{"id":3285,"date":"2025-09-28T00:18:08","date_gmt":"2025-09-27T15:18:08","guid":{"rendered":"https:\/\/math-friend.com\/?p=3285"},"modified":"2025-09-28T00:18:11","modified_gmt":"2025-09-27T15:18:11","slug":"%e3%80%90%e6%9d%b1%e4%ba%ac%e5%a4%a7%e5%ad%a6-%e6%96%87%e7%b3%bb4-2009%e3%80%91%e7%b5%b6%e5%af%be%e5%80%a4%e3%81%8c%e5%85%a5%e3%82%8b%e7%a9%8d%e5%88%86%e3%81%ae%e5%a0%b4%e5%90%88%e5%88%86%e3%81%91","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=3285","title":{"rendered":"\u3010\u6771\u4eac\u5927\u5b66 \u6587\u7cfb4 (2009)\u3011\u7d76\u5bfe\u5024\u304c\u5165\u308b\u7a4d\u5206\u306e\u5834\u5408\u5206\u3051"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\\(2\\)\u6b21\u4ee5\u4e0b\u306e\u6574\u5f0f\\(f(x)=ax^2+bx+c\\)\u306b\u5bfe\u3057,$$
S=\\int_0^2\\left|f^\\prime(x)\\right|\\,dx
$$\u3092\u8003\u3048\u308b.
(1) \\(f(0)=0\\), \\(f(2)=2\\)\u306e\u3068\u304d, \\(S\\)\u3092\\(a\\)\u306e\u95a2\u6570\u3068\u3057\u3066\u8868\u305b.
(2) \\(f(0)=0\\), \\(f(2)=2\\)\u3092\u6e80\u305f\u3057\u306a\u304c\u3089\\(f\\)\u304c\u5909\u5316\u3059\u308b\u3068\u304d, \\(S\\)\u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.
(2009 \u6771\u4eac\u5927\u5b66 \u6587\u7cfb [4])<\/span><\/p>\n\n\n\n

\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n

\n

(1) \\(f(0)=0\\), \\(f(2)=2\\)\u3088\u308a,$$
\\begin{align}
f(0)&=c=0\\\\[1.5ex]
f(2)&=4a+2b+c=2\\\\[1.5ex]
\\end{align}
$$\u3068\u306a\u308a, \\(c=0\\), \\(b=1-2a\\)\u304c\u308f\u304b\u308b. \u3088\u3063\u3066\\(f(x)=ax^2+(1-2a)x\\)\u3068\u306a\u308a,$$
f^\\prime(x)=2ax+1-2a
$$\u3067\u3042\u308b.

\u2460 \\( \\displaystyle a\\leq-\\frac{1}{2}\\)\u306e\u3068\u304d
\\(y=|f^(x)|\\)\u306e\u30b0\u30e9\u30d5\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\\(S\\)\u306f\u659c\u7dda\u90e8\u306e\u9762\u7a4d\u306a\u306e\u3067,$$
\\begin{align}
S&=\\frac{1}{2}\\left(1-\\frac{1}{2a}\\right)(1-2a)+\\frac{1}{2}\\left(1+\\frac{1}{2a}\\right)(2a-1)\\\\[1.5ex]
&=\\frac{1}{2}\\left(1-2a-\\frac{1}{2a}+1-2a-1-1-\\frac{1}{2a}\\right)\\\\[1.5ex]
&=-2a-\\frac{1}{2a}
\\end{align}
$$\u3068\u306a\u308b.

\u2461 \\(\\displaystyle -\\frac{1}{2}< a < \\frac{1}{2}\\)\u306e\u3068\u304d
\\(y=|f^\\prime(x)|\\)\u306e\u30b0\u30e9\u30d5\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\\(S\\)\u306f\u659c\u7dda\u90e8\u306e\u9762\u7a4d\u306a\u306e\u3067,$$
S=\\left\\{(1-2a)+(2a+1)\\right\\}\\times 2 \\times \\frac{1}{2}=2
$$\u3068\u306a\u308b.

\u2462 \\(\\displaystyle a\\geq\\frac{1}{2}\\)\u306e\u3068\u304d
\\(y=|f^\\prime(x)|\\)\u306e\u30b0\u30e9\u30d5\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a,<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\\(S\\)\u306f\u659c\u7dda\u90e8\u306e\u9762\u7a4d\u306a\u306e\u3067,$$
\\begin{align}
S&=\\frac{1}{2}\\left(1-\\frac{1}{2a}\\right)(2a-1)+\\frac{1}{2}\\left(1+\\frac{1}{2a}\\right)(2a+1)\\\\[1.5ex]
&=\\frac{1}{2}\\left(2a-1-1+\\frac{1}{2a}+2a+1+1+\\frac{1}{2a}\\right)\\\\[1.5ex]
&=2a+\\frac{1}{2a}
\\end{align}
$$\u3068\u306a\u308b.

\u3088\u3063\u3066,$$
S= \\begin{cases}
\\displaystyle -2a-\\frac{1}{2a} & \\displaystyle \\left(a\\leq -\\frac{1}{2} \u306e\u3068\u304d\\right) \\\\[1.5ex]
2 & \\displaystyle \\left(-\\frac{1}{2} < a< \\frac{1}{2} \u306e\u3068\u304d\\right) \\\\[1.5ex]
\\displaystyle2a+\\frac{1}{2a} & \\displaystyle \\left(a\\geq \\frac{1}{2} \u306e\u3068\u304d\\right)
\\end{cases}
$$\u3068\u306a\u308b.<\/p>\n\n\n\n

(2) \\(S\\)\u306f\\(a\\)\u306b\u95a2\u3057\u3066\u5076\u95a2\u6570\u3068\u306a\u308b\u306e\u3067, \\(a\\geq 0\\)\u306e\u7bc4\u56f2\u3067\u8003\u3048\u308b. \\(\\displaystyle a\\geq \\frac{1}{2}\\)\u306e\u3068\u304d, \u76f8\u52a0\u76f8\u4e57\u5e73\u5747\u304b\u3089$$
S=2a+\\frac{1}{2a}\\geq 2\\sqrt{2a\\cdot\\frac{1}{2a}}=2
$$\u3067\u3042\u308a, \\(\\displaystyle a\\geq \\frac{1}{2}\\)\u306b\u6ce8\u610f\u3057\u3066, \u7b49\u53f7\u6210\u7acb\u306f,$$
\\begin{align}
& 2a=\\frac{1}{2a}\\\\[1.5ex]
\\iff & a^2=\\frac{1}{4}\\\\[1.5ex]
\\iff & a=\\frac{1}{2}\\\\[1.5ex]
\\end{align}
$$\u306e\u3068\u304d\u3067\u3042\u308b.

\u5076\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u306b\u3082\u6ce8\u610f\u3057\u3066, \\(S(a)\\)\u306e\u30b0\u30e9\u30d5\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a, \\(S(a)\\)\u306e\u6700\u5c0f\u5024\u306f\\(2\\)\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n\n\n\n

\"\"<\/figure>\n<\/div><\/div>\n\n\n\n

youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n