u306f
$$ {y\\neq 0} $$<\/div> ) <\/dd>
$$ {\\sqrt{x^2} = \\left|x\\right|} $$<\/div>\u3053\u3053\u3067 | \u00d7<\/i> |\u306f x<\/i>\u306e\u7d76\u5bfe\u5024<\/a><\/span>\u3092\u793a\u3057\u307e\u3059\u3002<\/dd><\/dl>

\u30eb\u30fc\u30c8\u95a2\u6570\u306f\u3001\u3059\u3079\u3066\u306e<\/a><\/span>\u6b63\u306e\u5b9f\u6570x<\/i>\u3067\u9023\u7d9a\u3067\u3059 ( y \u304c<\/i>x<\/i>\u306b\u8fd1\u3044\u5834\u5408\u3001

$$ {\\sqrt{y}} $$<\/div>\u306b\u8fd1\u3044\u3067\u3059
$$ {\\sqrt{x}} $$<\/div> \uff09\u3002\u3055\u3089\u306b\u3001\u3053\u308c\u306f\u53b3\u5bc6\u306b\u6b63\u306e\u5b9f\u6570x<\/i>\u3067\u5fae\u5206\u53ef\u80fd\u3067\u3042\u308b\u305f\u3081\u3001 x<\/i> =0 \u3067\u306f\u5fae\u5206\u53ef\u80fd\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u3053\u306e\u70b9<\/a><\/span>\u3067\u3001\u63a5\u7dda\u306e\u50be\u304d\u306f\u7121\u9650\u5927\u3067\u3059\u3002\u4ee3\u8868\u7684\u306a\u66f2\u7dda\u3067\u306f<\/a><\/span>\u30010 \u3067\u5782\u76f4\u306e<\/a><\/span>\u534a\u63a5\u7dda\u304c\u8a8d\u3081\u3089\u308c\u307e\u3059\u3002<\/p>

\u304b\u3089\u6d3e\u751f\u3057\u305f<\/a><\/span>\u95a2\u6570

$$ {x\\mapsto \\sqrt{x}} $$<\/div>\u306f\u6b21\u306e\u3088\u3046\u306b\u4e0e\u3048\u3089\u308c\u307e\u3059\u3002 <\/p>
$$ {\\frac{\\mathrm d}{\\mathrm dx}\\sqrt{x}={1 \\over 2\\sqrt{x}}} $$<\/div><\/dd><\/dl>

\u30eb\u30fc\u30c8\u95a2\u6570\u306f\u5b9f\u969b\u306b\u306f\u30af\u30e9\u30b9\u3067\u3059

$$ {C^{\\infty}} $$<\/div>\u306e\u4e0a
$$ {\\R_+^*} $$<\/div> \u3002 <\/p>
$$ {\\frac{\\mathrm d^n}{\\mathrm dx^n}\\sqrt{x}={(-1)}^{n+1} {(2n-2)! \\over n! (n-1)! 2^{2n-1}} \\frac{1}{x^{n-1\/2}}} $$<\/div><\/dd><\/dl>

\u3055\u3089\u306b\u826f\u3044\u3053\u3068\u306b\u3001\u30eb\u30fc\u30c8\u95a2\u6570\u306f\u6574\u6570\u5217\u306b\u5c55\u958b\u3067\u304d\u307e\u3059\u3002\u70b9 1 \u306b\u304a\u3051\u308b\u5e73\u65b9\u6839\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u7d1a\u6570\u5c55\u958b\u306f\u3001\u4e00\u822c\u5316\u3055\u308c\u305f\u4e8c\u9805<\/span>\u516c\u5f0f\u304b\u3089\u3059\u3050\u306b\u5f97\u3089\u308c\u307e\u3059\u3002 <\/p>

$$ {\\sqrt{1+h}=1 + \\sum_{n=1}^{\\infty}(-1)^{n+1} {(2n-2)! \\over n! (n-1)! 2^{2n-1}}h^n} $$<\/div>
$$ {=1 + \\sum_{n=1}^{\\infty}(-1)^{n+1} {(2n)! \\over (n!)^2 (2n-1) 2^{2n}}h^n} $$<\/div><\/dd>
$$ {=1 + \\sum_{n=1}^{\\infty}(-1)^{n+1} {C_{2n}^n \\over (2n-1)2^{2n}}h^n} $$<\/div><\/dd>
$$ {= 1 + \\frac{1}{2}h – \\frac{1}{8}h^2 + \\frac{1}{16} h^3 – \\frac{5}{128} h^4 + \\dots} $$<\/div><\/dd><\/dl><\/dd><\/dl>

\u3059\u3079\u3066\u306e\u672c\u5f53\u306e\u3053\u3068\u306e\u305f\u3081\u306b | h<\/i> | <1.<\/p>

\u3064\u3044\u3067\u306b\u6ce8\u610f\u3057\u3066\u304a\u304d\u307e\u3059\u304c\u3001<\/span> <\/p>

$$ {{C_{2n}^n \\over (2n-1)}=2\\left[C_{2n-2}^{n-1}-C_{2n-2}^n \\right]} $$<\/div><\/dd><\/dl>

\u3057\u305f\u304c\u3063\u3066\u3001 \u306f\u81ea\u7136\u6570<\/a><\/span>\u3067\u3059[ref.\u5fc5\u8981]<\/sup> \u3002<\/p>

\u5e73\u65b9\u6839\u306e\u5e7e\u4f55\u5b66\u7684\u69cb\u9020<\/span><\/h3>
\n\"AO<\/figure>
AO = 1\u3001OB = a\u3001OH = x<\/div><\/div><\/div>

\u6b21\u306e\u5e7e\u4f55\u5b66\u7684\u306a\u69cb\u7bc9\u306f\u3001\u5b9a\u898f\u3068\u30b3\u30f3\u30d1\u30b9\u3092\u4f7f\u7528\u3057\u3066\u5b9f\u884c\u3055\u308c\u3001\u9577\u3055a \u306e\u30bb\u30b0\u30e1\u30f3\u30c8 0B \u304c\u4e0e\u3048\u3089\u308c\u308b\u3068\u3001<\/i>\u9577\u3055\u306e\u30bb\u30b0\u30e1\u30f3\u30c8\u3092\u69cb\u7bc9\u3067\u304d\u307e\u3059\u3002

$$ {\\sqrt{a}} $$<\/div> :<\/p>