{"id":9486,"date":"2025-10-20T18:55:25","date_gmt":"2025-10-20T09:55:25","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9486"},"modified":"2025-10-21T16:08:19","modified_gmt":"2025-10-21T07:08:19","slug":"ch05-differentiation-of-functions-of-one-variable","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch05-differentiation-of-functions-of-one-variable\/","title":{"rendered":"\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84"},"content":{"rendered":"
<\/p>\n

<\/p>\n

\uc774 \uc7a5\uc5d0\uc11c\ub294 \uc2e4\ud568\uc218\uc758 \ubbf8\ubd84 \uac00\ub2a5\uc131\uacfc \uad00\ub828\ub41c \uc911\uc694\ud55c \uc815\ub9ac\ub4e4\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ubbf8\ubd84\uc758 \uae30\ud558\ud559\uc801 \uc758\ubbf8\ub85c\ubd80\ud130 \uc2dc\uc791\ud558\uc5ec \ud3c9\uade0\uac12 \uc815\ub9ac, \ud14c\uc77c\ub7ec \uc815\ub9ac \ub4f1 \ud574\uc11d\ud559\uc758 \uc8fc\uc694 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

\ubbf8\ubd84 \uac00\ub2a5\uc131<\/h3>\n

\\(X\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(c\\in X\\cap X’\\)\uc774\ub77c\uace0 \ud558\uc790. \ud568\uc218 \\(f: X \\to \\mathbb{R}\\)\uc774 \uc810 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5<\/span>\ud558\ub2e4(differentiable)\ub294 \uac83\uc740 \ub2e4\uc74c \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4.
\n\\[f'(c) = \\lim_{h \\to 0} \\frac{f(c + h) – f(c)}{h}.\\]
\n\uc774 \uadf9\ud55c\uac12 \\(f'(c)\\)\ub97c \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \ubbf8\ubd84\uacc4\uc218<\/span>(derivative)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\ud568\uc218 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\ub294 \uac83\uc744 \ub2e4\ub974\uac8c \ud45c\ud604\ud558\uba74, \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 \\(L\\)\uacfc \ud568\uc218 \\(r(h)\\)\uac00 \uc874\uc7ac\ud558\ub294 \uac83\uc774\ub2e4.
\n\\[f(c + h) = f(c) + Lh + r(h), \\quad \\lim_{h \\to 0} \\frac{r(h)}{h} = 0 .\\]
\n\uc774\ub54c \\(L = f'(c)\\)\uac00 \ub41c\ub2e4.<\/p>\n

\\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c, \uc9c1\uc120 \\(y = f'(c)(x – c)+f(c)\\)\ub97c \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \uc811\uc120<\/span>(tangent line)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\ud568\uc218 \\(f:X\\rightarrow \\mathbb{R}\\)\uc758 \uc815\uc758\uc5ed\uc758 \uc810 \uc911\uc5d0\uc11c \\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud55c \uc810\uc758 \uc9d1\ud569\uc744 \\(D\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f ‘ \\)\uc740 \\(D\\)\uc758 \uc810 \\(x\\)\ub97c \\( f’ (x)\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\uc774\ub2e4. \uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \\(f ‘\\)\uc744 \\(f\\)\uc758 \ub3c4\ud568\uc218<\/span>(derivative function)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\n

\ubb38\uc81c 5.1.<\/span>
\n\ubbf8\ubd84\uacc4\uc218\uc640 \ub3c4\ud568\uc218\uc758 \ucc28\uc774\ub97c \uc124\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

\n

\uc815\ub9ac 5.1. (\ubbf8\ubd84 \uac00\ub2a5\uc131\uacfc \uc5f0\uc18d\uc131\uc758 \uad00\uacc4)<\/span><\/p>\n

\ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74, \\(f\\)\ub294 \uc810 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/p>\n

\\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74
\n\\[
\n\\lim_{x \\to c} f(x) = \\lim_{x \\to c} \\left[ f(c) + \\frac{f(x) – f(c)}{x – c}(x – c) \\right] = f(c) + f'(c) \\cdot 0 = f(c)
\n\\]
\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

\uadf8\ub7ec\ub098 \uc5f0\uc18d\uc778 \ud568\uc218\uac00 \ubaa8\ub450 \ubbf8\ubd84 \uac00\ub2a5\ud55c \uac83\uc740 \uc544\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uc5b4, \\(f(x) = |x|\\)\ub294 \\(x = 0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uc9c0\ub9cc \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\ub2e4.<\/p>\n

\uc2e4\ud568\uc218\uc758 \uadf9\ud55c\uc5d0\uc11c \uc88c\uadf9\ud55c\uacfc \uc6b0\uadf9\ud55c\uc744 \uc815\uc758\ud55c \uac83\ucc98\ub7fc, \ubbf8\ubd84\uacc4\uc218\ub3c4 \ud55c\ubc29\ud5a5 \ubbf8\ubd84\uacc4\uc218<\/span>\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n

    \n
  • \uc88c\ubbf8\ubd84\uacc4\uc218: \\(\\displaystyle f’_-(c) = \\lim_{h \\to 0^-} \\frac{f(c + h) – f(c)}{h}\\)<\/li>\n
  • \uc6b0\ubbf8\ubd84\uacc4\uc218: \\(\\displaystyle f’_+(c) = \\lim_{h \\to 0^+} \\frac{f(c + h) – f(c)}{h}\\)<\/li>\n<\/ul>\n

    \ub2eb\ud78c\uad6c\uac04\uc758 \ub05d\uc810\uc5d0\uc11c\ub294 \ud55c\ubc29\ud5a5 \ubbf8\ubd84\uacc4\uc218\ub85c \ubbf8\ubd84\uc744 \uc815\uc758\ud55c\ub2e4. \uc989 \ud568\uc218 \\(f:[a,\\,b] \\rightarrow \\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec, \\(f'(a)\\)\uc640 \\(f'(b)\\)\ub294 \uac01\uac01 \uc6b0\ubbf8\ubd84\uacc4\uc218\uc640 \uc88c\ubbf8\ubd84\uacc4\uc218\ub85c \uc815\uc758\ub41c\ub2e4.<\/p>\n

    \uad6c\uac04 \ub0b4\ubd80\uc758 \uc810\uc5d0\uc11c \ud568\uc218\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc88c\ubbf8\ubd84\uacc4\uc218\uc640 \uc6b0\ubbf8\ubd84\uacc4\uc218\uac00 \ubaa8\ub450 \uc874\uc7ac\ud558\uace0 \ub450 \ubbf8\ubd84\uacc4\uc218\uac00 \uc77c\uce58\ud558\ub294 \uac83\uc774\ub2e4. [\uc88c\ubbf8\ubd84\uacc4\uc218\uc640 \ub3c4\ud568\uc218\uc758 \uc88c\uadf9\ud55c\uc740 \uc11c\ub85c \ub2e4\ub97c \uc218 \uc788\ub2e4.]<\/p>\n

    \ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \uadf8\ub7ec\ub098 \\(f’\\)\uc740 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2d0 \uc218 \uc788\ub2e4. \uc608\ub97c \ub4e4\uc5b4
    \n\\[f(x) = \\begin{cases} x^2 \\sin\\frac{1}{x} & \\;\\text{if }\\, x \\neq 0, \\\\[6pt] 0 & \\;\\text{if }\\, x = 0 \\end{cases}\\]
    \n\uc774\ub77c\uace0 \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\ub294 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc \\(f’\\)\uc740 \\(x = 0\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\ub2e4.<\/p>\n

    \\(a\\)\uc640 \\(b\\)\uac00 \uc0c1\uc218\uc774\uace0 \\(f\\)\uc640 \\(g\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

      \n
    • \uc120\ud615\uc131: \\((af + bg)’ = af’ + bg’\\)<\/li>\n
    • \uacf1\uc758 \ubc95\uce59: \\((fg)’ = f’g + fg’\\)<\/li>\n
    • \ubaab\uc758 \ubc95\uce59: \\((f\/g)’ = (f’g – fg’)\/g^2\\) (\ub2e8, \\(g \\neq 0\\))<\/li>\n
    • \uc5f0\uc1c4\ubc95\uce59: \\((g \\circ f)’ = (g’ \\circ f) \\cdot f’\\)<\/li>\n<\/ul>\n

      \ud2b9\ud788 \ub2e4\ud56d\ud568\uc218 \\(f(x) = a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x^1 + a_0 \\)\uc758 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
      \n\\[f’ (x) = na_n x^{n-1} + (n-1)a_{n-1} x^{n-2} + \\cdots + a_1 .\\]<\/p>\n

      \n

      \ubb38\uc81c 5.2.<\/span>
      \n\ubbf8\ubd84\uc758 \uc120\ud615\uc131\uacfc \uacf1\uc758 \ubc95\uce59, \ubaab\uc758 \ubc95\uce59, \uc5f0\uc1c4\ubc95\uce59\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

      \n

      \ubb38\uc81c 5.3.<\/span>
      \n\\(n\\)\uc774 \uc790\uc5f0\uc218\uc774\uace0 \\(f(x)=x^n\\)\uc77c \ub54c \\(f ‘ (x) = nx^{n-1}\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. (\ub2e8, \\(n=1\\)\uc774\uace0 \\(x=0\\)\uc77c \ub54c \\(f'(0)=0^0\\)\uc774 \ub418\ub294\ub370, \uc774 \uacbd\uc6b0\uc5d0 \ud55c\ud558\uc5ec \\(0^0 = 1\\)\ub85c \uacc4\uc0b0\ud55c\ub2e4.)<\/p>\n<\/div>\n

      \n

      \ubb38\uc81c 5.4.<\/span>
      \n\\(m\\)\uc774 \\(0\\)\uc774 \uc544\ub2cc \uc815\uc218\uc774\uace0 \\(f(x)=x^m\\)\uc77c \ub54c \\(f ‘ (x) = mx^{m-1}\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

      \n

      \ubb38\uc81c 5.5.<\/span>
      \n\ubbf8\ubd84\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n

        \n
      1. \\(f(x)=\\sin x\\)<\/li>\n
      2. \\(f(x)=\\cos x\\)<\/li>\n
      3. \\(f(x)=\\tan x\\) (\ub2e8, \\(x\\neq\\frac{2k+1}{2}\\pi\\), \\(k\\in\\mathbb{Z}\\))<\/li>\n
      4. \\(f(x)=\\sec x\\) (\ub2e8, \\(x\\neq\\frac{2k+1}{2}\\pi\\), \\(k\\in\\mathbb{Z}\\))<\/li>\n
      5. \\(f(x)=\\csc x\\) (\ub2e8, \\(x\\neq k\\pi\\), \\(k\\in\\mathbb{Z}\\))<\/li>\n
      6. \\(f(x)=\\cot x\\) (\ub2e8, \\(x\\neq k\\pi\\), \\(k\\in\\mathbb{Z}\\))<\/li>\n<\/ol>\n<\/div>\n

        \ud3c9\uade0\uac12 \uc815\ub9ac\uc640 \uadf8 \ub530\ub984\uc815\ub9ac<\/h3>\n

        \ud3c9\uade0\uac12 \uc815\ub9ac\ub294 \ub3c4\ud568\uc218\uc758 \ud575\uc2ec\uc801\uc778 \uc131\uc9c8\uc744 \ub098\ud0c0\ub0b4\uba70, \ub3c4\ud568\uc218\ub97c \ud65c\uc6a9\ud558\ub294 \uc815\ub9ac\ub97c \uc99d\uba85\ud560 \ub54c \uc790\uc8fc \uc0ac\uc6a9\ub41c\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 5.2. (\uadf9\uac12\uc5d0 \ub300\ud55c \ud398\ub974\ub9c8\uc758 \uc815\ub9ac)<\/span><\/p>\n

        \\(c\\)\uac00 \uad6c\uac04 \\(I\\)\uc758 \ub0b4\uc810\uc774\uace0 \ud568\uc218 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uc73c\uba70 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uadf9\uac12\uc744 \uac00\uc9c0\uace0 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74 \\(f'(c) = 0\\)\uc774\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c4\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\delta>0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\( | x-c | < \\delta\\)\uc77c \ub54c \\(f(x) \\le f(c)\\)\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc989 \\(|h|<\\delta\\)\uc77c \ub54c \\(f(c+h)-f(c)\\le 0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c\n\\[\\begin{aligned}\nf_- ' (c) &= \\lim_{h\\rightarrow 0-} \\frac{f(c+h)-f(c)}{h} \\ge 0,\\\\[6pt]\nf_+ ' (c) &= \\lim_{h\\rightarrow 0+} \\frac{f(c+h)-f(c)}{h} \\le 0\n\\end{aligned}\\]\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \\(f'(c) = f_- ' (c) = f_+ ' (c)\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f'(c)=0\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

        \n

        \uc815\ub9ac 5.3. (\ub864\uc758 \uc815\ub9ac)<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,\\, b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(f(a) = f(b)\\)\uc774\uba74, \uc5b4\ub5a4 \\(c \\in (a,\\, b)\\)\uc5d0 \ub300\ud574 \\(f'(c) = 0\\)\uc774\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \ub9cc\uc57d \\(f\\)\uac00 \uc0c1\uc218\ud568\uc218\uc774\uba74 \\((a,\\,b)\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f’ = 0\\)\uc774\ub2e4. \\(f\\)\uac00 \uc0c1\uc218\ud568\uc218\uac00 \uc544\ub2cc \uacbd\uc6b0 \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \\(f\\)\uac00 \ucd5c\ub313\uac12 \ub610\ub294 \ucd5c\uc19f\uac12\uc774 \ub418\ub294 \uc810 \\(c\\)\uac00 \\((a,\\,b)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub54c \uadf9\uac12\uc5d0 \ub300\ud55c \ud398\ub974\ub9c8\uc758 \uc815\ub9ac(\uc815\ub9ac 5.2)\uc5d0 \uc758\ud558\uc5ec \\(f'(c)=0\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

        \n

        \uc815\ub9ac 5.4. (\ub77c\uadf8\ub791\uc8fc \ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,\\, b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74, \uc801\ub2f9\ud55c \\(c \\in (a,\\, b)\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
        \n\\[f'(c) = \\frac{f(b) – f(a)}{b – a} .\\]<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \uad6c\uac04 \\([a,\\,b]\\) \uc704\uc5d0\uc11c \ud568\uc218 \\(g\\)\ub97c
        \n\\[g(x) = f(x) – \\frac{f(b) – f(a)}{b – a}(x – a)\\]
        \n\ub77c\uace0 \uc815\uc758\ud558\uba74 \\(g(a) = g(b) = f(a)\\)\uc774\ub2e4. \ub864\uc758 \uc815\ub9ac\ub97c \\(g\\)\uc5d0 \uc801\uc6a9\ud558\uba74 \ubc14\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

        \ud3c9\uade0\uac12 \uc815\ub9ac\uc758 \uae30\ud558\ud559\uc801 \uc758\ubbf8\ub294 \uace1\uc120 \uc704\uc758 \uc5b4\ub5a4 \uc810\uc5d0\uc11c \uc811\uc120\uc758 \uae30\uc6b8\uae30\uac00 \ud560\uc120\uc758 \uae30\uc6b8\uae30\uc640 \uac19\ub2e4\ub294 \uac83\uc774\ub2e4.<\/p>\n

        \n

        \ubb38\uc81c 5.6.<\/span>
        \n\ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc784\uc758\uc758 \uc2e4\uc218 \\(x\\), \\(y\\)\uc5d0 \ub300\ud558\uc5ec \ubd80\ub4f1\uc2dd \\(|\\sin x – \\sin y| \\leq |x – y|\\)\uac00 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

        \n

        \uc815\ub9ac 5.5. (\ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,\\, b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(g'(x) \\neq 0\\)\uc774\uba74, \uc801\ub2f9\ud55c \\(c \\in (a,\\, b)\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
        \n\\[\\frac{f'(c)}{g'(c)} = \\frac{f(b) – f(a)}{g(b) – g(a)} .\\]<\/p>\n<\/div>\n

        \n

        \uc99d\uba85<\/p>\n

        \uba3c\uc800 \\(g(b) \\neq g(a)\\)\uc784\uc744 \ud655\uc778\ud558\uc790. \ub9cc\uc57d \\(g(b) = g(a)\\)\ub77c\uba74 \ub864\uc758 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc5b4\ub5a4 \\(c \\in (a,\\, b)\\)\uc5d0\uc11c \\(g'(c) = 0\\)\uc774 \ub418\ub294\ub370, \uc774\ub294 \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774\ub2e4.<\/p>\n

        \uc774\uc81c \ud568\uc218 \\(h\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.
        \n\\[h(x) = f(x) – \\frac{f(b) – f(a)}{g(b) – g(a)} \\cdot g(x).\\]<\/p>\n

        \uadf8\ub7ec\uba74 \\(h\\)\ub294 \\([a,\\, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\((a,\\, b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub610\ud55c
        \n\\[h(a) = f(a) – \\frac{f(b) – f(a)}{g(b) – g(a)} \\cdot g(a), \\quad
        \nh(b) = f(b) – \\frac{f(b) – f(a)}{g(b) – g(a)} \\cdot g(b)
        \n\\]
        \n\uc774\ubbc0\ub85c \\(h(a) = h(b)\\)\uc774\ub2e4. \\(h\\)\uc5d0 \ub864\uc758 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74, \uc5b4\ub5a4 \\(c \\in (a,\\, b)\\)\uc5d0 \ub300\ud574 \\(h'(c) = 0\\)\uc774\ub2e4. \uc989,
        \n\\[f'(c) – \\frac{f(b) – f(a)}{g(b) – g(a)} \\cdot g'(c) = 0.\\]
        \n\uc5ec\uae30\uc11c \\(g'(c) \\neq 0\\)\uc774\ubbc0\ub85c
        \n\\[\\frac{f'(c)}{g'(c)} = \\frac{f(b) – f(a)}{g(b) – g(a)}\\]
        \n\ub97c \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

        \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \ud568\uc22b\uac12\uc758 \ubcc0\ud654\uc640 \uad00\ub828\ub41c \uc720\uc6a9\ud55c \uacf5\uc2dd\uc744 \uc720\ub3c4\ud560 \uc218 \uc788\ub2e4.<\/p>\n

        \n

        \uc815\ub9ac 5.6. (\ud568\uc218\uc758 \uc99d\uac10\uc5d0 \ub300\ud55c \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

          \n
        1. \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f'(x) = 0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc0c1\uc218\ud568\uc218\uc774\ub2e4.<\/li>\n
        2. \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f'(x) > 0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc21c\uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\ub2e4.<\/li>\n
        3. \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f'(x) < 0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc21c\uac10\uc18c\ud558\ub294 \ud568\uc218\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \ubb38\uc81c 5.7.<\/span>
          \n\uc815\ub9ac 5.6\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

          \n

          \ubb38\uc81c 5.8.<\/span>
          \n\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc591\uc218 \\(M\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(|f'(x)| \\leq M\\)\uc774\uba74, \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \ub9bd\uc2dc\uce20 \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

          \ud14c\uc77c\ub7ec \uc815\ub9ac<\/h3>\n

          \ud568\uc218 \\(f\\)\uac00 \uc810 \\(a\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0, \ub3c4\ud568\uc218 \\(f ‘\\)\uc774 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c \\(a\\)\uc5d0\uc11c \\(f’\\)\uc758 \ubbf8\ubd84\uacc4\uc218\ub97c \\(f”(a)\\) \ub610\ub294 \\(\\frac{d^2}{dx^2}f(a)\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec, \\(f\\)\uac00 \\(a\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \\(n-1\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f\\)\ub97c \\(n-1\\)\ubc88 \ubbf8\ubd84\ud55c \ud568\uc218\uac00 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c, \\(a\\)\uc5d0\uc11c \\(f\\)\uc758 \\(n-1\\)\uacc4\ub3c4\ud568\uc218\uc758 \ubbf8\ubd84\uacc4\uc218\ub97c \\(a\\)\uc5d0\uc11c \\(f\\)\uc758 \\(n\\)\uacc4\ubbf8\ubd84\uacc4\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.
          \n\\[f^{(n)}(a) \\quad \\text{\ub610\ub294} \\quad \\frac{d^n}{dx^n}f(a).\\]
          \n\uc810 \\(a\\)\uc5d0\uc11c \\(f\\)\uc758 \\(n\\)\uacc4\ubbf8\ubd84\uacc4\uc218\uac00 \uc874\uc7ac\ud560 \ub54c “\\(f\\)\uac00 \\(a\\)\uc5d0\uc11c \\(n\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4”\ub77c\uace0 \ud45c\ud604\ud55c\ub2e4.<\/p>\n

          \\(f\\)\uc758 \uc815\uc758\uc5ed\uc758 \uc810 \\(x\\)\ub97c \\(x\\)\uc5d0\uc11c \\(f\\)\uc758 \\(n\\)\uacc4 \ubbf8\ubd84\uacc4\uc218\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ud568\uc218\ub97c \\(f\\)\uc758 \\(n\\)\uacc4\ub3c4\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0 \\(f^{(n)}\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

          \uad6c\uac04 \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f’\\)\uc774 \uc5f0\uc18d\uc77c \ub54c, “\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \\(C^1\\)<\/span>\uc774\ub2e4” \ub610\ub294 “\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc801\uc73c\ub85c \ubbf8\ubd84 \uac00\ub2a5<\/span>\ud558\ub2e4(continuously differentiable)”\ub77c\uace0 \ub9d0\ud55c\ub2e4. [“\uc5f0\uc18d\uc801\uc73c\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4”\ub77c\ub294 \ub9d0\uc774 “\uc784\uc758 \ud69f\uc218\ub85c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4”\ub77c\ub294 \ub73b\uc774 \uc544\ub2c8\ub2e4. \uc774\ub7ec\ud55c \ud63c\ub3d9\uc744 \ud53c\ud558\ub824\uba74 “\uc5f0\uc18d\uc778 \ub3c4\ud568\uc218\ub97c \uac00\uc9c4\ub2e4”\ub77c\ub294 \ud45c\ud604\uc744 \uc0ac\uc6a9\ud558\ub294 \ud3b8\uc774 \ub0ab\ub2e4.] \ub610\ud55c \uad6c\uac04 \\(I\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \\(f\\)\uc758 \\(n\\)\uacc4\ub3c4\ud568\uc218\uac00 \uc874\uc7ac\ud558\uace0 \\(f^{(n)}\\)\uc774 \uc5f0\uc18d\uc77c \ub54c, “\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \\(C^n\\)<\/span>\uc774\ub2e4”\ub77c\uace0 \ub9d0\ud55c\ub2e4. \ub9cc\uc57d \uc784\uc758\uc758 \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec, \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \\(C^n\\)\uc774\uba74, “\\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \\(C^{\\infty}\\)\uc774\ub2e4”\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

          \\(n\\)\uc774 \uc790\uc5f0\uc218\uc77c \ub54c, \uc810 \\(a\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \\(n\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218 \\(f\\)\uc5d0 \ub300\ud558\uc5ec, \\(a\\) \uadfc\ucc98\uc5d0\uc11c \ub2e4\ud56d\ud568\uc218\ub97c \uc0ac\uc6a9\ud558\uc5ec \\(f\\)\uc758 \uadfc\uc0ac\ud568\uc218\ub97c \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/p>\n

          \n

          \uc815\uc758 5.1. (\ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd)<\/span><\/p>\n

          \ud568\uc218 \\(f\\)\uac00 \uc810 \\(a\\)\uc5d0\uc11c \\(n\\)\ubc88 \uc774\uc0c1 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \uc810 \\(a\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc758 \\(n\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd<\/span>(Taylor polynomial)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
          \n\\[P_n(x) = \\sum_{k=0}^{n} \\frac{f^{(k)}(a)}{k!}(x – a)^k .\\]<\/p>\n<\/div>\n

          \ub9ce\uc740 \uacbd\uc6b0\uc5d0 \\(P_n (x)\\)\uc758 \ucc28\uc218\uac00 \ud074\uc218\ub85d, \uadf8\ub9ac\uace0 \\(x\\)\uac00 \\(a\\)\uc5d0 \uac00\uae4c\uc6b8\uc218\ub85d, \\(P_n(x)\\)\uc758 \uac12\uc774 \\(f(x)\\)\uc758 \uac12\uc5d0 \uac00\uae4c\uc6cc\uc9c4\ub2e4. \uc774\ub7ec\ud55c \uc0c1\ud669\uc5d0\uc11c \\(P_n (x)\\)\uc758 \uac12\uc744 \\(f(x)\\)\uc758 \uadfc\uc0bf\uac12\uc73c\ub85c \uc0ac\uc6a9\ud558\ub824\uba74 \ub450 \uac12\uc774 \uc5bc\ub9c8\ub098 \uac00\uae4c\uc6b4\uc9c0\ub97c \uac00\ub2a0\ud560 \uc218 \uc788\ub294 \uacf5\uc2dd\uc774 \ud544\uc694\ud558\ub2e4.<\/p>\n

          \n

          \uc815\ub9ac 5.7. (\ud14c\uc77c\ub7ec \uc815\ub9ac)<\/span><\/p>\n

          \ud568\uc218 \\(f\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \\(n\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\((a,\\, b)\\)\uc5d0\uc11c \\((n+1)\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba74, \uc801\ub2f9\ud55c \\(c \\in (a,\\, b)\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
          \n\\[f(b) = \\sum_{k=0}^{n} \\frac{f^{(k)}(a)}{k!}(b – a)^k + \\frac{f^{(n+1)}(c)}{(n+1)!}(b – a)^{n+1} .\\]
          \n\uc5ec\uae30\uc11c \uc6b0\ubcc0\uc758 \ub9c8\uc9c0\ub9c9 \ud56d\uc744 \ub77c\uadf8\ub791\uc8fc \ub098\uba38\uc9c0<\/span>(Lagrange remainder)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

          \n

          \uc99d\uba85<\/p>\n

          \\(a\\)\uc5d0\uc11c \\(f\\)\uc758 \\(n\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b4\uc790.
          \n\\[P_n(x) = \\sum_{k=0}^{n} \\frac{f^{(k)}(a)}{k!}(x – a)^k .\\]
          \n\ub098\uba38\uc9c0 \\(R_n(x) = f(x) – P_n(x)\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc0c1\uc218 \\(M\\)\uc774 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4.
          \n\\[f(b) = P_n(b) + M(b – a)^{n+1}.\\]
          \n\uc774\uc81c \ud568\uc218 \\(g\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
          \n\\[g(t) = f(t) – P_n(t) – M(t – a)^{n+1}, \\quad t \\in [a,\\, b].\\]<\/p>\n

          \uadf8\ub7ec\uba74 \\(g\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

            \n
          • \\(g(a) = f(a) – P_n(a) = 0\\) (\ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc758 \uc815\uc758\uc5d0 \uc758\ud574)<\/li>\n
          • \\(g(b) = f(b) – P_n(b) – M(b – a)^{n+1} = 0\\) (\\(M\\)\uc758 \uc815\uc758\uc5d0 \uc758\ud574)<\/li>\n
          • \\(g\\)\ub294 \\([a,\\, b]\\)\uc5d0\uc11c \\(n\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\((a,\\, b)\\)\uc5d0\uc11c \\((n+1)\\)\ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/li>\n<\/ul>\n

            \ub610\ud55c \\(k = 0,\\, 1,\\, \\cdots,\\, n\\)\uc5d0 \ub300\ud574 \\(g^{(k)}(a) = 0\\)\uc784\uc744 \ud655\uc778\ud560 \uc218 \uc788\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(P_n^{(k)}(a) = f^{(k)}(a)\\)\uc774\uace0 \\((t – a)^{n+1}\\)\uc758 \\(k\\)\uacc4 \ub3c4\ud568\uc218\uac00 \\(t = a\\)\uc5d0\uc11c 0\uc774 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n

            \uc774\uc81c \ub864\uc758 \uc815\ub9ac\ub97c \ubc18\ubcf5\ud558\uc5ec \uc801\uc6a9\ud558\uc790. \uc989 \\(g(a) = g(b) = 0\\)\uc774\ubbc0\ub85c, \uc5b4\ub5a4 \\(c_1 \\in (a,\\, b)\\)\uc5d0\uc11c \\(g'(c_1) = 0\\)\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(g'(a) = g'(c_1) = 0\\)\uc774\ubbc0\ub85c, \uc5b4\ub5a4 \\(c_2 \\in (a,\\, c_1)\\)\uc5d0\uc11c \\(g”(c_2) = 0\\)\uc774\ub2e4. \uc774 \uacfc\uc815\uc744 \uacc4\uc18d\ud558\uba74, \uc5b4\ub5a4 \\(c \\in (a,\\, b)\\)\uc5d0\uc11c \\(g^{(n+1)}(c) = 0\\)\uc774\ub2e4.<\/p>\n

            \\(g^{(n+1)}(t) = f^{(n+1)}(t) – M(n+1)!\\)\uc774\ubbc0\ub85c, \\(g^{(n+1)}(c) = 0\\)\uc73c\ub85c\ubd80\ud130 \ub2e4\uc74c \uc2dd\uc744 \uc5bb\ub294\ub2e4.
            \n\\[M = \\frac{f^{(n+1)}(c)}{(n+1)!}.\\]
            \n\ub530\ub77c\uc11c
            \n\\[f(b) = P_n(b) + \\frac{f^{(n+1)}(c)}{(n+1)!}(b – a)^{n+1}\\]
            \n\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

            \ud14c\uc77c\ub7ec \uc815\ub9ac\uc5d0\uc11c \ub098\uba38\uc9c0\ub294 \ub77c\uadf8\ub791\uc8fc\uc758 \ub098\uba38\uc9c0 \uc678\uc5d0\ub3c4 \ub2e4\uc74c\uacfc \uac19\uc740 \ud615\ud0dc\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n

              \n
            • \ucf54\uc2dc \ub098\uba38\uc9c0: \\(\\displaystyle R_n = \\frac{f^{(n+1)}(c)}{n!}(b – c)^n(b – a)\\).<\/li>\n
            • \uc801\ubd84\uc73c\ub85c \ud45c\ud604\ud55c \ub098\uba38\uc9c0: \\(\\displaystyle R_n = \\int_a^b \\frac{f^{(n+1)}(t)}{n!}(b – t)^n dt\\).<\/li>\n<\/ul>\n
              \n

              \ubb38\uc81c 5.9.<\/span>
              \n\ud568\uc218 \\(f(x) = e^x\\)\uc758 \\(0\\)\uc5d0\uc11c\uc758 \\(n\\)\ucc28 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(x=1\\)\uc5d0\uc11c \\(e\\)\uc758 \uadfc\uc0bf\uac12\uc744 \uad6c\ud560 \ub54c, \uc624\ucc28\uac00 \\(\\frac{3}{(n+1)!}\\)\ubcf4\ub2e4 \uc791\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

              \ud568\uc218\uc758 \uadf9\uac12\uacfc \ubcfc\ub85d\uc131<\/h3>\n

              \ud568\uc218 \\(f\\)\uac00 \uc810 \\(c\\)\uc5d0\uc11c \uad6d\uc18c\uadf9\ub313\uac12<\/span>(local maximum)\uc744 \uac00\uc9c4\ub2e4\ub294 \uac83\uc740, \uc5b4\ub5a4 \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(x\\in B(c,\\,\\delta )\\)\uc778 \ubaa8\ub4e0 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\leq f(c)\\)\uc778 \uac83\uc774\ub2e4. \uad6d\uc18c\uadf9\uc19f\uac12<\/span>(local minimum)\ub3c4 \ube44\uc2b7\ud558\uac8c \uc815\uc758\ub41c\ub2e4. \uad6d\uc18c\uadf9\ub313\uac12\uacfc \uad6d\uc18c\uadf9\uc19f\uac12\uc744 \uac04\ub2e8\ud788 \uadf9\ub313\uac12<\/span>\uacfc \uadf9\uc19f\uac12<\/span>\uc774\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4. \uadf9\ub313\uac12\uacfc \uadf9\uc19f\uac12\uc744 \ud1b5\ud2c0\uc5b4 \uad6d\uc18c\uadf9\uac12<\/span> \ub610\ub294 \uadf9\uac12<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

              \uadf9\uac12\uc744 \ud310\uc815\ud558\ub294 \ubc29\ubc95\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n

              \n

              \uc815\ub9ac 5.8. (\uadf9\uac12\uc5d0 \ub300\ud55c \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

              \ud568\uc218 \\(f\\)\uac00 \\(c\\)\ub97c \ub0b4\uc810\uc73c\ub85c \uac16\ub294 \uad6c\uac04\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f'(c) = 0\\)\uc774\uba70, \\(x\\)\uc758 \uac12\uc774 \\(c\\)\ub97c \ud1b5\uacfc\ud560 \ub54c \\(f'(x)\\)\uc758 \ubd80\ud638\uac00 \ubc14\ub00c\uba74, \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\uac12\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<\/div>\n

              \n

              \uc815\ub9ac 5.9. (\uadf9\uac12\uc5d0 \ub300\ud55c \uc774\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95)<\/span><\/p>\n

              \ud568\uc218 \\(f\\)\uac00 \\(c\\)\ub97c \ub0b4\uc810\uc73c\ub85c \uac16\ub294 \uad6c\uac04\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(c\\)\uc5d0\uc11c \ub450 \ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \\(f'(c) = 0\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

                \n
              1. \\(f”(c) > 0\\)\uc774\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n
              2. \\(f”(c) < 0\\)\uc774\uba74 \\(f\\)\ub294 \\(c\\)\uc5d0\uc11c \uadf9\ub313\uac12\uc744 \uac00\uc9c4\ub2e4.<\/li>\n
              3. \\(f”(c) = 0\\)\uc774\uba74 \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uadf9\uac12\uc744 \uac00\uc9c0\ub294\uc9c0 \uc5ec\ubd80\ub97c \ud310\uc815\ud560 \uc218 \uc5c6\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
                \n

                \ubb38\uc81c 5.10.<\/span>
                \n\uadf9\uac12\uc5d0 \ub300\ud55c \uc77c\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95(\uc815\ub9ac 5.8)\uacfc \uc774\uacc4\ub3c4\ud568\uc218 \ud310\uc815\ubc95(\uc815\ub9ac 5.9)\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                \n

                \ubb38\uc81c 5.11.<\/span>
                \n\ud568\uc218 \\(f(x) = x^3 – 3x + 1\\)\uc758 \uadf9\uac12\uc744 \uad6c\ud558\uace0, \uac01 \uadf9\uac12\uc774 \uadf9\ub313\uac12\uc778\uc9c0 \uadf9\uc19f\uac12\uc778\uc9c0 \ud310\uc815\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                \n

                \ubb38\uc81c 5.12.<\/span>
                \n\uc784\uc758\uc758 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(1+x \\le e^x\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. (\uc5ec\uae30\uc11c \\(e\\)\ub294 \uc790\uc5f0\uc0c1\uc218\uc774\ub2e4.)<\/p>\n<\/div>\n

                \n

                \ubb38\uc81c 5.13.<\/span>
                \n\ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n

                  \n
                1. \\(x \\ge -1\\)\uc774\uace0 \\(r\\ge 1\\)\uc77c \ub54c \\((1+x)^r \\ge 1+rx\\)\uc774\ub2e4.<\/li>\n
                2. \\(x\\ge -1\\)\uc774\uace0 \\(0\\le r\\le 1\\)\uc77c \ub54c \\((1+x)^r \\le 1+rx\\)\uc774\ub2e4.<\/li>\n<\/ol>\n

                  \uc774 \ubd80\ub4f1\uc2dd\uc744 \ubca0\ub974\ub204\uc774 \ubd80\ub4f1\uc2dd<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

                  \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d<\/span>\ud558\ub2e4(convex)\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(x,\\, y \\in I\\)\uc640 \\(t \\in [0,\\, 1]\\)\uc5d0 \ub300\ud574 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.
                  \n\\[f(tx + (1-t)y) \\leq tf(x) + (1-t)f(y).\\tag{5.1}\\]
                  \n\ub9cc\uc57d \ubd80\ub4f1\ud638\ub97c \ubc18\ub300\ub85c \ubc14\uafbc \uc870\uac74\uc774 \uc131\ub9bd\ud558\uba74 “\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc624\ubaa9<\/span>\ud558\ub2e4(concave)”\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

                  \ubd80\ub4f1\uc2dd (5.1)\uc740 \\(I\\)\uc758 \uc784\uc758\uc758 \uc11c\ub85c \ub2e4\ub978 \ub450 \uc810 \\(x\\), \\(y\\)\uc5d0 \ub300\ud558\uc5ec, \\(f\\)\uc758 \uadf8\ub798\ud504 \uc704\uc758 \ub450 \uc810 \\((x,\\,f(x))\\)\uc640 \\((y,\\,f(y))\\)\uc744 \uc774\uc740 \uc120\ubd84\uc774 \\(f\\)\uc758 \uadf8\ub798\ud504\ubcf4\ub2e4 \uc544\ub798\ucabd\uc5d0 \uc788\uc9c0 \uc54a\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n

                  \n

                  \uc815\ub9ac 5.10.<\/span><\/p>\n

                  \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ub450 \ubc88 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790.<\/p>\n

                    \n
                  1. \ub9cc\uc57d \\(I\\)\uc5d0\uc11c \\(f” \\geq 0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud558\ub2e4.<\/li>\n
                  2. \ub9cc\uc57d \\(I\\)\uc5d0\uc11c \\(f” \\leq 0\\)\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc624\ubaa9\ud558\ub2e4.<\/li>\n<\/ol>\n<\/div>\n
                    \n

                    \uc99d\uba85<\/p>\n

                    (1)\ub9cc \uc99d\uba85\ud558\uba74 \ucda9\ubd84\ud558\ub2e4. ((2)\ub294 (1)\uc5d0\uc11c \\(f\\)\ub97c \\(-f\\)\ub85c \ubc14\uafb8\uba74 \ub41c\ub2e4.)<\/p>\n

                    \\(I\\)\uc5d0\uc11c \\(f” \\geq 0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(x < y\\)\uc778 \uc784\uc758\uc758 \\(x,\\, y \\in I\\)\uc640 \\(t \\in (0,\\, 1)\\)\uc5d0 \ub300\ud574\n\\[z = tx + (1-t)y\\]\n\ub77c\uace0 \ub193\uc790. \uadf8\ub7ec\uba74 \\(x < z < y\\)\uc774\ub2e4. \ud14c\uc77c\ub7ec \uc815\ub9ac(\uc815\ub9ac 5.7)\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc810 \\(z\\)\uc5d0\uc11c \ud14c\uc77c\ub7ec 1\ucc28 \ub2e4\ud56d\uc2dd\uc744 \uad6c\ud558\uba74, \ub2e4\uc74c\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub450 \uc810 \\(c_1 \\in (x,\\, z)\\)\uc640 \\(c_2 \\in (z,\\, y)\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.\n\\[\\begin{aligned}\nf(x) &= f(z) + f'(z)(x - z) + \\frac{f''(c_1)}{2}(x - z)^2 ,\\\\[6pt]\nf(y) &= f(z) + f'(z)(y - z) + \\frac{f''(c_2)}{2}(y - z)^2 .\n\\end{aligned}\\]\n\\(f'' \\geq 0\\)\uc774\ubbc0\ub85c \\(f''(c_1) \\geq 0\\)\uc774\uace0 \\(f''(c_2) \\geq 0\\)\uc774\ub2e4. \ub530\ub77c\uc11c\n\\[\\begin{aligned}\nf(x) &\\geq f(z) + f'(z)(x - z),\\\\[6pt]\nf(y) &\\geq f(z) + f'(z)(y - z)\n\\end{aligned}\\]\n\uc774\ub2e4. \ub450 \ubd80\ub4f1\uc2dd\uc5d0 \uac01\uac01 \\(t\\)\uc640 \\((1-t)\\)\ub97c \uacf1\ud558\uc5ec \ub354\ud558\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.\n\\[\\begin{aligned}\ntf(x) + (1-t)f(y) &\\geq f(z) + f'(z)[t(x - z) + (1-t)(y - z)] \\\\[6pt]\n&= f(z) + f'(z)[tx + (1-t)y - z] \\\\[6pt]\n&= f(z) + f'(z) \\cdot 0 \\\\[6pt]\n&= f(z) = f(tx + (1-t)y).\n\\end{aligned}\\]\n\ub530\ub77c\uc11c \\(f\\)\ub294 \ubcfc\ub85d\ud558\ub2e4.<\/span><\/p>\n<\/div>\n

                    \n

                    \ubb38\uc81c 5.14.<\/span>
                    \n\\(I\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc5f4\ub9b0\uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud55c \ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

                    \n

                    \ubb38\uc81c 5.15.<\/span>
                    \n\ud568\uc218 \\(f\\)\uac00 \uc5f4\ub9b0\uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud568\uc218\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(f’\\)\uc774 \\(I\\)\uc5d0\uc11c \ub2e8\uc870\uc99d\uac00\ud568\uc218\uc778 \uac83\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

                    \n

                    \uc815\ub9ac 5.11. (\uc60c\uc13c \ubd80\ub4f1\uc2dd)<\/span><\/p>\n

                    \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubcfc\ub85d\ud55c \ud568\uc218\uc774\uace0 \\(i=1,\\,2,\\,\\cdots,\\,n\\)\uc5d0 \ub300\ud558\uc5ec \\(x_i\\in I\\), \\(\\lambda_i \\geq 0\\)\uc774\uba70 \\(\\sum_{i=1}^{n} \\lambda_i = 1\\)\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
                    \n\\[f\\left(\\sum_{i=1}^{n} \\lambda_i x_i\\right) \\leq \\sum_{i=1}^{n} \\lambda_i f(x_i) .\\]<\/p>\n<\/div>\n

                    \n

                    \uc99d\uba85<\/p>\n

                    \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc73c\ub85c \uc99d\uba85\ud558\uc790. \uc6b0\uc120 \\(n = 2\\)\uc77c \ub54c\ub294 \ubcfc\ub85d\ud568\uc218\uc758 \uc815\uc758\uc5d0 \uc758\ud574 \uc790\uba85\ud558\ub2e4.<\/p>\n

                    \ub2e4\uc74c\uc73c\ub85c \\(n = k\\)\uc77c \ub54c \uc131\ub9bd\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uace0, \\(n = k+1\\)\uc77c \ub54c\ub97c \ubcf4\uc774\uc790.<\/p>\n

                    \\(\\lambda_1,\\, \\ldots,\\, \\lambda_{k+1} \\geq 0\\)\uc774\uace0 \\(\\sum_{i=1}^{k+1} \\lambda_i = 1\\)\uc774\ub77c\uace0 \ud558\uc790. \\(\\lambda_{k+1} = 1\\)\uc778 \uacbd\uc6b0\ub294 \uc790\uba85\ud558\ubbc0\ub85c, \\(\\lambda_{k+1} < 1\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \\(\\mu_i = \\frac{\\lambda_i}{1 - \\lambda_{k+1}}\\) (\\(i = 1,\\, \\ldots,\\, k\\))\ub77c\uace0 \uc815\uc758\ud558\uba74, \\(\\sum_{i=1}^{k} \\mu_i = 1\\)\uc774\ub2e4. \uadc0\ub0a9\uc801 \uac00\uc815\uc5d0 \uc758\ud574 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[f\\left(\\sum_{i=1}^{k} \\mu_i x_i\\right) \\leq \\sum_{i=1}^{k} \\mu_i f(x_i).\\]\n\uc774\uc81c \\(y = \\sum_{i=1}^{k} \\mu_i x_i\\)\ub77c\uace0 \ub193\uc73c\uba74 \ub2e4\uc74c\uacfc \uac19\uc740 \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.\n\\[\\begin{aligned}\n\\sum_{i=1}^{k+1} \\lambda_i x_i &= \\sum_{i=1}^{k} \\lambda_i x_i + \\lambda_{k+1} x_{k+1} \\\\[6pt]\n&= (1 - \\lambda_{k+1}) \\sum_{i=1}^{k} \\mu_i x_i + \\lambda_{k+1} x_{k+1} \\\\[6pt]\n&= (1 - \\lambda_{k+1}) y + \\lambda_{k+1} x_{k+1}.\n\\end{aligned}\\]\n\ubcfc\ub85d\ud568\uc218\uc758 \uc815\uc758\uc5d0 \uc758\ud574 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.\n\\[\\begin{aligned}\nf\\left(\\sum_{i=1}^{k+1} \\lambda_i x_i\\right) &= f((1 - \\lambda_{k+1}) y + \\lambda_{k+1} x_{k+1}) \\\\[6pt]\n&\\leq (1 - \\lambda_{k+1}) f(y) + \\lambda_{k+1} f(x_{k+1}) \\\\[6pt]\n&\\leq (1 - \\lambda_{k+1}) \\sum_{i=1}^{k} \\mu_i f(x_i) + \\lambda_{k+1} f(x_{k+1}) \\\\[6pt]\n&= \\sum_{i=1}^{k} \\lambda_i f(x_i) + \\lambda_{k+1} f(x_{k+1})\n= \\sum_{i=1}^{k+1} \\lambda_i f(x_i).\n\\end{aligned}\\]\n\uadf8\ub7ec\ubbc0\ub85c \uc218\ud559\uc801 \uadc0\ub0a9\ubc95\uc5d0 \uc758\ud558\uc5ec \ubc14\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

                    \ud568\uc218 \\(f\\)\uac00 \\(c\\)\ub97c \ub0b4\uc810\uc73c\ub85c \uac16\ub294 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(f”(c) = 0\\)\uc774\uace0 \\(x\\)\uac00 \\(c\\)\ub97c \ud1b5\uacfc\ud560 \ub54c \\(f”(x)\\)\uc758 \ubd80\ud638\uac00 \ubc14\ub00c\uba74, \\((c,\\,f(c))\\)\ub97c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \ubcc0\uace1\uc810<\/span>(inflection point)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

                    \ub3c5\ud2b9\ud55c \uc815\ub9ac\ub4e4<\/h3>\n

                    \ud568\uc218\uac00 \uad6c\uac04\uc5d0\uc11c \uc77c\ub300\uc77c\ub300\uc751\uc774\uace0 \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c \uadf8 \uc5ed\ud568\uc218\uc758 \ubbf8\ubd84 \uac00\ub2a5\uc131\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4.<\/p>\n

                    \n

                    \uc815\ub9ac 5.12. (\uc5ed\ud568\uc218 \uc815\ub9ac (1\ucc28\uc6d0))<\/span><\/p>\n

                    \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0, \uc21c\uc99d\uac00\ud558\uac70\ub098 \uc21c\uac10\uc18c\ud558\uba74, \uc5ed\ud568\uc218 \\(f^{-1}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \ub610\ud55c \\(c\\in I\\)\uc774\uace0 \\(f'(c) \\neq 0\\)\uc774\uba74
                    \n\\[(f^{-1})'(f(c)) = \\frac{1}{f'(c)}\\]
                    \n\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n

                    \n

                    \uc99d\uba85<\/p>\n

                    \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(f\\)\uac00 \uc21c\uc99d\uac00\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790.<\/p>\n

                    \\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uc21c\uc99d\uac00\ud558\ubbc0\ub85c, \\(f: I \\to f(I)\\)\ub294 \uc77c\ub300\uc77c\ub300\uc751\uc774\ub2e4. \uc0ac\uc787\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \\(f(I)\\)\ub294 \uad6c\uac04\uc774\uace0, \ub530\ub77c\uc11c \uc5ed\ud568\uc218 \\(f^{-1}: f(I) \\to I\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n

                    \\(c\\in K\\subseteq I\\)\uc778 \ucef4\ud329\ud2b8 \uadfc\ubc29 \\(K\\)\ub97c \ud0dd\ud558\uba74 \\(f(K)\\)\ub294 \\(f(c)\\)\uc758 \ucef4\ud329\ud2b8 \uadfc\ubc29\uc774\uba70, \\(f\\)\ub294 \\(K\\)\uc5d0\uc11c \\(f(K)\\)\ub85c\uc758 \uc77c\ub300\uc77c\ub300\uc751\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f^{-1}\\)\ub294 \\(f(K)\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n

                    \uc774\uc81c \uc5ed\ud568\uc218\uc758 \ubbf8\ubd84 \uac00\ub2a5\uc131\uc744 \uc99d\uba85\ud558\uc790. \\(f'(c) \\neq 0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(d = f(c)\\)\ub77c\uace0 \ub193\uace0, \\(k \\to 0\\)\uc77c \ub54c
                    \n\\[\\frac{f^{-1}(d + k) – f^{-1}(d)}{k}\\]
                    \n\uc758 \uadf9\ud55c\uc744 \uad6c\ud558\uc790. \\(h = f^{-1}(d + k) – f^{-1}(d)\\)\ub77c\uace0 \ub193\uc73c\uba74, \\(f^{-1}\\)\uc758 \uc5f0\uc18d\uc131\uc5d0 \uc758\ud574 \\(k \\to 0\\)\uc77c \ub54c \\(h \\to 0\\)\uc774\ub2e4. \ub610\ud55c
                    \n\\[f(c + h) = f(f^{-1}(d + k)) = d + k = f(c) + k\\]
                    \n\uc774\ub2e4. \ub530\ub77c\uc11c \\(k = f(c + h) – f(c)\\)\uc774\uace0,
                    \n\\[\\frac{f^{-1}(d + k) – f^{-1}(d)}{k} = \\frac{h}{f(c + h) – f(c)} = \\frac{1}{\\frac{f(c + h) – f(c)}{h}}\\]
                    \n\uc774\ub2e4. \\(h \\to 0\\)\uc77c \ub54c \uc704 \uc2dd\uc758 \ub9c8\uc9c0\ub9c9 \uc2dd\uc774 \\(\\frac{1}{f'(c)}\\)\ub85c \uc218\ub834\ud558\ubbc0\ub85c
                    \n\\[(f^{-1})'(f(c)) = \\frac{1}{f'(c)}\\]
                    \n\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

                    \n

                    \ubb38\uc81c 5.16.<\/span>
                    \n\\(r\\)\uc774 \\(0\\)\uc544\ub2cc \uc720\ub9ac\uc218\uc774\uace0 \\(x>0\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x)=x^r\\)\uc774\ub77c\uace0 \ud560 \ub54c \\(f'(x)=rx^{r-1}\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

                    \n

                    \ubb38\uc81c 5.17.<\/span>
                    \n\ubbf8\ubd84\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218 \\(f\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n

                      \n
                    1. \\(f(x)=e^x\\)<\/li>\n
                    2. \\(f(x)=\\ln x\\) (\ub2e8, \\(x>0\\).)<\/li>\n
                    3. \\(f(x)=a^x\\) (\ub2e8, \\(a>0\\), \\(a\\ne 1\\).)<\/li>\n
                    4. \\(f(x)=\\log_a x\\) (\ub2e8, \\(a>0\\), \\(a\\ne 1\\), \\(x>0\\).)<\/li>\n<\/ol>\n<\/div>\n
                      \n

                      \ubb38\uc81c 5.18.<\/span>
                      \n\\(a > 0\\), \\(a\\ne 1\\)\uc774\uace0 \\(x\\)\uac00 \uc2e4\uc218\uc77c \ub54c, \\(a^x = e^{x\\ln a}\\)\uac00 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

                      \n

                      \ubb38\uc81c 5.19.<\/span>
                      \n\\(\\alpha\\)\uac00 \ubb34\ub9ac\uc218\uc774\uace0 \\(x>0\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x)=x^\\alpha\\)\uc774\ub77c\uace0 \ud560 \ub54c \\(f'(x) = \\alpha x^{\\alpha -1}\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

                      \n

                      \ubb38\uc81c 5.20.<\/span>
                      \n\uc5ed\ud568\uc218 \uc815\ub9ac(\uc815\ub9ac 5.12)\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n

                        \n
                      1. \\(f(x)=\\sin^{-1} x\\) (\ub2e8, \\(-1 < x < 1\\), \\(-\\frac{\\pi}{2} < y < \\frac{\\pi}{2}\\).)<\/li>\n
                      2. \\(f(x)=\\cos^{-1} x\\) (\ub2e8, \\(-1 < x < 1\\), \\(0 < y < \\pi\\).)<\/li>\n
                      3. \\(f(x)=\\tan^{-1} x\\) (\ub2e8, \\(x\\in\\mathbb{R}\\), \\(-\\frac{\\pi}{2} < y < \\frac{\\pi}{2}\\).)<\/li>\n<\/ol>\n<\/div>\n
                        \n

                        \ubb38\uc81c 5.21.<\/span>
                        \n\ud568\uc218 \\(f(x) = \\sum_{n=0}^{\\infty} \\frac{\\cos(3^n x)}{2^n}\\)\uac00 \uc5f0\uc18d\uc774\uc9c0\ub9cc \uc5b4\ub5a4 \uc810\uc5d0\uc11c\ub3c4 \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624. (\uc774 \ud568\uc218\ub97c \ubc14\uc774\uc5b4\uc288\ud2b8\ub77c\uc2a4 \ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.)<\/p>\n<\/div>\n

                        \ud568\uc218 \\(f\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub354\ub77c\ub3c4 \\(f’\\)\uc740 \uc5f0\uc18d\uc774 \uc544\ub2d0 \uc218 \uc788\ub2e4. \uadf8\ub7fc\uc5d0\ub3c4 \ubd88\uad6c\ud558\uace0 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc0ac\uc787\uac12 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

                        \n

                        \uc815\ub9ac 5.13. (\ub2e4\ub974\ubd80\uc758 \uc815\ub9ac)<\/span><\/p>\n

                        \\(f\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f'(a) \\neq f'(b)\\)\uc774\uba74, \\(f'(a)\\)\uc640 \\(f'(b)\\) \uc0ac\uc774\uc758 \ubaa8\ub4e0 \uac12\uc774 \\(f’\\)\uc758 \uce58\uc5ed\uc5d0 \ud3ec\ud568\ub41c\ub2e4.<\/p>\n<\/div>\n

                        \n

                        \uc99d\uba85<\/p>\n

                        \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(f'(a) < k < f'(b)\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

                        \\(g(x) = f(x) – kx\\)\ub77c\uace0 \ud558\uc790. \\(g'(x) = f'(x) – k\\)\uc774\ubbc0\ub85c
                        \n\\[g'(a) = f'(a) – k < 0, \\quad g'(b) = f'(b) - k > 0\\]
                        \n\uc774\ub2e4. \\(g'(a) < 0\\)\uc774\ubbc0\ub85c, \ucda9\ubd84\ud788 \uc791\uc740 \\(h > 0\\)\uc5d0 \ub300\ud574
                        \n\\[\\frac{g(a + h) – g(a)}{h} < 0\\]\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(g(a + h) < g(a)\\)\uc774\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(g'(b) > 0\\)\uc774\ubbc0\ub85c \ucda9\ubd84\ud788 \uc791\uc740 \\(h > 0\\)\uc5d0 \ub300\ud574 \\(g(b – h) < g(b)\\)\uc774\ub2e4. \\(g\\)\uac00 \ucef4\ud329\ud2b8 \uc9d1\ud569 \\([a,\\, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \uc774 \uc9d1\ud569\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac00\uc9c4\ub2e4. \\(g\\)\ub294 \\(a\\)\ub098 \\(b\\)\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac16\uc9c0 \uc54a\uc73c\ubbc0\ub85c, \uc5b4\ub5a4 \\(c \\in (a,\\, b)\\)\uc5d0 \ub300\ud558\uc5ec \\(g(c)\\)\uac00 \ucd5c\uc19f\uac12\uc774 \ub41c\ub2e4. \ud398\ub974\ub9c8\uc758 \uc815\ub9ac(\uc815\ub9ac 5.2)\uc5d0 \uc758\ud574 \\(g'(c) = 0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(f'(c) = k\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

                        \ud568\uc218\uc758 \uadf9\ud55c\uc774 \ubd80\uc815\ud615\uc77c \ub54c, \ub3c4\ud568\uc218\ub97c \uc0ac\uc6a9\ud558\uc5ec \uadf9\ud55c\uc744 \uad6c\ud558\ub294 \uc720\uc6a9\ud55c \uacf5\uc2dd\uc774 \uc788\ub2e4.<\/p>\n

                        \n

                        \uc815\ub9ac 5.14. (\ub85c\ud53c\ud0c8\uc758 \uc815\ub9ac)<\/span><\/p>\n

                        \ud568\uc218 \\(f\\)\uac00 \\(a\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uad6c\uac04\uc5d0\uc11c \uc815\uc758\ub418\uc5b4 \uc788\uace0,
                        \n\\[\\lim_{x \\to a} f(x) = \\lim_{x \\to a} g(x) = 0\\tag{5.2}\\]
                        \n\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \uadf9\ud55c
                        \n\\[\\lim_{x \\to a} \\frac{f'(x)}{g'(x)}\\]
                        \n\uac00 \uc874\uc7ac\ud558\uba74, \\(x\\rightarrow a\\)\uc77c \ub54c \\(f(x)\/g(x)\\)\uc758 \uadf9\ud55c\ub3c4 \uc874\uc7ac\ud558\uace0, \uadf8 \uadf9\ud55c\uac12\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
                        \n\\[\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\lim_{x \\to a} \\frac{f'(x)}{g'(x)}\\tag{5.3}\\]
                        \n\uadf9\ud55c (5.2)\uc640 (5.3)\uc5d0\uc11c \uacf5\ud1b5\uc801\uc73c\ub85c \\(x\\rightarrow a\\)\ub97c \uc88c\uadf9\ud55c\uc774\ub098 \uc6b0\uadf9\ud55c\uc73c\ub85c \ubc14\uafb8\uac70\ub098, \ub610\ub294 \\(x\\rightarrow \\pm\\infty\\)\ub85c \ubc14\uafb8\uc5b4\ub3c4 \uc815\ub9ac\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n

                        \n

                        \uc99d\uba85 \uac1c\uc694<\/span>
                        \n\uc5ec\uae30\uc11c\ub294 \\(0\/0\\) \uaf34\uc758 \uacbd\uc6b0\uc5d0 \ub300\ud55c \uc99d\uba85\ub9cc \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n

                        \uc6b0\uc120 \\(f(a) = g(a) = 0\\)\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \\(a\\)\uc5d0\uc11c\uc758 \uadf9\ud55c\uac12\uc740 \ud568\uc22b\uac12\uc5d0 \uc601\ud5a5\uc744 \ubc1b\uc9c0 \uc54a\uc73c\ubbc0\ub85c, \uc774\uc640 \uac19\uc774 \ud568\uc22b\uac12\uc744 \uc7ac\uc815\uc758\ud558\uc5ec\ub3c4 \ub41c\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(x\\ne a\\)\uc77c \ub54c, \ucf54\uc2dc\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac\ub97c \uc801\uc6a9\ud558\uba74
                        \n\\[\\frac{f(x)}{g(x)}=\\frac{f(x) – f(a)}{g(x) – g(a)} = \\frac{f'(c_x)}{g'(c_x)}\\]
                        \n\uc778 \\(c_x\\)\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(x \\to a\\)\uc77c \ub54c \\(c_x \\to a\\)\uc774\ubbc0\ub85c \ubc14\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

                        \n

                        \ubb38\uc81c 5.22.<\/span>
                        \n\ub2e4\uc74c \uadf9\ud55c\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n

                          \n
                        1. \\(\\lim_{x\\rightarrow\\infty} \\frac{x^2}{e^x}\\)<\/li>\n
                        2. \\(\\lim_{x\\rightarrow 0+} x\\ln x\\)<\/li>\n
                        3. \\(\\lim_{x\\rightarrow 0+} (1+3x)^{\\frac{1}{x}}\\)<\/li>\n
                        4. \\(\\lim_{x\\rightarrow 1} (\\ln x)^{1-x}\\)<\/li>\n<\/ol>\n<\/div>\n
                          \n

                          \ubb38\uc81c 5.23.<\/span>
                          \n\ud568\uc218 \\(f\\)\uac00 \\(\\mathbb{R}\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(A\\)\uac00 \uc2e4\uc218\uc774\uba70 \\(\\lim_{x\\rightarrow\\infty}(f(x)+f'(x))=A\\)\uc774\uba74 \\(\\lim_{x\\rightarrow\\infty} f(x)=A\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                          \n

                          \ud574\uc11d\ud559 \ud575\uc2ec\uc815\ub9ac \ub178\ud2b8<\/a><\/p>\n

                            \n
                          1. \uc2e4\uc218\uacc4\uc758 \uc131\uc9c8<\/a><\/li>\n
                          2. \uac70\ub9ac\uacf5\uac04<\/a><\/li>\n
                          3. \uc218\uc5f4\uc758 \uadf9\ud55c<\/a><\/li>\n
                          4. \ud568\uc218\uc758 \uadf9\ud55c\uacfc \uc5f0\uc18d\uc131<\/a><\/li>\n
                          5. \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n
                          6. \uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84<\/a><\/li>\n
                          7. \ubb34\ud55c\uae09\uc218<\/a><\/li>\n
                          8. \uc2e4\ud574\uc11d\uc801 \ud568\uc218<\/a><\/li>\n
                          9. \ub2e4\ubcc0\uc218 \ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li>\n
                          10. \uc911\uc801\ubd84<\/a><\/li>\n
                          11. \ubca1\ud130\uc7a5\uacfc \uc801\ubd84 \uc815\ub9ac<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                            <\/p>\n","protected":false},"excerpt":{"rendered":"

                            \uc774 \uc7a5\uc5d0\uc11c\ub294 \uc2e4\ud568\uc218\uc758 \ubbf8\ubd84 \uac00\ub2a5\uc131\uacfc \uad00\ub828\ub41c \uc911\uc694\ud55c \uc815\ub9ac\ub4e4\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ubbf8\ubd84\uc758 \uae30\ud558\ud559\uc801 \uc758\ubbf8\ub85c\ubd80\ud130 \uc2dc\uc791\ud558\uc5ec \ud3c9\uade0\uac12 \uc815\ub9ac, \ud14c\uc77c\ub7ec \uc815\ub9ac \ub4f1 \ud574\uc11d\ud559\uc758 \uc8fc\uc694 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. \ubbf8\ubd84 \uac00\ub2a5\uc131 \\(X\\)\uac00 \\(\\mathbb{R}\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc774\uace0 \\(c\\in X\\cap X’\\)\uc774\ub77c\uace0 \ud558\uc790. \ud568\uc218 \\(f: X \\to \\mathbb{R}\\)\uc774 \uc810 \\(c\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4(differentiable)\ub294 \uac83\uc740 \ub2e4\uc74c \uadf9\ud55c\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \\(f'(c) = \\lim_{h \\to 0} \\frac{f(c + h) – f(c)}{h}.\\) \uc774 \uadf9\ud55c\uac12 \\(f'(c)\\)\ub97c \\(c\\)\uc5d0\uc11c \\(f\\)\uc758 \ubbf8\ubd84\uacc4\uc218(derivative)\ub77c\uace0 \ubd80\ub978\ub2e4. \ud568\uc218…<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":9470,"menu_order":105,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"class_list":["post-9486","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9486","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=9486"}],"version-history":[{"count":11,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9486\/revisions"}],"predecessor-version":[{"id":9610,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9486\/revisions\/9610"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/9470"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9486"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}