{"id":1958,"date":"2019-03-17T12:06:36","date_gmt":"2019-03-17T03:06:36","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1958"},"modified":"2019-09-05T19:50:24","modified_gmt":"2019-09-05T10:50:24","slug":"calculus-linearization-and-differentials","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-linearization-and-differentials\/","title":{"rendered":"\uc77c\ucc28\uadfc\uc0ac\uc640 \ubbf8\ubd84\uc18c"},"content":{"rendered":"

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc77c\ucc28\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc758 \uadfc\uc0ac\ud568\uc218\ub97c \ub9cc\ub4dc\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ub610\ud55c \ubbf8\ubd84\uc18c\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uace0 \ubcc0\ud654\ub7c9\uc758 \uadfc\uc0bf\uac12\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

<\/p>\n

\ud560\uc120, \uc811\uc120, \ubc95\uc120<\/h3>\n

\ud568\uc218 \\(y=f(x)\\)\uac00 \uc11c\ub85c \ub2e4\ub978 \ub450 \uc810 \\(a,\\) \\(b\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uad6c\uac04\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub450 \uc810 \\((a,\\,f(a)),\\) \\((b,\\,f(b))\\)\ub97c \ubaa8\ub450 \uc9c0\ub098\ub294 \uc9c1\uc120\uc744 \ud568\uc218 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \ud560\uc120<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \\(f\\)\uc758 \uadf8\ub798\ud504 \uc704\uc5d0\uc11c \uc810 \\((b,\\,f(b))\\)\uac00 \uc810 \\((a,\\,f(a))\\)\uc5d0 \uac00\uae4c\uc774 \ub2e4\uac00\uac10\uc5d0 \ub530\ub77c \uc774 \ub450 \uc810\uc744 \uc9c0\ub098\ub294 \ud560\uc120\uc774 \ud558\ub098\uc758 \uc9c1\uc120 \\(L\\)\uc5d0 \uac00\uae4c\uc774 \ub2e4\uac00\uac00\uba74, \uc774 \uc9c1\uc120 \\(L\\)\uc744 \\((a,\\,f(a))\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc811\uc120<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\ubbf8\ubd84\uc744 \uc774\uc6a9\ud558\uc5ec \uadf8\ub798\ud504\uc758 \uc811\uc120\uc744 \uc815\ud655\ud558\uac8c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \ud568\uc218 \\(f\\)\uac00 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ubc29\uc815\uc2dd
\n\\[y= f ‘ (a) (x-a) + f(a)\\tag{1}\\]
\n\uc73c\ub85c \ud45c\ud604\ub418\ub294 \uc9c1\uc120\uc744 \uc810 \\((a,\\,f(a))\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \uc811\uc120<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \uc810 \\((a,\\,f(a))\\)\ub97c \uc9c0\ub098\uace0 \uc9c1\uc120 (1)\uacfc \uc218\uc9c1\uc778 \uc9c1\uc120\uc744 \uc810 \\((a,\\,f(a))\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \ubc95\uc120<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(f ‘ (a) \\ne 0\\)\uc77c \ub54c \uc810 \\((a,\\,f(a))\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \ubc95\uc120\uc758 \ubc29\uc815\uc2dd\uc740
\n\\[y= -\\frac{1}{f ‘ (a)} (x-a) + f(a)\\]
\n\uc774\uba70, \\(f ‘ (a)=0\\)\uc77c \ub54c \uc810 \\((a,\\,f(a))\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \ubc95\uc120\uc758 \ubc29\uc815\uc2dd\uc740 \\(x=a\\)\uc774\ub2e4.<\/p>\n

<\/p>\n

\uc77c\ucc28\uadfc\uc0ac<\/h3>\n

\ud568\uc218 \\(f\\)\uac00 \\(a\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \uc810 \\((a,\\,f(a))\\)\uc758 \uadfc\ucc98\uc5d0\uc11c \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\ub294 \uc774 \uc810\uc5d0\uc11c \uadf8\ub798\ud504\uc758 \uc811\uc120\uc5d0 \uac00\uae5d\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc73c\ub85c \ud45c\ud604\ub418\ub294 \ud568\uc218
\n\\[L (x) = f(a) + f ‘ (a)(x-a)\\tag{2}\\]
\n\ub294 \\(a\\) \uadfc\ucc98\uc5d0\uc11c \\(f(x)\\)\uc758 \uac12\uc5d0 \uac00\uae4c\uc6b4 \uac12\uc744 \uac16\ub294 \ud568\uc218, \uc989 \uadfc\uc0ac\ud568\uc218<\/span>(approximation)\uac00 \ub41c\ub2e4. \ud2b9\ud788 (2)\ub294 \uc77c\ucc28\ud568\uc218\uc774\uae30 \ub54c\ubb38\uc5d0, (2)\uc640 \uac19\uc740 \uadfc\uc0ac\ud568\uc218\ub97c \\(a\\)\uc5d0\uc11c \\(f\\)\uc758 \ud45c\uc900\uc77c\ucc28\uadfc\uc0ac\ud568\uc218<\/span>(standard linear approximation)\ub77c\uace0 \ubd80\ub974\uba70, \uc810 \\(a\\)\ub97c \uc774 \uadfc\uc0ac\ud568\uc218\uc758 \uc911\uc2ec<\/span>(center)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ucc45\uc5d0 \ub530\ub77c\uc11c\ub294 \uc77c\ucc28\uadfc\uc0ac\ub97c \uc120\ud615\uadfc\uc0ac<\/span>\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n

\n

\uc608\uc81c 1.<\/span>
\n\uc810 \\(x=2\\)\uc5d0\uc11c \ud568\uc218 \\(y=\\sqrt{2+x}\\)\uc758 \uc77c\ucc28\uadfc\uc0ac\ud568\uc218\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n

\ud480\uc774.<\/span>
\n\\[y ‘ = \\frac{1}{2\\sqrt{2+x}}\\]
\n\uc774\ubbc0\ub85c \\(2\\)\uc5d0\uc11c \uc774 \ud568\uc218\uc758 \ubbf8\ubd84\uacc4\uc218\ub294 \\(\\frac{1}{4}\\)\uc774\uba70, \uc77c\ucc28\uadfc\uc0ac\ud568\uc218\ub294
\n\\[L(x) = 2+ \\frac{1}{4} (x-2) = \\frac{1}{4} x + \\frac{3}{2}\\]
\n\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\n

\uc608\uc81c 2.<\/span>
\n\ud568\uc218 \\(f(x)=\\sqrt{2+x}\\)\uc758 \uc77c\ucc28\uadfc\uc0ac\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \\(\\sqrt{4.1}\\)\uc758 \uadfc\uc0bf\uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n

\ud480\uc774.<\/span>
\n\\(\\sqrt{4.1} = f(2.1)\\)\uc774\ub2e4. \uc5ec\uae30\uc11c \\(f(2)\\)\uc758 \uac12\uc740 \uc27d\uac8c \uad6c\ud560 \uc218 \uc788\uc73c\ubbc0\ub85c \\(2\\)\ub97c \uc911\uc2ec\uc73c\ub85c \ud558\ub294 \\(f(x)\\)\uc758 \uc77c\ucc28\uadfc\uc0ac\ud568\uc218 \\(L(x)\\)\ub97c \uad6c\ud55c \ub4a4 \uc774 \ud568\uc218\uc5d0 \\(x=2.1\\)\uc744 \ub300\uc785\ud558\uba74 \ub41c\ub2e4. \uadf8\ub7f0\ub370 \uc608\uc81c 1\uc5d0\uc11c
\n\\[L(x) = \\frac{1}{4}x+ \\frac{3}{2}\\]
\n\uc774\ubbc0\ub85c
\n\\[\\sqrt{4.1} = f(2.1) \\approx L(2.1) = 2.025\\]
\n\uc774\ub2e4. \ucc38\uace0\ub85c \\(\\sqrt{4.1}\\)\uc758 \ucc38\uac12\uc740
\n\\[\\sqrt{4.1} = 2.024845673131659\\cdots \\]
\n\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\uc77c\ucc28\uadfc\uc0ac\ub294 \ubb3c\ub9ac\ud559\uc774\ub098 \uacf5\ud559\uc5d0\uc11c \uc2e4\uc81c\ub85c \uc790\uc8fc \uc0ac\uc6a9\ub41c\ub2e4. \uadf8 \uc911 \uac00\uc7a5 \ub9ce\uc774 \uc0ac\uc6a9\ub418\ub294 \uc77c\ucc28\uadfc\uc0ac\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

\n

\n\\(k\\)\uac00 \uc2e4\uc218\uc77c \ub54c \\(0\\)\uc5d0 \uac00\uae4c\uc6b4 \\(x\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[(1+x)^k \\,\\approx\\, 1+kx.\\tag{3}\\]\n<\/p>\n<\/div>\n

\uc608\ucee8\ub300 \\(x\\)\uac00 \\(0\\)\uc5d0 \uac00\uae4c\uc6b8 \ub54c \ub2e4\uc74c\uacfc \uac19\uc740 \uadfc\uc0ac\uc2dd\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4.
\n\\[\\begin{align}
\n\\sqrt{1+x} \\,&\\approx\\, 1+\\frac{1}{2}x ,\\\\[6pt]
\n\\frac{1}{1-x} \\,&\\approx\\, 1+x ,\\\\[6pt]
\n\\frac{1}{\\sqrt{1-x^2}} \\,&\\approx\\, 1+ \\frac{1}{2} x^2 .
\n\\end{align}\\]\n<\/p>\n

\n

\ubcf4\uae30 3.<\/span>
\n\uc544\uc778\uc288\ud0c0\uc778\uc758 \uc0c1\ub300\uc131 \uc774\ub860\uc5d0 \uc758\ud558\uba74, \uc815\uc9c0\uc0c1\ud0dc\uc5d0\uc11c \uc9c8\ub7c9\uc774 \\(m_0\\)\uc778 \ubb3c\uccb4\uac00 \\(v\\)\uc758 \uc18d\ub825\uc73c\ub85c \uc6c0\uc9c1\uc77c \ub54c, \uc774 \ubb3c\uccb4\uc758 \uc9c8\ub7c9 \\(m\\)\uc740
\n\\[m = \\frac{m_0}{\\sqrt{1-v^2 \/ c^2}}\\tag{4}\\]
\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(c\\)\ub294 \ube5b\uc758 \uc18d\ub825\uc73c\ub85c\uc11c \uc57d 299792458 m\/s\uc774\ub2e4. \uc77c\ucc28\uadfc\uc0ac\uc2dd (3)\uc5d0 \\(x= -v^2\/c^2 ,\\) \\(k=-1\/2\\)\uc744 \ub300\uc785\ud558\uba74
\n\\[\\frac{1}{\\sqrt{1-v^2\/c^2}} \\,\\approx\\, 1+ \\frac{1}{2} \\cdot \\frac{v^2}{c^2}\\]
\n\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubb3c\uccb4\uc758 \uc18d\ub825 \\(v\\)\uac00 \\(0\\)\uc5d0 \uac00\uae4c\uc6b8 \ub54c(\uc77c\uc0c1\uc801\uc778 \uc0c1\ud669\uc5d0\uc11c \ub208\uc73c\ub85c \uad00\ucc30\ud560 \uc218 \uc788\ub294 \uc815\ub3c4\uc758 \ube60\ub974\uae30\uc77c \ub54c) \uc9c8\ub7c9 (4)\uc758 \uadfc\uc0bf\uac12\uc740
\n\\[m = \\frac{m_0}{\\sqrt{1-v^2\/c^2}} \\,\\approx\\, m_0 + \\frac{1}{2}m_0 \\frac{v^2}{c^2}\\]
\n\uc774 \ub41c\ub2e4. \uc774 \uc2dd\uc744 \ubcc0\ud615\ud558\uba74
\n\\[(m-m_0 )c^2 \\,\\approx\\, \\frac{1}{2}m_0 v^2\\tag{5}\\]
\n\uc778\ub370, \uc774 \uc2dd\uc758 \uc6b0\ubcc0\uc740
\n\\[\\frac{1}{2}m_0 v^2 = \\frac{1}{2}m_0 v^2 – \\frac{1}{2} m_0 (0)^2 = \\Delta (\\text{KE})\\]
\n\uc774\ubbc0\ub85c (5)\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc2dd\uc774 \ub41c\ub2e4.
\n\\[(\\Delta m)c^2 \\,\\approx\\, \\Delta (\\text{KE}).\\]
\n\uc5ec\uae30\uc11c \\((\\text{KE})\\)\ub294 \uc6b4\ub3d9\uc5d0\ub108\uc9c0(kinetic energy)\ub97c \uc758\ubbf8\ud55c\ub2e4.
\n\uc989 \ubb3c\uccb4\uac00 \uc815\uc9c0\uc0c1\ud0dc\uc5d0\uc11c \uc18d\ub825\uc774 \\(v\\)\ub85c \ubcc0\ud558\ub294 \ub3d9\uc548 \uc6b4\ub3d9\uc5d0\ub108\uc9c0\uc758 \ubcc0\ud654\ub7c9 \\(\\Delta (\\text{KE})\\)\uc758 \uadfc\uc0bf\uac12\uc740 \\((\\Delta m )c^2\\)\uc774\ub2e4.
\n\\(c \\approx \\)\\(3.0 \\times 10^8 \\text{ m\/s}\\)\uc784\uc744 \uc0dd\uac01\ud558\uba74 \uc791\uc740 \uc9c8\ub7c9\uc758 \ubcc0\ud654\ub294 \uc5c4\uccad\ub09c \uc6b4\ub3d9\uc5d0\ub108\uc9c0\uc758 \ubcc0\ud654\uc5d0 \ub9de\uba39\ub294\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc54c \uc218 \uc788\ub2e4.<\/p>\n

(Thomas’ Calculus International Edition 13\ud310 3.11\uc808\uc5d0\uc11c \ubc1c\ucdcc\ud568.)
\n<\/span><\/p>\n<\/div>\n

<\/p>\n

\ubbf8\ubd84\uc18c<\/h3>\n

\ud568\uc218 \\(y=f(x)\\)\uac00 \uc5f4\ub9b0 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uac01 \uc810 \\(x\\in I\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf8\ub798\ud504 \uc704\uc758 \uc810 \\(\\mathrm{P}(x,\\,f(x))\\)\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\uc73c\uba70, \uc810 \\(\\mathrm{P}\\)\uc5d0\uc11c \\(f\\)\uc758 \uadf8\ub798\ud504\uc5d0 \uc811\ud558\ub294 \uc9c1\uc120 \\(L\\)\uc744 \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \uc774 \uc9c1\uc120 \\(L\\) \uc704\uc758 \uc810 \\(\\mathrm{v}\\)\ub97c, \uc2dc\uc810\uc774 \\(\\mathrm{P}\\)\uc774\uace0 \uc885\uc810\uc774 \\(\\mathrm{v}\\)\uc778 \ubca1\ud130 \\(\\vec{v}\\)\uc640 \ub3d9\uc77c\uc2dc\ud558\uc790. \uc774\ub7ec\ud55c \ubca1\ud130\ub4e4\uc744 \ubaa8\uc740 \uc9d1\ud569\uc740 1\ucc28\uc6d0 \ubca1\ud130\uacf5\uac04\uc774 \ub41c\ub2e4. \uc774 \ubca1\ud130\uacf5\uac04\uc744 \\(\\mathrm{P}\\)\uc5d0\uc11c\uc758 \uc811\uacf5\uac04<\/span>(tangent space)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(T_{\\mathrm{P}}\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n

\uc811\uacf5\uac04 \\(T_{\\mathrm{P}}\\)\uc758 \ubca1\ud130 \\(\\vec{v}\\)\uc5d0 \ub300\ud558\uc5ec, \\(\\vec{v}\\)\uc758 \\(x\\)\ucd95 \ubc29\ud5a5\uc758 \ubcc0\ud654\ub7c9\uc744 \\(dx(\\vec{v})\\)\ub85c \ub098\ud0c0\ub0b4\uace0 \\(\\vec{v}\\)\uc758 \\(y\\)\ucd95 \ubc29\ud5a5\uc758 \ubcc0\ud654\ub7c9\uc744 \\(dy(\\vec{v})\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \\(\\mathrm{v}\\)\ub294 \uc9c1\uc120 \\(L\\) \uc704\uc5d0 \ub193\uc5ec \uc788\uace0, \\(\\vec{v}\\)\ub294 \uc2dc\uc810\uc774 \uc811\uc810 \\(\\mathrm{P}\\)\uc774\uace0 \uc885\uc810\uc774 \\(\\mathrm{v}\\)\uc778 \ubca1\ud130\uc774\ubbc0\ub85c, \\(T_{\\mathrm{P}}\\)\uc758 \uc784\uc758\uc758 \ubca1\ud130 \\(\\vec{v}\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[dy(\\vec{v}) = f ‘ (a) dx(\\vec{v})\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc989 \uc774 \ub4f1\uc2dd\uc740 \ub450 \ud568\uc218 \\(dx : T_{\\mathrm{P}} \\to \\mathbb{R}\\)\uc640 \\(dy : T_{\\mathrm{P}} \\to \\mathbb{R}\\)\uac00 \uc815\ube44\ub840 \uad00\uacc4\uc784\uc744 \ub098\ud0c0\ub0b8\ub2e4. (\ubc15\ubd80\uc131 \uad50\uc218\ub2d8\uc758 \ube14\ub85c\uadf8<\/a>\ub97c \ucc38\uace0\ud568.)<\/p>\n

\uc774\uc640 \uac19\uc740 \uad00\uc810\uc5d0\uc11c \\(dx\\)\uc640 \\(dy\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n

\n

\uc815\uc758 1. (\ubbf8\ubd84\uc18c)<\/span><\/p>\n

\ud568\uc218 \\(y=f(x)\\)\uac00 \uc5f4\ub9b0 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ubbf8\ubd84\uc18c \\(dx\\)<\/span>\ub97c \ub3c5\ub9bd\ubcc0\uc218\ub85c \uc815\uc758\ud558\uace0, \ubbf8\ubd84\uc18c \\(dy\\)<\/span>\ub97c
\n\\[dy = f ‘ (x) dx\\tag{6}\\]
\n\uc758 \uad00\uacc4\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubcc0\uc218\ub85c \uc815\uc758\ud55c\ub2e4. \uc989 \\(dy\\)\ub294 \\(dx,\\) \\(f,\\) \\(x\\)\uc5d0 \uc758\ud558\uc5ec \ubcc0\ud558\ub294 \ubcc0\uc218\uc774\ub2e4.<\/p>\n

\\(dy\\)\ub97c \\(df\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4. \uc774\ub54c \\(df\\)\ub97c \\(f\\)\uc758 \ubbf8\ubd84\uc18c<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4.\n<\/div>\n

\ubbf8\ubd84\uc18c\ub97c \uad6c\ud560 \ub54c\uc5d0\ub294 \ubbf8\ubd84 \ubc95\uce59\uc744 \uadf8\ub300\ub85c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \ub2e4\uc74c \ubcf4\uae30\ub97c \ubcf4\uc790.<\/p>\n

\n

\ubcf4\uae30 4.<\/span> (\ubbf8\ubd84\uc18c\ub97c \uad6c\ud558\ub294 \ub2e4\uc591\ud55c \uc608)<\/p>\n

    \n
  1. \\(y=x^4 + 7x\\)\uc77c \ub54c \\(dy = (4x^3 + 7)dx\\)\uc774\ub2e4.<\/li>\n
  2. \\(d(\\sin 2x) = 2 \\cos 2x \\,dx .\\)<\/li>\n
  3. \\(f(x) = \\frac{x}{x-1}\\)\uc77c \ub54c
    \n\\[df = \\frac{-1}{(x-1)^2} dx\\]
    \n\uc774\ub2e4.<\/span><\/li>\n<\/ol>\n<\/div>\n

    \uc751\uc6a9 \ubb38\uc81c\uc5d0\uc11c \\(dx\\)\ub294 \\(x\\)\uac12\uc758 \uc791\uc740 \ubcc0\ud654\ub7c9\uc744 \ub098\ud0c0\ub0b4\uba70, \\(dy\\)\ub294 \\(x\\)\uac12\uc758 \ubcc0\ud654\ub7c9\uc5d0 \ub530\ub978 \\(y\\)\uac12\uc758 \uc791\uc740 \ubcc0\ud654\ub7c9\uc744 \ub098\ud0c0\ub0b8\ub2e4. \uc989 \uc9c1\uad00\uc801\uc73c\ub85c\ub294 \\(\\Delta x\\)\uc758 \uac12\uc774 \\(0\\)\uc5d0 \uac00\uae4c\uc6b8 \ub54c \\(\\Delta x\\)\ub97c \\(dx\\)\ub85c \ub450\uace0 (6)\uc744 \uc774\uc6a9\ud558\uc5ec \\(dy\\)\ub97c \uad6c\ud55c\ub2e4. \uc774\uac83\uc744 \uc2dd\uc73c\ub85c \ub098\ud0c0\ub0b4\uba74, \ud568\uc218 \\(f\\)\uac00 \\(a\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud560 \ub54c \\(dx \\approx \\Delta\\)\ub85c \uadfc\uc0ac\ud558\uba74
    \n\\[f(a+dx) = f(a) + \\Delta y\\]
    \n\ub610\ub294
    \n\\[f(a+dx) \\,\\approx\\, f(a) + dy\\tag{7}\\]
    \n\uc774\ub2e4.<\/p>\n

    \n

    \uc608\uc81c 5.<\/span>
    \n\uc6d0\uc758 \ubc18\uc9c0\ub984 \\(r\\)\uac00 \\(20\\,\\text{m}\\)\uc5d0\uc11c \\(20.1\\,\\text{m}\\)\ub85c \uc99d\uac00\ud560 \ub54c, \ubbf8\ubd84\uc18c\ub97c \uc774\uc6a9\ud558\uc5ec \uc774 \uc6d0\uc758 \ub113\uc774\uc758 \uc99d\uac00\ub7c9\uc758 \uadfc\uc0bf\uac12\uacfc \ub354 \ucee4\uc9c4 \uc6d0\uc758 \ub113\uc774\uc758 \uadfc\uc0bf\uac12\uc744 \uad6c\ud558\uc2dc\uc624.<\/p>\n

    \ud480\uc774.<\/span>
    \n\\(a = 20 \\,(\\mathrm{m})\\)\ub85c \ub450\uace0, \ubc18\uc9c0\ub984\uc774 \\(r\\)\uc778 \uc6d0\uc758 \ub113\uc774\ub97c \\(A(r)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(A(r) = \\pi r^2\\)\uc774\uace0
    \n\\[dA = A ‘ (a) dr = 2\\pi a \\,dr = 2\\pi \\times 20 \\times 0.1 = 4\\pi \\,(\\mathrm{m}^2 )\\]
    \n\uc774\ub2e4. (7)\uc744 \uc774\uc6a9\ud558\uba74
    \n\\[A(r+\\Delta r) \\approx A(r) + dA\\]
    \n\uc774\ubbc0\ub85c
    \n\\[A(20+0.1) \\approx A(20) + 4\\pi = 400 \\pi + 4\\pi = 404 \\pi \\,(\\mathrm{m}^2 )\\]
    \n\uc774\ub2e4. \ucc38\uace0\ub85c \ub354 \ucee4\uc9c4 \uc6d0\uc758 \ub113\uc774\uc758 \ucc38\uac12\uc740
    \n\\[20.1 \\times 20.1 \\times \\pi = 404.01 \\pi \\,(\\mathrm{m}^2 )\\]
    \n\uc774\ub2e4.
    \n<\/span><\/p>\n<\/div>\n

    \ud568\uc218 \\(f\\)\uac00 \\(a\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \\(x\\)\uc758 \uac12\uc774 \\(a\\)\uc5d0\uc11c \\(a+\\Delta x\\)\ub85c \ubcc0\ud558\ub294 \ub3d9\uc548 \\(f\\)\uc758 \uac12\uc758 \uc99d\ubd84\uc744 \ub2e4\uc74c\uacfc \uac19\uc740 \ub450 \uac00\uc9c0 \ubc29\ubc95\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n

      \n
    • \\(f\\)\uc758 \ubcc0\ud654\ub7c9\uc758 \ucc38\uac12 : \\(\\Delta f = f(a+\\Delta x) – f(a) ,\\)<\/li>\n
    • \\(f\\)\uc758 \ubcc0\ud654\ub7c9\uc758 \uadfc\uc0bf\uac12 : \\(df = f ‘ (a) \\Delta x .\\)<\/li>\n<\/ul>\n

      \uc5ec\uae30\uc11c \\(df\\)\uc640 \\(\\Delta f\\)\uc758 \ucc28\uc774\ub97c \uac00\ub2a0\ud574 \ubcf4\uc790.
      \n\\[\\begin{align}
      \n\\Delta f – df
      \n&= \\Delta f – f ‘ (a) \\Delta x \\\\[8pt]
      \n&= f(a+ \\Delta x) – f(a) – f ‘ (a) \\Delta x \\\\[6pt]
      \n&= \\left( \\frac{f(a+\\Delta x)-f(a)}{\\Delta x} – f ‘ (a) \\right) \\cdot \\Delta x.\\tag{8}\\\\[6pt]
      \n\\end{align}\\]
      \n\uc704 \uc2dd\uc5d0\uc11c
      \n\\[\\epsilon = \\frac{f(a+\\Delta x)-f(a)}{\\Delta x} – f ‘ (a)\\]
      \n\ub77c\uace0 \ub450\uba74 \\(\\epsilon\\)\uc740 \\(\\Delta x\\)\uc758 \uac12\uc5d0 \ub530\ub77c \ubcc0\ud558\ub294 \ud568\uc218\uc774\uba70, (8)\uc740
      \n\\[\\Delta f – df = \\epsilon \\cdot \\Delta x\\]
      \n\ub85c \uc4f8 \uc218 \uc788\ub2e4. \\(\\Delta x \\to 0\\)\uc77c \ub54c
      \n\\[\\frac{f(a+\\Delta x)-f(a)}{\\Delta x} \\,\\to\\, f ‘ (a)\\]
      \n\uc774\ubbc0\ub85c, (8)\uc5d0\uc11c \uad04\ud638 \uc548\uc758 \uc2dd\uc740 \\(0\\)\uc5d0 \uc218\ub834\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\Delta x \\to 0\\)\uc77c \ub54c \\(\\epsilon \\to 0\\)\uc774\ub2e4. \uc774\ub85c\uc368 \ub2e4\uc74c \uacf5\uc2dd\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

      \n

      \\(\\boldsymbol{x=a}\\) \uadfc\ucc98\uc5d0\uc11c \ud568\uc218 \\(\\boldsymbol{y=f(x)}\\)\uc758 \uc99d\ubd84<\/span><\/p>\n

      \ud568\uc218 \\(f\\)\uac00 \\(a\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \\(x\\)\uc758 \uac12\uc774 \\(a\\)\uc5d0\uc11c \\(a+\\Delta x\\)\ub85c \ubcc0\ud558\ub294 \ub3d9\uc548 \\(f\\)\uc758 \ud568\uc22b\uac12\uc758 \uc99d\ubd84 \\(\\Delta y\\)\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
      \n\\[\\Delta y \\,=\\, f ‘ (a) \\Delta x + \\epsilon \\Delta x\\tag{9}\\]
      \n\ub2e8, \uc5ec\uae30\uc11c \\(\\epsilon\\)\uc740 \\(\\Delta x \\to 0\\)\uc77c \ub54c \\(\\epsilon \\to 0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\uc774\ub2e4.<\/p>\n<\/div>\n

      \uc608\ucee8\ub300 \uc608\uc81c 5\uc5d0\uc11c
      \n\\[dA
      \n= \\pi \\cdot 20.1^2 – \\pi \\cdot 20^2 = (404.01 – 400)\\pi = (4\\pi + 0.01 \\pi ) \\,(\\mathrm{m}^2 )\\]
      \n\uc774\ubbc0\ub85c, \ub113\uc774\uc758 \ubcc0\ud654\ub7c9\uc758 \uadfc\uc0bf\uac12\uc758 \uc624\ucc28\ub294
      \n\\[\\Delta A – dA = \\epsilon \\Delta r = 0.01 \\pi\\,(\\mathrm{m}^2)\\]
      \n\uc774\uba70, \ubc18\uc9c0\ub984\uc758 \ubcc0\ud654\ub7c9\uc5d0 \ub300\ud55c<\/span> \ub113\uc774\uc758 \ubcc0\ud654\ub7c9\uc758 \uadfc\uc0bf\uac12\uc758 \uc624\ucc28\uc728\uc740
      \n\\[\\epsilon = \\frac{0.01\\pi}{\\Delta r} = \\frac{0.01 \\pi}{0.1} = 0.1 \\pi \\,(\\mathrm{m})\\]
      \n\uc774\ub2e4.<\/p>\n

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

      \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc77c\ucc28\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc758 \uadfc\uc0ac\ud568\uc218\ub97c \ub9cc\ub4dc\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ub610\ud55c \ubbf8\ubd84\uc18c\uc758 \uac1c\ub150\uc744 \ub3c4\uc785\ud558\uace0 \ubcc0\ud654\ub7c9\uc758 \uadfc\uc0bf\uac12\uc744 \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ud560\uc120, \uc811\uc120, \ubc95\uc120 \ud568\uc218 \\(y=f(x)\\)\uac00 \uc11c\ub85c \ub2e4\ub978 \ub450 \uc810 \\(a,\\) \\(b\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uad6c\uac04\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \ub450 \uc810 \\((a,\\,f(a)),\\) \\((b,\\,f(b))\\)\ub97c \ubaa8\ub450 \uc9c0\ub098\ub294 \uc9c1\uc120\uc744 \ud568\uc218 \\(f\\)\uc758 \uadf8\ub798\ud504\uc758 \ud560\uc120\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub9cc\uc57d \\(f\\)\uc758 \uadf8\ub798\ud504 \uc704\uc5d0\uc11c \uc810 \\((b,\\,f(b))\\)\uac00 \uc810 \\((a,\\,f(a))\\)\uc5d0 \uac00\uae4c\uc774 \ub2e4\uac00\uac10\uc5d0 \ub530\ub77c \uc774 \ub450 \uc810\uc744…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[208,206,203,185,207,202,205,204,201,200],"class_list":["post-1958","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-differential","tag-linear-approximation","tag-tangent-line","tag-185","tag-207","tag-202","tag-205","tag-204","tag-201","tag-200"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1958","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"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=1958"}],"version-history":[{"count":61,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1958\/revisions"}],"predecessor-version":[{"id":3221,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1958\/revisions\/3221"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1958"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1958"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1958"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}