{"id":19656,"date":"2011-01-07T04:57:22","date_gmt":"2011-01-06T20:57:22","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=19656"},"modified":"2021-10-06T16:29:34","modified_gmt":"2021-10-06T08:29:34","slug":"%e5%be%ae%e7%a9%8d%e5%88%86%e5%88%9d%e9%9a%8e%ef%bc%8d%e6%ad%b7%e5%8f%b2%e7%99%bc%e5%b1%95%e7%9a%84%e7%9c%bc%e5%85%89%ef%bc%8812%ef%bc%89%e5%be%ae%e7%a9%8d%e5%88%86%e5%ad%b8%e6%a0%b9%e6%9c%ac%e5%ae%9a","status":"publish","type":"post","link":"http:\/\/localhost\/%e5%be%ae%e7%a9%8d%e5%88%86%e5%88%9d%e9%9a%8e%ef%bc%8d%e6%ad%b7%e5%8f%b2%e7%99%bc%e5%b1%95%e7%9a%84%e7%9c%bc%e5%85%89%ef%bc%8812%ef%bc%89%e5%be%ae%e7%a9%8d%e5%88%86%e5%ad%b8%e6%a0%b9%e6%9c%ac%e5%ae%9a\/","title":{"rendered":"\u5fae\u7a4d\u5206\u521d\u968e\uff0d\u6b77\u53f2\u767c\u5c55\u7684\u773c\u5149\uff0812\uff09\u5fae\u7a4d\u5206\u5b78\u6839\u672c\u5b9a\u7406\uff08First Course in Calculus\uff0dA Historical Approach 12. The Fundamental Theorem of Calculus\uff09"},"content":{"rendered":"<div class=\"pf-content\"><p><strong><span style=\"color: #ff6600;\">\u5fae\u7a4d\u5206\u521d\u968e\uff0d\u6b77\u53f2\u767c\u5c55\u7684\u773c\u5149\uff0812\uff09\u5fae\u7a4d\u5206\u5b78\u6839\u672c\u5b9a\u7406\uff08First Course in Calculus\uff0dA Historical Approach 12. The Fundamental Theorem of Calculus\uff09<\/span><\/strong><br \/>\n<strong><span style=\"color: #008000;\">\u570b\u7acb\u81fa\u7063\u5927\u5b78\u6578\u5b78\u7cfb\u8521\u8070\u660e\u526f\u6559\u6388\/\u570b\u7acb\u81fa\u7063\u5927\u5b78\u6578\u5b78\u7cfb\u8521\u8070\u660e\u526f\u6559\u6388\u8cac\u4efb\u7de8\u8f2f<\/span><\/strong><\/p>\n<p>\u9023\u7d50\uff1a<a href=\"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=19551\">\u5fae\u7a4d\u5206\u521d\u968e\uff0d\u6b77\u53f2\u767c\u5c55\u7684\u773c\u5149\uff0811\uff09\u5fae\u5206\u8207\u7a4d\u5206\u7684\u5b9a\u7fa9<\/a><\/p>\n<p>\u5fae\u7a4d\u5206\u6700\u91cd\u8981\u4e14\u6700\u6838\u5fc3\u7684\u300c\u5fae\u7a4d\u5206\u5b78\u6839\u672c\u5b9a\u7406\u300d\u3002\u6c42\u5207\u7dda\u8207\u6c42\u9762\u7a4d\u5169\u8005\u8868\u9762\u4e0a\u5f88\u4e0d\u540c\uff0c\u5be6\u5247\u95dc\u4fc2\u5bc6\u5207\u3002<\/p>\n<p><span style=\"color: #800080;\"><strong>\u3010\u5b9a\u74067\u3011<\/strong>\uff08\u5fae\u7a4d\u5206\u5b78\u6839\u672c\u5b9a\u7406, the Fundamental Theorem of Calculus, FTC\uff09<\/span><\/p>\n<p>\u5047\u8a2d\u51fd\u6578 $$f$$ \u5728\u9589\u5340\u9593 $$[a,b]$$ \u4e0a\u9023\u7e8c\uff0c\u90a3\u9ebc\u5c31\u6709\uff1a<\/p>\n<ul>\n<li>$$(\\mathrm{i})$$\u00a0<strong>\u5fae\u5206\u8207\u7a4d\u5206\u7684\u4e92\u9006\u6027<\/strong>\uff1a\u4ee4 $$G(x)={\\int}_a^x{f(t)}dt$$\uff0c\u5247 $$DG(x)=f(x),~~\\forall{x}\\in[a,b]$$<\/li>\n<li>$$(\\mathrm{ii})$$\u00a0<strong>N-L\u516c\u5f0f<\/strong>\uff1a\u82e5 $$DF(x)=f(x),\\forall{x}\\in[a,b]$$\uff0c\u5247\u00a0$${\\int}_a^b{f(x)}dx=F(b)-F(a)$$<!--more--><\/li>\n<\/ul>\n<p>\u5982\u679c\u6211\u5011\u63a5\u53d7\u5e95\u4e0b\u5169\u500b\u91cd\u8981\u6975\u9650\u516c\u5f0f\uff0c\u5c31\u53ef\u4ee5\u6c42\u5f97\u6b63\u5f26\u51fd\u6578\u8207\u9918\u5f26\u51fd\u6578\u7684\u5fae\u5206\u516c\u5f0f\u3002\u5f9e\u800c\u5f97\u5230\u66f4\u591a\u7684\u5b9a\u7a4d\u5206\u516c\u5f0f\u3002<\/p>\n<p><strong>\u3010\u88dc\u984c\uff13\u3011<\/strong>$$(\\mathrm{i})\\displaystyle\\lim_{x\\to 0}\\frac{\\sin x}{x}=1~~~~~~(\\mathrm{ii})\\lim_{x\\to 0}\\frac{1-\\cos x}{x}=0$$<\/p>\n<p><strong>\u3010\u5fae\u5206\u516c\u5f0f\u3011<\/strong>$$D\\sin{x}=\\cos{x}$$<\/p>\n<p><strong>\u3010\u8b49\u660e\u3011<\/strong><\/p>\n<p>$$\\begin{array}{ll}D\\sin x&amp;=\\displaystyle\\lim_{\\Delta x\\to 0}\\frac{\\sin(x+\\Delta x)-\\sin x}{\\Delta x}\\\\&amp;=\\displaystyle\\lim_{\\Delta x\\to 0}\\frac{\\sin x\\cos\\Delta x+\\cos x\\sin\\Delta x-\\sin x}{\\Delta x}\\\\&amp;=\\displaystyle\\lim_{\\Delta x\\to 0}\\left[\\sin x\\left(\\frac{\\cos\\Delta x-1}{\\Delta x}\\right)+\\cos x\\left(\\frac{\\sin\\Delta x}{\\Delta x}\\right)\\right]\\\\&amp;=\\displaystyle \\sin x\\cdot\\lim_{\\Delta x\\to 0}\\frac{\\cos\\Delta x-1}{\\Delta x}+\\cos x\\cdot\\lim_{\\Delta x\\to 0}\\frac{\\sin\\Delta x}{\\Delta x}\\\\&amp;=(\\sin x)\\cdot 0+(\\cos x)\\cdot 1=\\cos x\\end{array}$$<\/p>\n<p><strong>\u3010\u5fae\u5206\u516c\u5f0f\u3011<\/strong>$$D\\cos{x}=-\\sin{x}$$.<br \/>\n<strong>\u3010\u8b49\u660e\u3011<\/strong>\u7576\u4f5c\u8b80\u8005\u7684\u7df4\u7fd2\u984c\u3002<\/p>\n<p><strong>\u3010\u4f8b13\u3011<\/strong>\u7531\u4e0a\u8ff0\u5169\u500b\u5fae\u5206\u516c\u5f0f\u8207\u5fae\u7a4d\u5206\u5b78\u6839\u672c\u5b9a\u7406\uff0c\u6211\u5011\u5c31\u5f97\u5230\uff1a<\/p>\n<p>$${\\int}_a^b\\cos{x}~dx=\\sin{b}-\\sin{a}$$ \u8207 $${\\int}_a^b\\sin{x}~dx=\\cos{a}-\\cos{b}$$<\/p>\n<p>\u9023\u7d50\uff1a<strong><a href=\"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=19664\" target=\"_blank\">\u5fae\u7a4d\u5206\u521d\u968e\uff0d\u6b77\u53f2\u767c\u5c55\u7684\u773c\u5149\uff0813\uff09\u4e00\u6cd5\u4e8c\u5ff5\u4e8c\u7fa9\u4e00\u7406<\/a><\/strong><\/p>\n<p>\u53c3\u8003\u6587\u737b\uff1a<\/p>\n<ol>\n<li>\u8521\u8070\u660e\uff1a\u5fae\u7a4d\u5206\u7684\u6b77\u53f2\u6b65\u9053\u3002\u4e09\u6c11\u66f8\u5c40\uff0c\u53f0\u5317\uff0c2009\u3002<\/li>\n<li>\u8521\u8070\u660e\uff1a\u6578\u5b78\u7684\u767c\u73fe\u8da3\u8ac7\uff0c\u7b2c\u4e8c\u7248\uff0c\u7b2c19\u7ae0\u3002\u4e09\u6c11\u66f8\u5c40\uff0c\u53f0\u5317\uff0c2010\u3002<\/li>\n<li>Edward\uff1a\u5fae\u7a4d\u5206\u767c\u5c55\u53f2\uff0c\u51e1\u7570\u51fa\u7248\u793e\uff0c\u6797\u8070\u6e90\u8b6f\u3002<\/li>\n<li>Simons\uff1aCalculus Gems, Brief Lives and Memorable Mathematics.McGraw-Hill, Inc.1992.<\/li>\n<li>Dunham\uff1aThe Calculus Gallery, Masterpieces from Newton to Lebesgue.Princeton University Press,2005.<\/li>\n<li>Toeplitz\uff1aThe Calculus,A Genetic Approach.The University of Chicago Press.1963<\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>\u5b8c\u6574\u9673\u8ff0\u5fae\u7a4d\u5206\u6839\u672c\u5b9a\u7406\uff0c\u4e26\u4ee5\u6b63\u5f26\u3001\u9918\u5f26\u51fd\u6578\u70ba\u4f8b\u8aaa\u660e\u4e4b\u3002\u5fae\u7a4d\u5206\u6700\u91cd\u8981\u4e14\u6700\u6838\u5fc3\u7684\u300c\u5fae\u7a4d\u5206\u5b78\u6839\u672c\u5b9a\u7406\u300d\u3002\u6c42\u5207\u7dda\u8207\u6c42\u9762\u7a4d\u5169\u8005\u8868\u9762\u4e0a\u5f88\u4e0d\u540c\uff0c\u5be6\u5247\u95dc\u4fc2\u5bc6\u5207\u3002<br \/>\n\u3010\u5b9a\u74067\u3011\uff08\u5fae\u7a4d\u5206\u5b78\u6839\u672c\u5b9a\u7406, the Fundamental Theorem of Calculus, FTC\uff09&#8230;.<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[233,111],"tags":[777],"class_list":["post-19656","post","type-post","status-publish","format-standard","hentry","category-math08","category-mathematics00","tag-777","loop-entry","cat-233","cat-111","no-thumbnail"],"views":5089,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/19656","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=19656"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/19656\/revisions"}],"predecessor-version":[{"id":88974,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/19656\/revisions\/88974"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=19656"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=19656"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=19656"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}