{"id":2715,"date":"2025-08-14T03:53:28","date_gmt":"2025-08-13T18:53:28","guid":{"rendered":"https:\/\/math-friend.com\/?p=2715"},"modified":"2025-08-14T10:16:37","modified_gmt":"2025-08-14T01:16:37","slug":"%e3%80%90%e4%ba%ac%e9%83%bd%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e7%b5%b6%e5%af%be%e5%80%a4%e3%81%ae%e5%85%a5%e3%81%a3%e3%81%9f%e9%96%a2%e6%95%b0%e3%81%ae%e3%82%b0%e3%83%a9%e3%83%95%e3%81%ae","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=2715","title":{"rendered":"\u3010\u4eac\u90fd\u5927\u5b66\u5165\u8a66\u3011\u7d76\u5bfe\u5024\u306e\u5165\u3063\u305f\u95a2\u6570\u306e\u30b0\u30e9\u30d5\u306e\u5171\u901a\u63a5\u7dda\u3068\u9818\u57df\u306e\u9762\u7a4d(2025)"},"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

\u66f2\u7dda\\(C_1:y=x^2-2|x|\\), \u66f2\u7dda\\(\\displaystyle C_2:y=x^2-5x+\\frac{7}{4}\\), \u76f4\u7dda\\(\\displaystyle l_1:x=\\frac{3}{2}\\)\u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.

(1) \u539f\u70b9\\(\\mathrm{O}(0,0)\\)\u3068\u7570\u306a\u308b\u70b9\u3067\\(C_1\\)\u3068\u63a5\u3057, \u3055\u3089\u306b\\(C_2\\)\u3068\u3082\u63a5\u3059\u308b\u3088\u3046\u306a\u76f4\u7dda\\(l_2\\)\u304c\u305f\u3060\u4e00\u3064\u5b58\u5728\u3059\u308b\u3053\u3068\u3092\u793a\u305b.
(2) \\(C_1\\)\u3068\\(l_2\\)\u306e\u5171\u6709\u70b9\u3092\\(\\mathrm{P}\\)\u3068\u3057, \u305d\u306e\\(x\\)\u5ea7\u6a19\u3092\\(\\alpha\\)\u3068\u3059\u308b. \u307e\u305f, \\(l_1\\)\u3068\\(l_2\\)\u306e\u5171\u6709\u70b9\u3092\\(\\mathrm{Q}\\)\u3068\u3057, \\(C_1\\)\u3068\\(l_1\\)\u306e\u5171\u6709\u70b9\u3092\\(\\mathrm{R}\\)\u3068\u3059\u308b. \u66f2\u7dda\\(C_1\\)\u306e\\(\\displaystyle\\alpha\\leq x\\leq \\frac{3}{2}\\)\u306e\u90e8\u5206, \u7dda\u5206\\(\\mathrm{PQ}\\), \u304a\u3088\u3073\u7dda\u5206\\(\\mathrm{QR}\\)\u3067\u56f2\u307e\u308c\u308b\u56f3\u5f62\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.
(2025 \u4eac\u90fd\u5927\u5b66 [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) \u307e\u305a\u4ee5\u4e0b\u306e\u901a\u308a\u95a2\u6570\u3092\u5b9a\u7fa9\u3059\u308b\u3068, $$
\\begin{align}
f_1(x)&=x^2-2x\\\\[1.5ex]
f_2(x)&=x^2+2x\\\\[1.5ex]
g(x)&=x^2-5x+\\frac{7}{4}
\\end{align}
$$\u66f2\u7dda\\(C_1\\)\u306e\u65b9\u7a0b\u5f0f\u306f,$$
y=x^2-2|x|=\\left\\{\\begin{array}{ll}f_1(x) & (x\\geq 0)\\\\[1.5ex]
f_2(x)& (x\\leq 0)
\\end{array}\\right.
$$\u3068\u8868\u305b\u308b.

\\(t\\)\u3092\u5b9f\u6570\u3068\u3057, \u66f2\u7dda\\(\\displaystyle C_2:y=g(x)=x^2-5x+\\frac{7}{4}\\)\u4e0a\u306e\u70b9\u306b\u304a\u3051\u308b\u63a5\u7dda\\(l(t)\\)\u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u308b\u3068, \\(g^\\prime(x)=2x-5\\)\u3088\u308a,$$
\\begin{align}
y&=g^\\prime(t)(x-t)+g(t)\\\\[1.5ex]
&=(2t-5)(x-t)+t^2-5t+\\frac{7}{4}\\\\[1.5ex]
&=(2t-5)x-t^2+\\frac{7}{4}
\\end{align}
$$\u3068\u306a\u308b. \u3053\u306e\u76f4\u7dda\\(l(t)\\)\u304c\\(y=f_1(x)\\), \\(y=f_2(x)\\)\u3068\u63a5\u3059\u308b\u3068\u304d\u306e\\(t\\)\u306e\u5024\u3092\u6c42\u3081\u308b.

\u2460 \\(y=f_1(x)\\)\u306b\u3064\u3044\u3066
\u5148\u306b\u6c42\u3081\u305f\\(l(t)\\)\u306e\u65b9\u7a0b\u5f0f\u3068, \\(y=f_1(x)\\)\u304b\u3089\\(y\\)\u3092\u6d88\u53bb\u3059\u308b\u3068,
$$
\\begin{align}
&(2t-5)x-t^2+\\frac{7}{4}=x^2-2x\\\\[1.5ex]
\\iff & x^2-(2t-3)x+t^2-\\frac{7}{4}=0
\\end{align}
$$\u3068\u306a\u308b. \\(l(t)\\)\u304c\\(y=f_1(x)\\)\u3068\u63a5\u3059\u308b\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f, \u3053\u306e2\u6b21\u65b9\u7a0b\u5f0f\u304c\u91cd\u89e3\u3092\u6301\u3064\u3053\u3068\u3067\u3042\u308b\u304b\u3089, \u5224\u5225\u5f0f\u3092\\(D\\)\u3068\u3057\u3066, $$
\\begin{align}
D&=(2t-3)^2-4\\left(t^2-\\frac{7}{4}\\right)=0\\\\[1.5ex]
\\iff &-12t+9+7=0\\\\[1.5ex]
\\iff & t=\\frac{4}{3}
\\end{align}
$$ \u3068\u306a\u308a, \\(\\displaystyle t=\\frac{4}{3}\\)\u306e\u3068\u304d, \\(l(t)\\)\u306f\\(y=f_1(x)\\)\u306b\u63a5\u3059\u308b. \\(y=f_1(x)\\)\u3068\u306e\u63a5\u70b9\u306e\\(x\\)\u5ea7\u6a19\u306f2\u6b21\u65b9\u7a0b\u5f0f\u306e\u91cd\u89e3\u3068\u306a\u308b\u306e\u3067, \\(\\displaystyle t=\\frac{4}{3}\\)\u3068\u3057\u3066, \u305d\u306e\\(x\\)\u5ea7\u6a19\u306f$$
\\begin{align}
& x^2-\\left(2\\cdot\\frac{4}{3}-3\\right)x+\\left(\\frac{4}{3}\\right)^2-\\frac{7}{4}=0\\\\[1.5ex]
\\iff & x^2+\\frac{1}{3}+\\frac{1}{36}=0\\\\[1.5ex]
\\iff & \\left(x+\\frac{1}{6}\\right)^2=0\\\\[1.5ex]
\\iff & x=-\\frac{1}{6}
\\end{align}
$$\u3068\u3057\u3066\u6c42\u307e\u308b. \u3057\u304b\u3057, \\(\\displaystyle -\\frac{1}{6}<0\\)\u3088\u308a, \\(y=f_1(x)\\)\u3068\u306e\u63a5\u70b9\u306f\\(C_1\\)\u306b\u542b\u307e\u308c\u306a\u3044. \u3088\u3063\u3066, \\(x\\geq 0\\)\u306e\u7bc4\u56f2\u3067\\(C_1\\)\u3068\u63a5\u3059\u308b\u3088\u3046\u306a\\(C_2\\)\u3068\u306e\u5171\u901a\u63a5\u7dda\u306f\u5b58\u5728\u3057\u306a\u3044.

\u2461 \\(y=f_2(x)\\)\u306b\u3064\u3044\u3066
\u5148\u306b\u6c42\u3081\u305f\\(l(t)\\)\u306e\u65b9\u7a0b\u5f0f\u3068, \\(y=f_2(x)\\)\u304b\u3089\\(y\\)\u3092\u6d88\u53bb\u3059\u308b\u3068,
$$
\\begin{align}
&(2t-5)x-t^2+\\frac{7}{4}=x^2+2x\\\\[1.5ex]
\\iff & x^2-(2t-7)x+t^2-\\frac{7}{4}=0
\\end{align}
$$\u3068\u306a\u308b. \u2460\u3068\u540c\u69d8\u306b, \\(l(t)\\)\u304c\\(y=f_2(x)\\)\u3068\u3082\u63a5\u3059\u308b\\(t\\)\u3092\u6c42\u3081\u308b\u3068, \u5224\u5225\u5f0f\\(D=0\\)\u304b\u3089, $$
\\begin{align}
D&=(2t-7)^2-4\\left(t^2-\\frac{7}{4}\\right)=0\\\\[1.5ex]
\\iff &-28t+49+7=0\\\\[1.5ex]
\\iff & t=2
\\end{align}
$$ \u3068\u306a\u308b. \\(y=f_2(x)\\)\u3068\u306e\u63a5\u70b9\u306e\\(x\\)\u5ea7\u6a19\u306f2\u6b21\u65b9\u7a0b\u5f0f\u306e\u91cd\u89e3\u3068\u306a\u308b\u306e\u3067, \\(t=2\\)\u3068\u3057\u3066, \u305d\u306e\\(x\\)\u5ea7\u6a19\u306f$$
\\begin{align}
& x^2-\\left(2\\cdot 2-7\\right)x+2^2-\\frac{7}{4}=0\\\\[1.5ex]
\\iff & x^2+3x+\\frac{9}{4}=0\\\\[1.5ex]
\\iff & \\left(x+\\frac{3}{2}\\right)^2=0\\\\[1.5ex]
\\iff & x=-\\frac{3}{2}
\\end{align}
$$\u3068\u3057\u3066\u6c42\u307e\u308a, \\(\\displaystyle -\\frac{3}{2}<0\\)\u3088\u308a, \\(y=f_2(x)\\)\u3068\u306e\u63a5\u70b9\u306f\\(C_1\\)\u306b\u542b\u307e\u308c\u308b. \u3088\u3063\u3066, \u3053\u308c\u304c\u552f\u4e00\u306e\\(C_1\\), \\(C_2\\)\u306e\u5171\u901a\u63a5\u7dda\u3068\u306a\u308a, \u3053\u308c\u304c\\(l_2\\)\u3067\u3042\u308b. \\(t=2\\)\u304b\u3089\\(l_2\\)\u306e\u65b9\u7a0b\u5f0f\u306f,
$$
y=(2\\cdot 2-5)x-2^2+\\frac{7}{4}=-x-\\frac{9}{4}
$$\u3068\u306a\u308b.<\/p>\n\n\n\n

(2) \u5404\u66f2\u7dda, \u76f4\u7dda\u306e\u30b0\u30e9\u30d5\u3068, \u9762\u7a4d\u3092\u6c42\u3081\u308b\u56f3\u5f62\u3092\u63cf\u304f\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\n\n\n\n

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

(1)\u304b\u3089, \\(\\displaystyle \\alpha=-\\frac{3}{2}\\)\u3067\u3042\u308a, \u56f3\u5f62\u306e\u9762\u7a4d\\(S\\)\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u6c42\u307e\u308b.
$$
\\begin{align}
S&=\\int_{-\\frac{3}{2}}^0\\left\\{x^2+2x-\\left(-x-\\frac{9}{4}\\right)\\right\\}\\,dx\\\\[1.5ex]
&\\qquad +\\int_0^{\\frac{3}{2}}\\left\\{x^2-2x-\\left(-x-\\frac{9}{4}\\right)\\right\\}\\,dx\\\\[1.5ex]
&=\\int_{-\\frac{3}{2}}^0\\left(x^2+3x+\\frac{9}{4}\\right)\\,dx\\\\[1.5ex]
&\\qquad + \\int_0^{\\frac{3}{2}}\\left(x^2-x+\\frac{9}{4}\\right)\\,dx\\\\[1.5ex]
&=\\left[\\frac{x^3}{3}+\\frac{3}{2}x^2+\\frac{9}{4}x\\right]_{-\\frac{3}{2}}^0\\\\[1.5ex]
&\\qquad +\\left[\\frac{x^3}{3}-\\frac{1}{2}x^2+\\frac{9}{4}x\\right]_0^{\\frac{3}{2}}\\\\[1.5ex]
&=-\\left\\{\\frac{1}{3}\\left(-\\frac{3}{2}\\right)^3+\\frac{3}{2}\\left(-\\frac{3}{2}\\right)^2+\\frac{9}{4}\\cdot\\left(-\\frac{3}{2}\\right)\\right\\}\\\\[1.5ex]
&\\qquad +\\left\\{\\frac{1}{3}\\left(\\frac{3}{2}\\right)^3-\\frac{1}{2}\\left(\\frac{3}{2}\\right)^2+\\frac{9}{4}\\cdot\\left(\\frac{3}{2}\\right)\\right\\}\\\\[1.5ex]
&=-\\left(-\\frac{9}{8}+\\frac{27}{8}-\\frac{27}{8}\\right)\\\\[1.5ex]
&\\qquad +\\left(\\frac{9}{8}-\\frac{9}{8}+\\frac{27}{8}\\right)\\\\[1.5ex]
&=\\frac{9}{8}+\\frac{27}{8}=\\frac{36}{8}=\\frac{9}{2}
\\end{align}
$$\u3068\u306a\u308a, \u9762\u7a4d\u304c\u6c42\u307e\u3063\u305f.<\/p>\n<\/div><\/div>\n\n\n\n

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