\u95a2\u6570 \\(f(x) =x^3-x^2+x\\) \u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(f(x)\\) \u306f\u3064\u306d\u306b\u5897\u52a0\u3059\u308b\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(f(x)\\) \u306e\u9006\u95a2\u6570\u3092 \\(g(x)\\) \u3068\u304a\u304f. \\(x \\gt 0\\) \u306b\u3064\u3044\u3066\r\n\\[\r\n\\sqrt[3]{x} -1 \\lt g(x) \\lt \\sqrt[3]{x} +1\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(b \\gt a \\gt 0\\) \u306b\u3064\u3044\u3066\r\n\\[\r\n0 \\lt \\displaystyle\\int _ a^b \\dfrac{1}{x^2+1} \\, dx \\lt \\dfrac{1}{a}\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (4)<\/strong>\u3000\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066, (2)<\/strong> \u3067\u5b9a\u7fa9\u3055\u308c\u305f \\(g(x)\\) \u3092\u7528\u3044\u3066\r\n\\[\r\nA _ n =\\displaystyle\\int _ n^{2n} \\dfrac{1}{\\{ g(x) \\}^3 +g(x)} \\, dx\r\n\\]\r\n\u3068\u304a\u304f\u3068\u304d, \u6975\u9650\u5024 \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} A _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\\begin{align}\r\nf'(x) & = 3x^2 -2x +1 \\\\\r\n& =3 \\left( x -\\dfrac{1}{3} \\right)^2 +\\dfrac{2}{3} \\gt 0\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u95a2\u6570\u3067\u3042\u308b.<\/p>\r\n (2)<\/strong><\/p>\r\n \\(y = g(x)\\) \u3068\u304a\u304f\u3068, \\(x = f(y)\\) . (3)<\/strong><\/p>\r\n \\(I =\\displaystyle\\int _ a^b \\dfrac{1}{x^2+1} \\, dx\\) \u3068\u304a\u304f. (4)<\/strong><\/p>\r\n (2)<\/strong> \u3068\u540c\u69d8\u306b, \\(y = g(x)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nA _ n & = \\displaystyle\\int _ {g(n)}^{g(2n)} \\dfrac{1}{y^3+y} \\cdot f'(y) \\, dy \\\\\r\n& = \\displaystyle\\int _ {g(n)}^{g(2n)} \\dfrac{3y^2 -2y +1}{y^3+y} \\, dy \\\\\r\n& = \\displaystyle\\int _ {g(n)}^{g(2n)} \\left( \\dfrac{2y-2}{y^2+1} +\\dfrac{1}{y} \\right) \\, dy \\\\\r\n& = \\left[ \\log y +\\log (y^2+1) \\right] _ {g(n)}^{g(2n)} -2 \\displaystyle\\int _ {g(n)}^{g(2n)} \\dfrac{1}{y^2+1} \\, dy \\\\\r\n& = \\log \\underline{\\dfrac{g(2n) \\left[ \\left\\{ g(2n) \\right\\}^2 +1 \\right]}{g(n) \\left[ \\left\\{ g(n) \\right\\}^2 +1 \\right]}} _ {[2]} -2 \\underline{\\displaystyle\\int _ {g(n)}^{g(2n)} \\dfrac{1}{y^2+1} \\, dy} _ {[3]} \\\\\n\\end{align}\\]\r\n[2] \u306b\u3064\u3044\u3066, (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\n\\dfrac{\\sqrt[3]{2n} -1}{\\sqrt[3]{n} +1} & \\lt \\dfrac{g(2n)}{g(n)} \\lt \\dfrac{\\sqrt[3]{2n} +1}{\\sqrt[3]{n} -1} \\\\\r\n\\text{\u2234} \\quad \\dfrac{\\sqrt[3]{2} -\\frac{1}{\\sqrt[3]{n}}}{1 +\\frac{1}{\\sqrt[3]{n}}} & \\lt \\dfrac{g(2n)}{g(n)} \\lt \\dfrac{\\sqrt[3]{2} +\\frac{1}{\\sqrt[3]{n}}}{1 -\\frac{1}{\\sqrt[3]{n}}}\n\\end{align}\\]\r\n\\(n \\rightarrow \\infty\\) \u306e\u3068\u304d, \u7b2c \\(1\\) \u8fba\u3068\u7b2c \\(3\\) \u8fba\u306f\u3068\u3082\u306b \\(\\sqrt[3]{2}\\) \u306b\u53ce\u675f\u3059\u308b\u306e\u3067, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{g(2n)}{g(n)} = \\sqrt[3]{2} \\quad ... [4]\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n[2] & = \\dfrac{g(2n)}{g(n)} \\cdot \\dfrac{\\left\\{ \\frac{g(2n)}{g(n)} \\right\\}^2 +\\frac{1}{\\{ g(n) \\}^2}}{1 +\\frac{1}{\\{ g(n) \\}^2}} \\\\\r\n& \\rightarrow \\sqrt[3]{2} \\cdot \\left\\{ \\sqrt[3]{2} \\right\\}^2 = 2 \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} )\n\\end{align}\\]\r\n[3] \u306b\u3064\u3044\u3066, (2)<\/strong> (3)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n0 \\lt [3] \\lt \\dfrac{1}{g(n)} \\lt \\dfrac{1}{\\sqrt[3]{n} -1}\n\\]\r\n\\(n \\rightarrow \\infty\\) \u306e\u3068\u304d, \u6700\u53f3\u8fba\u306f \\(0\\) \u306b\u53ce\u675f\u3059\u308b\u306e\u3067\r\n\u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} [3] = 0 \\quad ... [5]\n\\]\r\n\u3088\u3063\u3066, [4] [5] \u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} A _ n = \\log 2 -2 \\cdot 0 =\\underline{\\log 2}\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u95a2\u6570 \\(f(x) =x^3-x^2+x\\) \u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(f(x)\\) \u306f\u3064\u306d\u306b\u5897\u52a0\u3059\u308b\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b. (2)\u3000\\(f(x)\\) \u306e\u9006\u95a2\u6570\u3092 \\(g(x)\\) \u3068\u304a\u304f. \\(x … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\u793a\u3059\u3079\u304d\u4e0d\u7b49\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\sqrt[3]{f(y)} -1 & \\lt y \\lt \\sqrt[3]{f(y)} +1 \\\\\r\ny-1 & \\lt \\sqrt[3]{f(y)} \\lt y+1 \\\\\r\n\\text{\u2234} \\quad (y-1)^3 & \\lt f(y) \\lt (y+1)^3 \\quad ... [\\text{A}]\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, [A] \u3092\u793a\u305b\u3070\u3088\u3044.\r\n\\[\\begin{align}\r\nf(y) -(y-1)^3 & = 4y^2 -2y +1 \\\\\r\n& = 4 \\left( y -\\dfrac{1}{4} \\right)^2 +\\dfrac{3}{4} \\gt 0 , \\\\\r\n(y+1)^3 -f(y) & = 4y^2 +2y +1 \\\\\r\n& = 4 \\left( y +\\dfrac{1}{4} \\right)^2 +\\dfrac{3}{4} \\gt 0\n\\end{align}\\]\r\n\u3088\u3063\u3066, [A] \u304c\u793a\u3055\u308c\u3066, \u984c\u610f\u3082\u793a\u3055\u308c\u305f.<\/p>\r\n
\r\n\\(a \\lt x \\lt b\\) \u306b\u304a\u3044\u3066\r\n\\[\r\n0 \\lt \\dfrac{1}{x^2+1} \\lt \\dfrac{1}{x^2}\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nI \\gt 0\n\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nI & \\lt \\displaystyle\\int _ a^b \\dfrac{1}{x^2} \\, dx = \\left[ -\\dfrac{1}{x} \\right] _ a^b \\\\\r\n& = \\dfrac{1}{a} -\\dfrac{1}{b} \\lt \\dfrac{1}{a}\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n0 \\lt I \\lt \\dfrac{1}{a}\n\\]\r\n