{"id":10584,"date":"2023-08-03T20:44:11","date_gmt":"2023-08-03T12:44:11","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=10584"},"modified":"2025-04-06T22:13:30","modified_gmt":"2025-04-06T14:13:30","slug":"%e5%8f%af%e5%af%bc%e3%80%81%e5%8f%af%e5%be%ae%e5%92%8c%e8%bf%9e%e7%bb%ad","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=10584","title":{"rendered":"\u53ef\u5bfc\u3001\u53ef\u5fae\u548c\u8fde\u7eed"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_84 ez-toc-wrap-center counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">\u76ee\u5f55<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" 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href=\"https:\/\/blog.iyatt.com\/?p=10584\/#%E6%8C%AF%E8%8D%A1%E9%97%B4%E6%96%AD%E7%82%B9\" >\u632f\u8361\u95f4\u65ad\u70b9<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"%E5%85%B3%E7%B3%BB\"><\/span>\u5173\u7cfb<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u53ef\u5bfc\uff08\u53ef\u5fae\uff09\u4e00\u5b9a\u8fde\u7eed\uff0c\u8fde\u7eed\u4e0d\u4e00\u5b9a\u53ef\u5bfc\uff08\u53ef\u5fae\uff09\uff0c\u4e0d\u8fde\u7eed\u4e00\u5b9a\u4e0d\u53ef\u5bfc\uff08\u4e0d\u53ef\u5fae\uff09<br \/>\n<img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/08\/image-1691299878354.png\" alt=\"\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 408px; --smush-placeholder-aspect-ratio: 408\/328;\" \/><\/p>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/08\/image-1691299812343.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 408px; --smush-placeholder-aspect-ratio: 408\/153;\" \/><\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E5%8F%AF%E5%AF%BC\"><\/span>\u53ef\u5bfc<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>y=f(x) \u5728 <code class=\"katex-inline\">x_0<\/code> \u7684\u90bb\u57df\u5185\u6709\u5b9a\u4e49\u4e14 <code class=\"katex-inline\">\\lim_{\\Delta x\\to0}\\frac{\\Delta x}{\\Delta y}=\\lim_{\\Delta x\\to0}\\frac{f(x_0+\\Delta x)-f(x_0)}{\\Delta x}<\/code> \u5b58\u5728\uff0c\u5219 f(x) \u5728 <code class=\"katex-inline\">x_0<\/code> \u5904\u53ef\u5bfc\u3002<br \/>\n<code class=\"katex-inline\">f'(x_0)=y'|_{x=x_0}=\\frac{dy}{dx}|_{x=x_0}=\\lim_{\\Delta x\\to0}\\frac{f(x_0+\\Delta x)-f(x_0)}{\\Delta x}=\\lim_{x\\to x_0}\\frac{f(x)-f(x_0)}{x-x_0}<\/code><\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E5%8F%AF%E5%BE%AE\"><\/span>\u53ef\u5fae<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/08\/image-1691065207584.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 303px; --smush-placeholder-aspect-ratio: 303\/257;\" \/><\/p>\n<p>y=f(x) \u5728 <code class=\"katex-inline\">x_0<\/code> \u7684\u90bb\u57df\u5185\u6709\u5b9a\u4e49\uff0c\u589e\u91cf <code class=\"katex-inline\">\\Delta y=f(x_0+\\Delta x)-f(x_0)<\/code> \u53ef\u4ee5\u8868\u793a\u4e3a <code class=\"katex-inline\">\\Delta y=A\\Delta x+o(\\Delta x)<\/code>\uff0c\u5176\u4e2d <code class=\"katex-inline\">\\Delta x\\to0<\/code><\/p>\n<p><code class=\"katex-inline\">A\\Delta x<\/code> \u5c31\u662f y=f(x) \u5728 <code class=\"katex-inline\">x_0<\/code> \u5904\u7684\u5fae\u5206\uff0c\u5373 <code class=\"katex-inline\">dy=df(x)=A\\Delta x<\/code><br \/>\n\u53ef\u5fae\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u5c31\u662f\u53ef\u5bfc\uff0c\u6709 <code class=\"katex-inline\">dy=f'(x_0)\\Delta x=f'(x_0)dx<\/code><\/p>\n<h1><span class=\"ez-toc-section\" id=\"%E8%BF%9E%E7%BB%AD\"><\/span>\u8fde\u7eed<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>y=f(x) \u5728 <code class=\"katex-inline\">x_0<\/code> \u7684\u90bb\u57df\u5185\u6709\u5b9a\u4e49\uff0c<code class=\"katex-inline\">\\lim_{x\\to x_0}f(x)=f(x_0)<\/code> \u65f6\u8fde\u7eed\uff0c\u5373\u5de6\u53f3\u6781\u9650\u5b58\u5728\u4e14\u7b49\u4e8e\u51fd\u6570\u503c\u5c31\u8fde\u7eed<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E7%AC%AC%E4%B8%80%E7%B1%BB%E9%97%B4%E6%96%AD%E7%82%B9\"><\/span>\u7b2c\u4e00\u7c7b\u95f4\u65ad\u70b9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u53ea\u6709\u4e24\u79cd<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E5%8F%AF%E5%8E%BB%E9%97%B4%E6%96%AD%E7%82%B9\"><\/span>\u53ef\u53bb\u95f4\u65ad\u70b9<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u5de6\u53f3\u6781\u9650\u76f8\u7b49\uff0c\u4f46\u4e0d\u7b49\u4e8e\u51fd\u6570\u503c<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E8%B7%B3%E8%B7%83%E9%97%B4%E6%96%AD%E7%82%B9\"><\/span>\u8df3\u8dc3\u95f4\u65ad\u70b9<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u5de6\u53f3\u6781\u9650\u4e0d\u76f8\u7b49<\/p>\n<h2><span class=\"ez-toc-section\" id=\"%E7%AC%AC%E4%BA%8C%E7%B1%BB%E9%97%B4%E6%96%AD%E7%82%B9\"><\/span>\u7b2c\u4e8c\u7c7b\u95f4\u65ad\u70b9<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5de6\u53f3\u6781\u9650\u81f3\u5c11\u6709\u4e00\u4e2a\u4e0d\u5b58\u5728\uff0c\u8fd9\u91cc\u53ea\u5217\u4e3e\u5e38\u89c1\u7684\u4e24\u79cd<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E6%97%A0%E7%A9%B7%E9%97%B4%E6%96%AD%E7%82%B9\"><\/span>\u65e0\u7a77\u95f4\u65ad\u70b9<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u5de6\u53f3\u6781\u9650\u6709\u4e00\u4e2a\u8d8b\u4e8e\u65e0\u7a77<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E6%8C%AF%E8%8D%A1%E9%97%B4%E6%96%AD%E7%82%B9\"><\/span>\u632f\u8361\u95f4\u65ad\u70b9<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u6709\u754c\uff0c\u4f46\u662f\u6ca1\u6709\u6781\u9650<\/p>\n<p>\u6bd4\u5982 <code class=\"katex-inline\">\\sin\\frac{1}{x}<\/code> \u5728 <code class=\"katex-inline\">x\\to0<\/code> \u65f6\uff0c\u503c\u4fdd\u6301\u5728 [-1,1] \u8303\u56f4\uff0c\u4f46\u662f\u6ca1\u6709\u6781\u9650\uff0c\u662f\u632f\u8361\u7684<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5173\u7cfb \u53ef\u5bfc\uff08\u53ef\u5fae\uff09\u4e00\u5b9a\u8fde\u7eed\uff0c\u8fde\u7eed\u4e0d\u4e00\u5b9a\u53ef\u5bfc\uff08\u53ef\u5fae\uff09\uff0c\u4e0d\u8fde\u7eed\u4e00\u5b9a\u4e0d\u53ef\u5bfc\uff08\u4e0d\u53ef\u5fae\uff09 \u53ef\u5bfc y=f(x) \u5728 x_0 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