{"id":4691,"date":"2020-06-01T11:41:14","date_gmt":"2020-06-01T02:41:14","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=4691"},"modified":"2020-07-02T00:05:20","modified_gmt":"2020-07-01T15:05:20","slug":"calculus-ii-prehomework-2020-summer","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-ii-prehomework-2020-summer\/","title":{"rendered":"\ubbf8\uc801\ubd84\ud559 II \uc790\uae30\uc8fc\ub3c4\uc801 \ud559\uc2b5 \uacfc\uc81c (\ud6c4\ubc18\ubd80)"},"content":{"rendered":"<style type=\"text\/css\">\nol.questions li {\nmargin-bottom: 0.7em;\n}<\/p>\n<\/style>\n<p>\u2018\uc790\uae30\uc8fc\ub3c4\uc801 \ud559\uc2b5 \uacfc\uc81c\u2019\ub294 \uc2a4\uc2a4\ub85c \uacf5\ubd80\ud558\ub294 \ud559\uc0dd\ub4e4\uc5d0\uac8c \ud559\uc2b5\uc758 \ubc29\ud5a5\uc744 \uc548\ub0b4\ud574\uc8fc\uae30 \uc704\ud55c \ubb38\uc81c\uc785\ub2c8\ub2e4. \ub9e4\uc8fc 5\ubb38\uc81c\uac00 \uc81c\uacf5\ub429\ub2c8\ub2e4. Thomas Calculus \uad00\ub828 \ub2e8\uc6d0\uc744 \uacf5\ubd80\ud55c \ud6c4 \ucda9\ubd84\ud788 \uc0dd\uac01\ud558\uba74\uc11c \ubb38\uc81c\ub97c \ud480\uc5b4\ubcf4\uc138\uc694. \uc5ec\ub7ec\ubd84\uc758 \uc2e4\ub825 \ud5a5\uc0c1\uc5d0 \ub3c4\uc6c0\uc774 \ub420 \uac83\uc785\ub2c8\ub2e4.<\/p>\n<p><a name=\"week09\"><\/a><\/p>\n<p class=\"aligncenter margintop2 marginbottom2\">**** **** ****<\/p>\n<h3>9\uc8fc\ucc28<\/h3>\n<div class=\"box\">\n<p>9\uc8fc\ucc28 \ubb38\uc81c\uc758 \uad00\ub828 \ub2e8\uc6d0\uc740 2.5, 10.2, 14.1\uc808\uc785\ub2c8\ub2e4. \ub2e4\uc74c \ubb38\uc81c\uc5d0\uc11c \\(D\\)\ub294 \\(\\mathbb{R}^2\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc744 \ub098\ud0c0\ub0c5\ub2c8\ub2e4.<\/p>\n<\/div>\n<ol class=\"questions\" start=\"41\">\n<li>\\(D\\)\uac00 \ub2eb\ud78c\uc9d1\ud569\uc774\ub77c\uace0 \ud569\uc2dc\ub2e4. \ub610\ud55c \uc218\uc5f4 \\(\\left\\{ \\textbf{x}_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \uc810\uc774 \\(D\\)\uc5d0 \uc18d\ud55c\ub2e4\uace0 \ud569\uc2dc\ub2e4. \ub9cc\uc57d \\(\\left\\{ \\textbf{x}_n \\right\\}\\)\uc774 \\(\\textbf{L}\\)\uc5d0 \uc218\ub834\ud558\uba74 \\(\\textbf{L} \\in D\\)\uc784\uc744 \uc99d\uba85\ud558\uc138\uc694. &nbsp;[\ub3c4\uc6c0\ub9d0: \\(\\textbf{L}\\notin D\\)\ub77c\uace0 \uac00\uc815\ud558\uba74 \\(\\textbf{L}\\)\uc740 \\(\\mathbb{R}^2 \\setminus D\\)\uc758 \ub0b4\ubd80\uc810\uc774 \ub429\ub2c8\ub2e4. \uadf8 \ub4a4 \uadf9\ud55c\uc758 \uc815\uc758\ub97c \uc774\uc6a9\ud558\uc5ec \\(\\left\\{ \\textbf{x}_n \\right\\}\\)\uc758 \ud56d \uc911\uc5d0\uc11c \\(D\\)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294 \uac83\uc774 \uc874\uc7ac\ud558\uac8c \ub428\uc744 \ubcf4\uc785\ub2c8\ub2e4.]<\/li>\n<li>\\(D\\)\uac00 \uc720\uacc4\uc778 \uc9d1\ud569\uc774\uace0 \uc218\uc5f4 \\(\\left\\{ \\textbf{x}_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \uc810\uc774 \\(D\\)\uc5d0 \uc18d\ud55c\ub2e4\uace0 \ud569\uc2dc\ub2e4. \uadf8\ub7ec\uba74 \\(\\left\\{ \\textbf{x}_n \\right\\}\\)\uc758 \ubd80\ubd84\uc218\uc5f4 \uc911\uc5d0\uc11c \uc218\ub834\ud558\ub294 \uac83\uc774 \uc874\uc7ac\ud568\uc744 \uc99d\uba85\ud558\uc138\uc694. &nbsp;[\ub3c4\uc6c0\ub9d0: \\(D\\)\uac00 \uc720\uacc4\uc774\ubbc0\ub85c \\(D\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc720\uacc4\uc778 \uc9c1\uc0ac\uac01\ud615 \uc9d1\ud569 \\(E\\)\uac00 \uc874\uc7ac\ud569\ub2c8\ub2e4. \\(E\\)\ub97c \ud569\ub3d9\uc778 \ub124 \uac1c\uc758 \ub2eb\ud78c \uc9c1\uc0ac\uac01\ud615\uc73c\ub85c \ucabc\uac1c\uba74(\uc774\ub4e4\uc740 \uacbd\uacc4\ub97c \uacf5\uc720\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4), \ub124 \uac1c\uc758 \uc9c1\uc0ac\uac01\ud615 \uc911 \ud558\ub098 \uc774\uc0c1\uc740 \\(\\left\\{ \\textbf{x}_n \\right\\}\\)\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \uac00\uc9d1\ub2c8\ub2e4. \uadf8\ub7ec\ud55c \uc9c1\uc0ac\uac01\ud615\uc744 \uace8\ub77c\uc11c \\(E_1\\)\uc774\ub77c\uace0 \ud569\uc2dc\ub2e4. \uc774\uc640 \uac19\uc740 \uacfc\uc815\uc744 \ubc18\ubcf5\ud558\uc5ec \\(\\left\\{ \\textbf{x}_n \\right\\}\\)\uc758 \ud56d\uc744 \ubb34\ud55c\ud788 \ub9ce\uc774 \uac16\ub294 \uc9c1\uc0ac\uac01\ud615\uc5f4 \\(E_k\\)\ub97c \ub9cc\ub4e4 \uc218 \uc788\uc2b5\ub2c8\ub2e4. \uac01 \\(E_k\\)\uc5d0\uc11c \ud56d \\(\\textbf{x}_{n_k} \\)\ub97c \ud558\ub098\uc529 \ud0dd\ud558\uc5ec \uc218\ub834\ud558\ub294 \ubd80\ubd84\uc218\uc5f4\uc744 \uad6c\uc131\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4. \uc774 \uba85\uc81c\ub97c <span class=\"defined\">\uc218\uc5f4\uc5d0 \ub300\ud55c Bolzano-Weierstrass \uc815\ub9ac<\/span>\ub77c\uace0 \ubd80\ub985\ub2c8\ub2e4.]<\/li>\n<li>\\(D\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \uc720\uacc4\uc778 \ub2eb\ud78c\uc9d1\ud569\uc774\uace0, \\(f\\)\uac00 \\(D\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud569\uc2dc\ub2e4. \uc774\ub54c \\(f\\)\uac00 \\(D\\)\uc5d0\uc11c \uc720\uacc4\uc784\uc744 \uc99d\uba85\ud558\uc138\uc694. &nbsp;[\ub3c4\uc6c0\ub9d0: \\(f\\)\uac00 \\(D\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud569\uc2dc\ub2e4. \uadf8\ub7ec\uba74 \\(D\\)\uc758 \ud55c \uc810 \\(\\textbf{p}\\)\uc5d0 \uc218\ub834\ud558\ub294 \uc218\uc5f4 \\(\\left\\{ \\textbf{x}_n \\right\\}\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(n \\to \\infty\\)\uc77c \ub54c \\(\\lvert f\\left(\\textbf{x}_n \\right) \\rvert \\to \\infty\\)\ub97c \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc785\ub2c8\ub2e4. \uadf8\ub9ac\uace0 \uc5f0\uc18d\ud568\uc218\uc758 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uc5ec \uc774\uac83\uc774 \ubaa8\uc21c\uc784\uc744 \ubcf4\uc785\ub2c8\ub2e4.]<\/li>\n<li>\ud568\uc218 \\(f : D \\to \\mathbb{R}^m\\)\uc774 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c \u2018\\(f\\)\ub294 <span class=\"defined\">\\(D\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d<\/span>\uc774\ub2e4\u2019\ub77c\uace0 \ub9d0\ud569\ub2c8\ub2e4.<br \/>\n\\[\\forall \\epsilon > 0 \\, \\exists \\delta > 0 \\, \\forall s \\in D \\, \\forall t \\in D \\, : \\,\\, ( \\lvert s-t \\rvert < \\delta \\,\\, \\rightarrow \\,\\, \\lvert f(s) - f(t) \\rvert < \\epsilon ) \\]\n\\(f(x,\\,y) = \\sin x + \\cos y\\)\uc77c \ub54c \\(f\\)\uac00 \\(\\mathbb{R}^2\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc784\uc744 \uc99d\uba85\ud558\uc138\uc694.\n<\/li>\n<li>\\(D\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2c8\uace0 \uc720\uacc4\uc778 \ub2eb\ud78c \uc9d1\ud569\uc774\uba70 \\(f\\)\uac00 \\(D\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud569\uc2dc\ub2e4. \ub9cc\uc57d \\(f\\)\uac00 \\(D\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \\(f\\)\ub294 \\(D\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc784\uc744 \uc99d\uba85\ud558\uc138\uc694. &nbsp;[\ub3c4\uc6c0\ub9d0: \\(f\\)\uac00 \\(D\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774 \uc544\ub2c8\ub77c\uace0 \uac00\uc815\ud558\uace0, \\(\\epsilon > 0\\)\uacfc \ub450 \uc218\uc5f4 \\(\\left\\{ \\textbf{s} _k \\right\\} ,\\) \\(\\left\\{ \\textbf{t} _k \\right\\} \\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\lvert \\textbf{s}_k &#8211; \\textbf{t}_k \\rvert \\rightarrow 0\\)\uc774\uba74\uc11c \ubaa8\ub4e0 \\(k\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert f(\\textbf{s}_k ) -f(\\textbf{t}_k) \\rvert \\ge \\epsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc785\ub2c8\ub2e4. \uadf8\ub9ac\uace0 \uc774 \uc0ac\uc2e4\ub85c\ubd80\ud130 \\(f\\)\uac00 \uc5f0\uc18d\uc774\ub77c\ub294 \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774 \ub418\ub294 \uba85\uc81c\ub97c \uc720\ub3c4\ud569\ub2c8\ub2e4.]<\/li>\n<\/ol>\n<p><a name=\"week10\"><\/a><\/p>\n<p class=\"aligncenter margintop2 marginbottom2\">**** **** ****<\/p>\n<h3>10\uc8fc\ucc28<\/h3>\n<div class=\"box\">\n<p>\ub2e4\uc74c \ubb38\uc81c\uc5d0\uc11c \\(I\\)\ub294 \uae38\uc774\uac00 \uc591\uc218\uc778 \ub2eb\ud78c \uad6c\uac04\uc744 \ub098\ud0c0\ub0b4\uba70, \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\uc22b\uac12 \ud568\uc218\ub97c \ub098\ud0c0\ub0c5\ub2c8\ub2e4.<\/p>\n<\/div>\n<ol class=\"questions\" start=\"46\">\n<li>\\(I\\)\uac00 \ub2eb\ud78c \uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud558\uc138\uc694.<\/li>\n<li>\uc801\ubd84\uc744 \uc815\uc758\ud560 \ub54c \uc0ac\uc6a9\ud55c \uac1c\ub150\uc778 \u2018\ubd84\ud560\u2019\uc744 \ub5a0\uc62c\ub824 \ubd05\uc2dc\ub2e4. \\(P\\)\uac00 \\(I\\)\uc758 \ubd84\ud560\uc774\uace0<br \/>\n\\[P = \\left\\{ x_0 ,\\, x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_n \\right\\}\\]<br \/>\n\uc774\ub77c\uace0 \ud569\uc2dc\ub2e4. \ubd84\ud560\uc744 \uc704\uc640 \uac19\uc774 \ub098\ud0c0\ub0bc \ub54c\ub294<br \/>\n\\[a = x_0 < x_1 < x_2 < \\cdots < x_n = b\\]\n\uac00 \uc131\ub9bd\ud558\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud569\ub2c8\ub2e4. \uc774\ub54c \\(i = 1,\\,2,\\,\\cdots,\\,n\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\begin{align}\nM_i &#038;= \\sup \\left\\{ f(c_i ) \\,\\vert\\, x_{i-1} \\le c_i \\le x_i \\right\\} , \\\\[6pt]\nm_i &#038;= \\inf \\left\\{ f(c_i ) \\,\\vert\\, x_{i-1} \\le c_i \\le x_i \\right\\} \n\\end{align}\\]\n\ub85c \uc815\uc758\ud569\ub2c8\ub2e4. \n\uc989 \uc9c1\uad00\uc801\uc73c\ub85c \uc0dd\uac01\ud588\uc744 \ub54c, \uc18c\uad6c\uac04 \\([x_{i-1} ,\\, x_i ]\\)\uc5d0\uc11c \\(f\\)\uc758 \ud568\uc22b\uac12 \uc911 \uac00\uc7a5 \ud070 \uac12\uc744 \\(M_i\\)\uc73c\ub85c \ub098\ud0c0\ub0b4\uace0, \uac00\uc7a5 \uc791\uc740 \uac12\uc744 \\(m_i\\)\ub85c \ub098\ud0c0\ub0c5\ub2c8\ub2e4. (\ubb3c\ub860 \\(f\\)\ub294 \uac01 \uc18c\uad6c\uac04\uc5d0\uc11c \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac16\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\uc2b5\ub2c8\ub2e4. \uadf8\ub798\uc11c \uc0c1\ud55c\uacfc \ud558\ud55c\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(M_i\\)\uc640 \\(m_i\\)\ub97c \uc815\uc758\ud569\ub2c8\ub2e4.)\n\uadf8\ub9ac\uace0 \\(I\\)\uc5d0\uc11c \\(P\\)\uc5d0 \uc758\ud55c \\(f\\)\uc758 <span class=\"defined\">\uc0c1\ud569<\/span>(upper sum) \\(U(f,\\,P)\\)\uc640 <span class=\"defined\">\ud558\ud569<\/span>(lower sum) \\(L(f,\\,P)\\)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud569\ub2c8\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nU(f,\\,P) &#038;= \\sum_{i=1}^n M_i \\Delta x_i , \\\\[6pt]<br \/>\nL(f,\\,P) &#038;= \\sum_{i=1}^n m_i \\Delta x_i .<br \/>\n\\end{align}\\]<br \/>\n\\(P\\)\uac00 \\(I\\)\uc758 \ubd84\ud560\uc77c \ub54c \ud56d\uc0c1 \\(L(f,\\,P) \\le U(F,\\,P)\\)\uac00 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud558\uc138\uc694.<\/li>\n<li>\\(P\\)\uc640 \\(Q\\)\uac00 \\(I\\)\uc758 \ubd84\ud560\uc774\uace0 \\(P\\subseteq Q\\)\uc77c \ub54c \\(Q\\)\ub97c \\(P\\)\uc758 <span class=\"defined\">\uc138\ub828\ubd84\ud560<\/span>(refinement)\uc774\ub77c\uace0 \ubd80\ub985\ub2c8\ub2e4. \uc989 \ubd84\ud560\uc774 \uad6c\uac04\uc744 \uc790\ub978 \uac83\uc73c\ub85c \ubcf8\ub2e4\uba74 \uc138\ub828\ubd84\ud560\uc774\ub780 \uc6d0\ub798\uc758 \ubd84\ud560\uc5d0\uc11c \uba87 \ubc88 \ub354 \uc790\ub978 \ubd84\ud560\uc774\ub77c\uace0 \uc0dd\uac01\ud558\uba74 \ub429\ub2c8\ub2e4. \uc774\ub54c<br \/>\n\\[\\begin{gather}<br \/>\nL(f,\\,P) \\le L(f,\\,Q) ,\\\\[6pt]<br \/>\nU(f,\\,P) \\ge U(f,\\,Q)<br \/>\n\\end{gather}\\]<br \/>\n\uac00 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud558\uc138\uc694.\n<\/li>\n<li>\ub9cc\uc57d \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \uc784\uc758\uc758 \uc591\uc218 \\(\\epsilon\\)\uc5d0 \ub300\ud558\uc5ec \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[U(f,\\,P) &#8211; L(f,\\,P) < \\epsilon\\]\n\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc774\uc138\uc694.<\/li>\n<li>\uad6c\uac04 \\(I\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc758 \uc0c1\ud569\ub4e4\uc758 \ubaa8\uc784\uc740 \uc544\ub798\ub85c \uc720\uacc4\uc774\ubbc0\ub85c \ud558\ud55c\uc744 \uac00\uc9d1\ub2c8\ub2e4. \uadf8 \uac12\uc744 \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc0c1\uc801\ubd84<\/span>(upper integral)\uc774\ub77c \ubd80\ub974\uace0<br \/>\n\\[U_f = \\inf \\left\\{ U(f,\\,P) \\,\\vert\\, P \\text{ is a partition of }I \\right\\}\\]<br \/>\n\ub85c \uc815\uc758\ud569\ub2c8\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \ud558\ud569\ub4e4\uc758 \ubaa8\uc784\uc758 \uc0c1\ud55c\uc744 \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\ud558\uc801\ubd84<\/span>(lower integral)\uc774\ub77c \ubd80\ub974\uace0<br \/>\n\\[L_f = \\sup \\left\\{ L(f,\\,P) \\,\\vert\\, P \\text{ is a partition of }I \\right\\}\\]<br \/>\n\ub85c \uc815\uc758\ud569\ub2c8\ub2e4. \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218\uc774\uba74 \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\uc801\ubd84\uc758 \uac12\uacfc \ud558\uc801\ubd84\uc758 \uac12\uc774 \uc77c\uce58\ud568\uc744 \uc99d\uba85\ud558\uc138\uc694.<\/li>\n<\/ol>\n<p><a name=\"week11\"><\/a><\/p>\n<p class=\"aligncenter margintop2 marginbottom2\">**** **** ****<\/p>\n<h3>11\uc8fc\ucc28<\/h3>\n<div class=\"box\">\n<p>\ub2e4\uc74c \ubb38\uc81c\uc5d0\uc11c \\(I=[a,\\,b]\\)\ub294 \uae38\uc774\uac00 \uc591\uc218\uc778 \ub2eb\ud78c \uad6c\uac04\uc744 \ub098\ud0c0\ub0b4\uba70, \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\uc22b\uac12 \ud568\uc218\ub97c \ub098\ud0c0\ub0c5\ub2c8\ub2e4.<\/p>\n<\/div>\n<ol class=\"questions\" start=\"51\">\n<li>\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \uc784\uc758\uc758 \uc591\uc218 \\(\\epsilon\\)\uc5d0 \ub300\ud558\uc5ec \uc591\uc218 \\(\\delta\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\lVert P \\rVert < \\delta\\)\uc778 \\(I\\)\uc758 \uc784\uc758\uc758 \ubd84\ud560 \\(P\\)\uc5d0 \ub300\ud558\uc5ec \\(U(f,\\,P) - L(f,\\,P) < \\epsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0b4\uc744 \ubcf4\uc774\uc138\uc694. [\ub3c4\uc6c0\ub9d0: \ubb38\uc81c 49\ubc88\uc744 \ucc38\uace0\ud558\uc138\uc694.]<\/li>\n<li>\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\uc801\ubd84\uac12\uc744 \\(J\\)\ub77c\uace0 \ud569\uc2dc\ub2e4(\ubb38\uc81c 50\ubc88\uc744 \ucc38\uace0\ud558\uc138\uc694). \uc774\ub54c \\(I\\)\uc5d0\uc11c \\(f\\)\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uace0, \uc801\ubd84\uac12\uc774 \\(J\\)\uc640 \uac19\uc74c\uc744 \uc99d\uba85\ud558\uc138\uc694. \uc989 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc77c \ub54c, \uc784\uc758\uc758 \uc591\uc218 \\(\\epsilon\\)\uc5d0 \ub300\ud558\uc5ec \uc591\uc218 \\(\\delta\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(\\lVert P \\rVert < \\delta\\)\uc778 \\(I\\)\uc758 \uc784\uc758\uc758 \ubd84\ud560 \\(P\\)\uc640, \\(P\\)\uc758 \uac01 \uc18c\uad6c\uac04\uc5d0\uc11c \uc6d0\uc18c\ub97c \ud558\ub098\uc529 \ud0dd\ud558\uc5ec \ub9cc\ub4e0 \uc218\uc5f4 \\(\\left\\{ c_i \\right\\},\\) \\(c_i \\in \\left[ x_{i-1},\\,x_o \\right]\\)\uc5d0 \ub300\ud558\uc5ec,\n\\[\\left\\lvert \\sum_{P} f( c_i ) \\Delta x_i - J \\right\\rvert < \\epsilon\\]\n\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc138\uc694. [\ub3c4\uc6c0\ub9d0: \ubb38\uc81c 45, 49, 50, 51\ubc88\uc744 \uc0ac\uc6a9\ud558\uc138\uc694.]<\/li>\n<li>\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uba74, \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \ub9ac\ub9cc \uc801\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ud560 \uc218 \uc788\uc74c\uc744 \ubcf4\uc774\uc138\uc694.<br \/>\n\\[\\int_a^b f(x) dx = \\lim_{n\\to\\infty} \\sum_{k=1}^{n} f \\left( a+ \\frac{b-a}{n} k \\right) \\frac{b-a}{n}. \\]<br \/>\n[\ub3c4\uc6c0\ub9d0: \\(I\\)\ub97c \\(n\\)\uac1c\uc758 \uc18c\uad6c\uac04\uc73c\ub85c \uade0\ub4f1\ud558\uac8c \uc790\ub978 \ubd84\ud560\uc744 \uc0dd\uac01\ud558\uace0, \uac01 \uc18c\uad6c\uac04\uc5d0\uc11c \uc624\ub978\ucabd \ub05d\uc810\uc744 \ud0dd\ud558\uc5ec \ub9ac\ub9cc \ud569\uc744 \uad6c\ud558\uba74 \uc704 \uc2dd\uc758 \uc6b0\ubcc0\uc758 \ud569\uc774 \ub429\ub2c8\ub2e4. \uc5ec\uae30\uc5d0 \\(n \\to \\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 \ubd84\ud560\uc758 \ub178\ub984\uc774 \\(0\\)\uc5d0 \uc218\ub834\ud569\ub2c8\ub2e4.] \ub610\ud55c, \uc774 \uba85\uc81c\uc758 \uc5ed\uc774 \uc131\ub9bd\ud558\uc9c0 \uc54a\uc74c\uc744 \uc99d\uba85\ud558\uc138\uc694. \uc989 \uc704 \ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc758 \uadf9\ud55c\uc774 \uc218\ub834\ud560\uc9c0\ub77c\ub3c4 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\uc740 \ud568\uc218 \\(f\\)\uc640 \uad6c\uac04 \\(I\\)\uc758 \uc608\ub97c \ub9cc\ub4e4\uc5b4 \ubcf4\uc138\uc694.\n<\/li>\n<li>\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\uba74, \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\uc801\ubd84\uc758 \uac12\uacfc \ud558\uc801\ubd84\uc758 \uac12\uc774 \uc77c\uce58\ud568\uc744 \uc99d\uba85\ud558\uc138\uc694. [\ub3c4\uc6c0\ub9d0: \ubb38\uc81c 49, 50\ubc88\uc744 \ud478\ub294 \ubc29\ubc95\uacfc \ube44\uc2b7\ud569\ub2c8\ub2e4.]<\/li>\n<li>\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc99d\uac00\ud558\ub294 \ud568\uc218\uc774\uace0, \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\uc801\ubd84\uac12\uc744 \\(J\\)\ub77c\uace0 \ud569\uc2dc\ub2e4. \uc774\ub54c \\(I\\)\uc5d0\uc11c \\(f\\)\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uace0, \uc801\ubd84\uac12\uc774 \\(J\\)\uc640 \uac19\uc74c\uc744 \uc99d\uba85\ud558\uc138\uc694. [\ub3c4\uc6c0\ub9d0: \ubb38\uc81c 52\ubc88\uc744 \ud478\ub294 \ubc29\ubc95\uacfc \ube44\uc2b7\ud569\ub2c8\ub2e4.]<\/li>\n<\/ol>\n<p><a name=\"week12\"><\/a><\/p>\n<p class=\"aligncenter margintop2 marginbottom2\">**** **** ****<\/p>\n<h3>12\uc8fc\ucc28<\/h3>\n<div class=\"box\">\n<p>\ub2e4\uc74c \ubb38\uc81c\uc5d0\uc11c \\(R = [a,\\,b] \\times [c,\\,d]\\)\ub294 \ub113\uc774\uac00 \uc591\uc218\uc778 \uc9c1\uc0ac\uac01\ud615 \uc9d1\ud569\uc744 \ub098\ud0c0\ub0b4\uba70, \\(f\\)\ub294 \\(R\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\uc22b\uac12 \ud568\uc218\ub97c \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \ub610\ud55c \u2018\ub9ac\ub9cc \uc801\ubd84\u2019\uc740 \u2018\uc911\uc801\ubd84\u2019\uc744 \uc758\ubbf8\ud569\ub2c8\ub2e4.<\/p>\n<\/div>\n<ol class=\"questions\" start=\"56\">\n<li>\\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\ud569, \ud558\ud569, \uc0c1\uc801\ubd84, \ud558\uc801\ubd84\uc744 \uc801\uc808\ud558\uac8c \uc815\uc758\ud574 \ubcf4\uc138\uc694. [\ubb38\uc81c 47-50\ubc88\uc744 \ucc38\uace0\ud558\uc138\uc694.]<\/li>\n<li>\\(R\\)\uc5d0\uc11c \\(f\\)\uac00 \uc5f0\uc18d\uc774\uba74, \\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\uc801\ubd84\uc758 \uac12\uacfc \ud558\uc801\ubd84\uc758 \uac12\uc774 \uc77c\uce58\ud568\uc744 \uc99d\uba85\ud558\uc138\uc694. [\ubb38\uc81c 50\ubc88\uc744 \ucc38\uace0\ud558\uc138\uc694.]<\/li>\n<li>\\(R\\)\uc5d0\uc11c \\(f\\)\uac00 \uc5f0\uc18d\uc774\uba74, \\(R\\)\uc5d0\uc11c \\(f\\)\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \uc99d\uba85\ud558\uc138\uc694. [\ubb38\uc81c 52\ubc88\uc744 \ucc38\uace0\ud558\uc138\uc694.]<\/li>\n<li>\\(R\\)\uc5d0\uc11c \\(f\\)\uac00 \uc5f0\uc18d\uc77c \ub54c, \ubb38\uc81c 53\uacfc \uac19\uc774 \ud569\uacfc \uadf9\ud55c\uc744 \uc774\uc6a9\ud558\uc5ec \\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \ub9ac\ub9cc \uc801\ubd84\uc758 \uac12\uc744 \uad6c\ud558\ub294 \uacf5\uc2dd\uc744 \ub9cc\ub4e4\uc5b4 \ubcf4\uc138\uc694.<\/li>\n<li>\uc784\uc758\uc758 \\(x_0 \\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x_0 ,\\,y )\\)\uac00 \\(y\\)\uc5d0 \ub300\ud55c \uc99d\uac00\ud568\uc218\uc774\uba70, \uc784\uc758\uc758 \\(y_0 \\in [c,\\,d]\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x,\\,y_0 )\\)\uac00 \\(x\\)\uc5d0 \ub300\ud55c \uc99d\uac00\ud568\uc218\ub77c\uace0 \ud569\uc2dc\ub2e4. \uc774\ub54c \\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc138\uc694. [\ubb38\uc81c 55\ubc88\uc744 \ucc38\uace0\ud558\uc138\uc694.]<\/li>\n<\/ol>\n<p><a name=\"week13\"><\/a><\/p>\n<p class=\"aligncenter margintop2 marginbottom2\">**** **** ****<\/p>\n<h3>13\uc8fc\ucc28<\/h3>\n<div class=\"box\">\n<p>\ub2e4\uc74c \ubb38\uc81c\uc5d0\uc11c \\(R = [a,\\,b] \\times [c,\\,d]\\)\ub294 \ub113\uc774\uac00 \uc591\uc218\uc778 \uc9c1\uc0ac\uac01\ud615 \uc9d1\ud569\uc744 \ub098\ud0c0\ub0b4\uba70, \\(f\\)\ub294 \\(R\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\uc22b\uac12 \ud568\uc218\ub97c \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \ub610\ud55c \u2018\ub9ac\ub9cc \uc801\ubd84\u2019\uc740 \u2018\uc911\uc801\ubd84\u2019\uc744 \uc758\ubbf8\ud569\ub2c8\ub2e4.<\/p>\n<\/div>\n<ol class=\"questions\" start=\"61\">\n<li>\ud568\uc218 \\(f\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud569\uc2dc\ub2e4.<br \/>\n\\[ f(x,\\,y) =<br \/>\n\\begin{cases}<br \/>\n1 \\quad &#038; \\quad \\text{if  } (x,\\,y)\\in\\mathbb{Q} \\times \\mathbb{Q} ,\\\\[8pt]<br \/>\n0 \\quad &#038; \\quad \\text{otherwise.}<br \/>\n\\end{cases}<br \/>\n\\]<br \/>\n\uc774\ub54c \\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \ubd88\uac00\ub2a5\ud568\uc744 \uc99d\uba85\ud558\uc138\uc694.\n<\/li>\n<li>\\(R\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \ubaa8\uc784\uc774 \ube44\uac00\uc0b0\uc774\uc9c0\ub9cc \\(R\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud55c \ud568\uc218 \\(f\\)\uc758 \uc608\ub97c \ub9cc\ub4e4\uc5b4 \ubcf4\uc138\uc694.<\/li>\n<li>\ud568\uc218 \\(f\\)\uac00 \\(R\\)\uc758 \ub0b4\ubd80\uc5d0\uc11c \uc5f0\uc18d\uc774\uc9c0\ub9cc<br \/>\n\\[\\int_c^d \\int_a^b f(x,\\,y) dx\\,dy \\ne \\int_a^b \\int_c^d f(x,\\,y) dy\\,dx\\]<br \/>\n\uc778 \ud568\uc218 \\(f\\)\uc758 \uc608\ub97c \ub9cc\ub4e4\uc5b4 \ubcf4\uc138\uc694.<\/li>\n<li>\ud568\uc218 \\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc \\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \ud3b8\uc801\ubd84(partial integral)\uc774 \ubd88\uac00\ub2a5\ud55c \ud568\uc218 \\(f\\)\uc758 \uc608\ub97c \ub9cc\ub4e4\uc5b4 \ubcf4\uc138\uc694. \uc989 \uc801\ub2f9\ud55c \\(y_0 \\in [c,\\,d]\\)\uac00 \uc874\uc7ac\ud558\uc5ec, \\(x\\)\uc5d0 \ub300\ud55c \ud568\uc218 \\(f(x,\\,y_0 )\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \ubd88\uac00\ub2a5\ud55c \ud568\uc218 \\(f\\)\uc758 \uc608\ub97c \ub9cc\ub4e4\uc5b4 \ubcf4\uc138\uc694.<\/li>\n<li>\\(R\\)\uc5d0\uc11c \\(f\\)\uac00 \uc5f0\uc18d\uc77c \ub54c, \ub2e4\uc74c \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud558\uc138\uc694.<br \/>\n\\[\\int_c^d \\int_a^b f(x,\\,y) dx\\,dy = \\int_a^b \\int_c^d f(x,\\,y) dy\\,dx\\]<br \/>\n[\ub3c4\uc6c0\ub9d0: \ubb38\uc81c 59\ubc88\uc744 \uc774\uc6a9\ud558\uc138\uc694.]<\/p>\n<\/li>\n<\/ol>\n<p><!--\n\n<a name=\"week1n\"><\/a>\n\n\n<p class=\"aligncenter margintop2 marginbottom2\">**** **** ****<\/p>\n\n\n\n\n\n<h3>n\uc8fc\ucc28<\/h3>\n\n\n\n\n<ol class=\"questions\" start=\"66\">\n\n\n<li><\/li>\n\n\n\n\n<li><\/li>\n\n\n\n\n<li><\/li>\n\n\n\n\n<li><\/li>\n\n\n\n\n<li><\/li>\n\n\n<\/ol>\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u2018\uc790\uae30\uc8fc\ub3c4\uc801 \ud559\uc2b5 \uacfc\uc81c\u2019\ub294 \uc2a4\uc2a4\ub85c \uacf5\ubd80\ud558\ub294 \ud559\uc0dd\ub4e4\uc5d0\uac8c \ud559\uc2b5\uc758 \ubc29\ud5a5\uc744 \uc548\ub0b4\ud574\uc8fc\uae30 \uc704\ud55c \ubb38\uc81c\uc785\ub2c8\ub2e4. \ub9e4\uc8fc 5\ubb38\uc81c\uac00 \uc81c\uacf5\ub429\ub2c8\ub2e4. Thomas Calculus \uad00\ub828 \ub2e8\uc6d0\uc744 \uacf5\ubd80\ud55c \ud6c4 \ucda9\ubd84\ud788 \uc0dd\uac01\ud558\uba74\uc11c \ubb38\uc81c\ub97c \ud480\uc5b4\ubcf4\uc138\uc694. \uc5ec\ub7ec\ubd84\uc758 \uc2e4\ub825 \ud5a5\uc0c1\uc5d0 \ub3c4\uc6c0\uc774 \ub420 \uac83\uc785\ub2c8\ub2e4. **** **** **** 9\uc8fc\ucc28 9\uc8fc\ucc28 \ubb38\uc81c\uc758 \uad00\ub828 \ub2e8\uc6d0\uc740 2.5, 10.2, 14.1\uc808\uc785\ub2c8\ub2e4. \ub2e4\uc74c \ubb38\uc81c\uc5d0\uc11c \\(D\\)\ub294 \\(\\mathbb{R}^2\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc744 \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \\(D\\)\uac00 \ub2eb\ud78c\uc9d1\ud569\uc774\ub77c\uace0 \ud569\uc2dc\ub2e4. \ub610\ud55c \uc218\uc5f4 \\(\\left\\{ \\textbf{x}_n \\right\\}\\)\uc758 \ubaa8\ub4e0 \uc810\uc774 \\(D\\)\uc5d0 \uc18d\ud55c\ub2e4\uace0 \ud569\uc2dc\ub2e4. \ub9cc\uc57d \\(\\left\\{ \\textbf{x}_n \\right\\}\\)\uc774&hellip;<\/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":[],"class_list":["post-4691","post","type-post","status-publish","format-standard","hentry","category-calculus-ap"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4691","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=4691"}],"version-history":[{"count":53,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4691\/revisions"}],"predecessor-version":[{"id":4858,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/4691\/revisions\/4858"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=4691"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=4691"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=4691"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}