{"id":12906,"date":"2024-01-12T21:17:27","date_gmt":"2024-01-12T13:17:27","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=12906"},"modified":"2025-08-22T09:55:05","modified_gmt":"2025-08-22T01:55:05","slug":"%e6%95%b0%e5%ad%a6%e4%b8%8e-python-1-%e9%ab%98%e7%ad%89%e6%95%b0%e5%ad%a6%ef%bc%88%e7%bc%96%e8%be%91%e4%b8%ad%ef%bc%89","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=12906","title":{"rendered":"\u6570\u5b66\u4e0e\u8ba1\u7b97\u673a\u3010\u9ad8\u7b49\u6570\u5b66\u3011"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 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\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#1_%E5%B7%A5%E5%85%B7\" >1 \u5de5\u5177<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#11_Python\" >1.1 Python<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#12_MATLAB\" >1.2 MATLAB<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#2_%E6%9E%81%E9%99%90\" >2 \u6781\u9650<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#21_%E4%BE%8B_1\" >2.1 \u4f8b 1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#22_%E4%BE%8B_2\" >2.2 \u4f8b 2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#22_%E4%BE%8B_3\" >2.2 \u4f8b 3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#24_%E4%BE%8B_4_%E6%97%A0%E7%A9%B7%E9%A1%B9\" >2.4 \u4f8b 4 \u65e0\u7a77\u9879<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#25_%E4%BE%8B_5\" >2.5 \u4f8b 5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#26_%E4%BE%8B6\" >2.6 \u4f8b6<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#3_%E5%AF%BC%E6%95%B0\" >3 \u5bfc\u6570<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#31_%E4%BE%8B_1\" >3.1 \u4f8b 1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#32_%E4%BE%8B_2_%E8%AE%A1%E7%AE%97%E5%AF%BC%E6%95%B0%E5%80%BC%EF%BC%88%E5%90%AB%E5%AE%9A%E4%B9%89%E6%B1%82%E5%AF%BC%EF%BC%89\" >3.2 \u4f8b 2 \u8ba1\u7b97\u5bfc\u6570\u503c\uff08\u542b\u5b9a\u4e49\u6c42\u5bfc\uff09<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#33_%E4%BE%8B_3\" >3.3 \u4f8b 3<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#4_%E5%81%8F%E5%AF%BC%E6%95%B0\" >4 \u504f\u5bfc\u6570<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#41_%E4%BE%8B_1\" >4.1 \u4f8b 1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#42_%E4%BE%8B_2\" >4.2 \u4f8b 2<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#5_%E6%96%B9%E5%90%91%E5%AF%BC%E6%95%B0\" >5 \u65b9\u5411\u5bfc\u6570<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#51_%E4%BE%8B_1\" >5.1 \u4f8b 1<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#6_%E6%A2%AF%E5%BA%A6\" >6 \u68af\u5ea6<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#61_%E4%BE%8B_1_%E6%A2%AF%E5%BA%A6%E4%B8%8B%E9%99%8D%E6%B3%95%E6%B1%82%E6%9C%80%E5%B0%8F%E5%80%BC_%E2%80%93_%E7%BB%98%E5%9B%BE\" >6.1 \u4f8b 1 \u68af\u5ea6\u4e0b\u964d\u6cd5\u6c42\u6700\u5c0f\u503c &#8211; \u7ed8\u56fe<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#7_%E7%A7%AF%E5%88%86\" >7 \u79ef\u5206<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#71_%E4%BE%8B_1\" >7.1 \u4f8b 1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#72_%E4%BE%8B_2\" >7.2 \u4f8b 2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#73_%E4%BE%8B_3\" >7.3 \u4f8b 3<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#74_%E4%BE%8B_4\" >7.4 \u4f8b 4<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#75_%E4%BE%8B_5_%E5%9F%BA%E4%BA%8E%E7%A7%AF%E5%88%86%E5%AE%9A%E4%B9%89%EF%BC%88%E9%BB%8E%E6%9B%BC%E5%92%8C%EF%BC%89%E8%AE%A1%E7%AE%97\" >7.5 \u4f8b 5 \u57fa\u4e8e\u79ef\u5206\u5b9a\u4e49\uff08\u9ece\u66fc\u548c\uff09\u8ba1\u7b97<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#8_%E6%B3%B0%E5%8B%92%E5%85%AC%E5%BC%8F\" >8 \u6cf0\u52d2\u516c\u5f0f<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#81_%E4%BE%8B_1_ex_%E9%BA%A6%E5%85%8B%E5%8A%B3%E6%9E%97%E5%B1%95%E5%BC%80\" >8.1 \u4f8b 1 e^x \u9ea6\u514b\u52b3\u6797\u5c55\u5f00<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#82_%E4%BE%8B_2_sin_x_%E9%BA%A6%E5%85%8B%E5%8A%B3%E6%9E%97%E5%B1%95%E5%BC%80\" >8.2 \u4f8b 2 sin x \u9ea6\u514b\u52b3\u6797\u5c55\u5f00<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#9_%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E4%B9%98%E5%AD%90%E6%B3%95\" >9 \u62c9\u683c\u6717\u65e5\u4e58\u5b50\u6cd5<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#91_%E4%BE%8B_1\" >9.1 \u4f8b 1<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#92_%E4%BE%8B_2\" >9.2 \u4f8b 2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-34\" href=\"https:\/\/blog.iyatt.com\/?p=12906\/#93_%E4%BE%8B_3\" >9.3 \u4f8b 3<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"1_%E5%B7%A5%E5%85%B7\"><\/span>1 \u5de5\u5177<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"11_Python\"><\/span>1.1 Python<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>\u57fa\u7840\u5de5\u5177<\/strong><\/p>\n<ul>\n<li>Python 3.11.2<\/li>\n<\/ul>\n<p><strong>\u6570\u5b66\u6a21\u5757<\/strong><\/p>\n<ul>\n<li>SymPy 1.12<\/li>\n<li>SciPy 1.11.4<\/li>\n<li>NumPy 1.26.3<\/li>\n<\/ul>\n<p>Scientific Python\uff08SciPy\uff09\u662f\u4e00\u4e2a\u57fa\u4e8e NumPy \u7684\u6570\u503c\u8ba1\u7b97\u5e93\uff0c\u800c Symbolic Python\uff08SymPy\uff09 \u662f\u4e00\u4e2a\u7b26\u53f7\u8ba1\u7b97\u5e93\u3002<\/p>\n<p><strong>\u4ea4\u4e92\u5de5\u5177<\/strong><\/p>\n<ul>\n<li>Jupyter Notebook 7.0.6<\/li>\n<\/ul>\n<p>JN \u5177\u6709\u7b14\u8bb0\u672c\u529f\u80fd\uff0c\u53ef\u4ee5\u4f7f\u7528 Markdown \u8bed\u6cd5\uff0c\u652f\u6301 LaTex \u516c\u5f0f\uff0c\u540c\u65f6\u53ef\u4ee5\u8fd0\u884c Python \u4ee3\u7801\uff0c\u8bb0\u5f55\u8fd0\u884c\u7ed3\u679c\uff0c\u652f\u6301\u5bf9\u5f0f\u5b50\u4ee5\u53ca\u5206\u6570\u7684\u663e\u793a\uff0c\u800c\u4e0d\u662f\u7ebf\u6027\u663e\u793a\u3002<br \/>\n\u53ef\u4ee5\u8fd0\u884c Jupyter Notebook\uff0c\u7136\u540e\u7f51\u9875\u8bbf\u95ee\uff0c\u6216\u8005\u4f7f\u7528 VScode \u4e5f\u53ef\u4ee5\u6253\u5f00 .ipynb \u6587\u4ef6\u3002<\/p>\n<p><strong>\u6570\u636e\u53ef\u89c6\u5316<\/strong><\/p>\n<ul>\n<li>Matplotlib 3.8.2<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"12_MATLAB\"><\/span>1.2 MATLAB<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>MATLAB 2023b<\/li>\n<\/ul>\n<h1><span class=\"ez-toc-section\" id=\"2_%E6%9E%81%E9%99%90\"><\/span>2 \u6781\u9650<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"21_%E4%BE%8B_1\"><\/span>2.1 \u4f8b 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6c42 <code class=\"katex-inline\">\\lim_{x\\to\\infty}\\frac{\\sin x}{x}<\/code><\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx = sy.Symbol(&#039;x&#039;) # \u5b9a\u4e49\u7b26\u53f7\uff0c\u7528 Python \u53d8\u91cf x \u8868\u793a\u7b26\u53f7 x\nf = sy.sin(x) \/ x # \u8868\u8fbe\u5f0f\nsy.limit(f, x, sy.oo) # \u8ba1\u7b97\u6781\u9650<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705199835405.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 63px; --smush-placeholder-aspect-ratio: 63\/44;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = sin(x) \/ x;\nlimit(f, x, inf)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705199875805.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 76px; --smush-placeholder-aspect-ratio: 76\/59;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"22_%E4%BE%8B_2\"><\/span>2.2 \u4f8b 2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6c42 <code class=\"katex-inline\">\\lim_{x\\to1}\\frac{x^2-1}{x-1}<\/code><\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx = sy.Symbol(&#039;x&#039;)\nf = (x**2 - 1) \/ (x - 1)\nsy.limit(f, x, 1)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705199901059.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 65px; --smush-placeholder-aspect-ratio: 65\/39;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = (x^2 - 1) \/ (x - 1);\nlimit(f, x, 1)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705199939958.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 74px; --smush-placeholder-aspect-ratio: 74\/54;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"22_%E4%BE%8B_3\"><\/span>2.2 \u4f8b 3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6c42 <code class=\"katex-inline\">\\lim_{x\\to0}\\frac{\\sin x}{3x+x^3}<\/code><\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx = sy.Symbol(&#039;x&#039;)\nf = sy.sin(x) \/ (3 * x + x**3)\nsy.limit(f, x, 0)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705199963373.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 53px; --smush-placeholder-aspect-ratio: 53\/49;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = sin(x) \/ (3 * x + x^3);\nlimit(f, x, 0)\n<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"24_%E4%BE%8B_4_%E6%97%A0%E7%A9%B7%E9%A1%B9\"><\/span>2.4 \u4f8b 4 \u65e0\u7a77\u9879<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6c42<code class=\"katex-inline\">\\lim_{n\\to\\infty}\\frac{1}{n^2}+\\frac{2}{n^2}+\\cdots+\\frac{n}{n^2}<\/code><\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nn, t = sy.symbols(&#039;n t&#039;)\nf = sy.summation(t \/ n**2, (t, 1, n))\nprint(&quot;\u8868\u8fbe\u5f0f\u4e3a\uff1a&quot;)\ndisplay(f)\nprint(&quot;\u8ba1\u7b97\u6781\u9650\u7ed3\u679c\u4e3a\uff1a&quot;)\nsy.limit(f, n, sy.oo)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705305392063.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 135px; --smush-placeholder-aspect-ratio: 135\/141;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-mtlab\">syms n t;\n\nf = symsum(t \/ n^2, t, 1, n)\nlimit(f, n, inf)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705198735465.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 114px; --smush-placeholder-aspect-ratio: 114\/168;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"25_%E4%BE%8B_5\"><\/span>2.5 \u4f8b 5<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6c42<code class=\"katex-inline\">\\lim_{x\\to1}\\sin(\\ln x))<\/code><\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx = sy.symbols(&#039;x&#039;)\nf = sy.sin(sy.ln(x))\nsy.limit(f, x, 1)<\/code><\/pre>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = sin(log(x))\nlimit(f, x, 1)<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"26_%E4%BE%8B6\"><\/span>2.6 \u4f8b6<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6c42<code class=\"katex-inline\">\\lim_{x\\to8}\\frac{\\sqrt[3]{x}-2}{x-8}<\/code><\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx = sy.symbols(&#039;x&#039;)\nf = (sy.root(x, 3) - 2) \/ (x - 8)\nsy.limit(f, x, 8)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705217546639.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 63px; --smush-placeholder-aspect-ratio: 63\/60;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = (x^(1\/3) - 2) \/ (x - 8);\nlimit(f, x, 8)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705217585692.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 74px; --smush-placeholder-aspect-ratio: 74\/92;\" \/><\/p>\n<h1><span class=\"ez-toc-section\" id=\"3_%E5%AF%BC%E6%95%B0\"><\/span>3 \u5bfc\u6570<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u4eba\u5de5\u6c42\u5bfc\u65b9\u6cd5\u53c2\u8003\uff1a<a href=\"https:\/\/blog.iyatt.com\/?p=7986\">https:\/\/blog.iyatt.com\/?p=7986<\/a><\/p>\n<h2><span class=\"ez-toc-section\" id=\"31_%E4%BE%8B_1\"><\/span>3.1 \u4f8b 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5bf9 <code class=\"katex-inline\">y=\\arcsin\\sqrt{\\sin x}<\/code> \u6c42\u5bfc<\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\nfrom sympy.abc import x # \u8fd9\u4e2a\u6a21\u5757\u5b9a\u4e49\u4e86\u5e38\u7528\u7b26\u53f7\u53d8\u91cf\uff0c\u6db5\u76d6\u5927\u5c0f\u5199\u5b57\u6bcd\n\nf = sy.asin(sy.sqrt(sy.sin(x)))\nsy.diff(f) # \u6c42\u5bfc<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705066120117.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 201px; --smush-placeholder-aspect-ratio: 201\/97;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = asin(sqrt(sin(x)));\ndiff(f)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705199457134.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 180px; --smush-placeholder-aspect-ratio: 180\/97;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"32_%E4%BE%8B_2_%E8%AE%A1%E7%AE%97%E5%AF%BC%E6%95%B0%E5%80%BC%EF%BC%88%E5%90%AB%E5%AE%9A%E4%B9%89%E6%B1%82%E5%AF%BC%EF%BC%89\"><\/span>3.2 \u4f8b 2 \u8ba1\u7b97\u5bfc\u6570\u503c\uff08\u542b\u5b9a\u4e49\u6c42\u5bfc\uff09<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><code class=\"katex-inline\">f(x)=x^5<\/code>\uff0c\u8ba1\u7b97f'(2)<\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx = sy.symbols(&#039;x&#039;)\nf = x**5\nsy.diff(f).evalf(subs={x:2})<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705205314736.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 66px; --smush-placeholder-aspect-ratio: 66\/38;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = x^5;\ndf = diff(f);\nsubs(df, x, 2)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705205338054.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 83px; --smush-placeholder-aspect-ratio: 83\/56;\" \/><\/p>\n<p>\u57fa\u4e8e\u5b9a\u4e49<\/p>\n<pre><code class=\"language-py\">def f(x):\n    return x**5\n\ndef derivative(x, h):\n    return (f(x + h) - f(x)) \/ h\n\nprint(derivative(2, 1e-1))\nprint(derivative(2, 1e-2))\nprint(derivative(2, 1e-3))\nprint(derivative(2, 1e-4))\nprint(derivative(2, 1e-5))\nprint(derivative(2, 1e-6))<\/code><\/pre>\n<p>\u53d6\u7684\u4e24\u4e2a\u70b9\u8d8a\u903c\u8fd1\uff0c\u8fd1\u4f3c\u503c\u8d8a\u63a5\u8fd1<br \/>\n<img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705287002661.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 148px; --smush-placeholder-aspect-ratio: 148\/107;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"33_%E4%BE%8B_3\"><\/span>3.3 \u4f8b 3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5bf9 <code class=\"katex-inline\">y=x^4-2x^3+5\\sin x+\\ln3<\/code> \u6c42\u5bfc<\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx = sy.symbols(&#039;x&#039;)\nf = x**4 - 2 * x**3 + 5 * sy.sin(x) + sy.ln(3)\nsy.diff(f)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705199762662.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 204px; --smush-placeholder-aspect-ratio: 204\/49;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = x^4 - 2 * x^3 + 5 * sin(x) +log(3);\ndiff(f)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705199796766.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 222px; --smush-placeholder-aspect-ratio: 222\/74;\" \/><\/p>\n<h1><span class=\"ez-toc-section\" id=\"4_%E5%81%8F%E5%AF%BC%E6%95%B0\"><\/span>4 \u504f\u5bfc\u6570<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"41_%E4%BE%8B_1\"><\/span>4.1 \u4f8b 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6c42 <code class=\"katex-inline\">f(x,y)=x^2+3xy+y^2<\/code>\u5728\u70b9(1,2)\u5904\u7684\u504f\u5bfc\u6570<\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\nfrom sympy.abc import x, y\n\nf = x**2 + 3 * x * y + y**2\n\nfx = sy.diff(f, x) # \u6c42 x \u504f\u5bfc\ndisplay(fx)\n\nfy = sy.diff(f, y) # \u6c42 y \u504f\u5bfc\ndisplay(fy)\n\nfxv = fx.evalf(subs={x:1, y:2}) # \u6c42 x \u504f\u5bfc\u503c\ndisplay(fxv)\n\nfyv = fy.evalf(subs={x:1, y:2}) # \u6c42 y \u504f\u5bfc\u503c\ndisplay(fyv)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705111068778.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 107px; --smush-placeholder-aspect-ratio: 107\/148;\" \/><\/p>\n<p>SciPy<\/p>\n<pre><code class=\"language-py\">from scipy.optimize import approx_fprime\n\ndef f(xy):\n    x, y = xy\n    return x**2 + 3 * x * y + y**2\n\napprox_fprime([1,2], f, 1e-6)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705159438740.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 228px; --smush-placeholder-aspect-ratio: 228\/43;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x y;\n\nf = x^2 + 3 * x * y + y^2;\n\nfx = diff(f, x)\nfy = diff(f, y)\n\nsubs(fx, [x, y], [1, 2])\nsubs(fy, [x, y], [1, 2])<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705208278232.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 122px; --smush-placeholder-aspect-ratio: 122\/153;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"42_%E4%BE%8B_2\"><\/span>4.2 \u4f8b 2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5df2\u77e5<code class=\"katex-inline\">z=(3x^2+y^2)^{4x+2y}<\/code>\uff0c\u6c42\u5728\u70b9(1,2)\u5904\u7684\u504f\u5bfc\u6570<\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx, y = sy.symbols(&quot;x y&quot;)\nf = (3 * x**2 + y**2)**(4 * x + 2 * y)\n\nfx = sy.diff(f, x)\ndisplay(fx)\n\nfy = sy.diff(f, y)\ndisplay(fy)\n\nfxv = fx.evalf(subs={x:1, y:2})\ndisplay(fxv)\n\nfyv = fy.evalf(subs={x:1, y:2})\ndisplay(fyv)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705159248718.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 404px; --smush-placeholder-aspect-ratio: 404\/188;\" \/><\/p>\n<p>SciPy<\/p>\n<pre><code class=\"language-py\">from scipy.optimize import approx_fprime\n\ndef f(xy):\n    x, y = xy\n    return (3 * x**2 + y**2)**(4 * x + 2 * y)\n\napprox_fprime([1, 2], f, 1e-100)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705189459508.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 288px; --smush-placeholder-aspect-ratio: 288\/41;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">format longG % \u4e0d\u4f7f\u7528\u79d1\u5b66\u8ba1\u6570\u6cd5\u663e\u793a\uff0c\u652f\u6301\u6700\u591a 15 \u4f4d\u6709\u6548\u6570\u5b57\nsyms x y;\n\nf = (3 * x^2 + y^2)^(4 * x + 2 * y);\n\nfx = diff(f, x)\nfy = diff(f, y)\n\neval(subs(fx, [x, y], [1, 2]))\neval(subs(fy, [x, y], [1, 2]))<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705209097706.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 237px; --smush-placeholder-aspect-ratio: 237\/219;\" \/><\/p>\n<h1><span class=\"ez-toc-section\" id=\"5_%E6%96%B9%E5%90%91%E5%AF%BC%E6%95%B0\"><\/span>5 \u65b9\u5411\u5bfc\u6570<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"51_%E4%BE%8B_1\"><\/span>5.1 \u4f8b 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u6c42\u51fd\u6570 <code class=\"katex-inline\">z=xe^{2y}<\/code>\u5728\u70b9P(1,0)\u5904\u6cbf\u4ece\u70b9P(1,0)\u5230\u70b9Q(2,1)\u65b9\u5411\u7684\u65b9\u5411\u5bfc\u6570<\/p>\n<p>SymPy<\/p>\n<p>\u8ba1\u7b97\u89d2\u5ea6<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nP = sy.Point(1, 0)\nQ = sy.Point(2, -1)\nPQ = sy.Line(P, Q).direction # \u76f4\u7ebf\u7684\u5411\u91cf\nangle_rad_xol = sy.atan(PQ.y \/ PQ.x) # \u4ece x \u8f74\u8f6c\u5230 l \u7684\u89d2\u5ea6\ndisplay(angle_rad_xol)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705115218647.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 74px; --smush-placeholder-aspect-ratio: 74\/66;\" \/><\/p>\n<p>\u8ba1\u7b97\u65b9\u5411\u5bfc\u6570<\/p>\n<pre><code class=\"language-py\">\nfrom sympy.abc import x, y\n\nf = x * sy.E**(2 * y)\n\nfxv = sy.diff(f, x).evalf(subs={x:1, y:0})\ndisplay(fxv) # x \u504f\u5bfc\u503c\n\nfyv = sy.diff(f, y).evalf(subs={x:1, y:0})\ndisplay(fyv) # y \u504f\u5bfc\u503c\n\nflv = fxv * sy.cos(angle_rad_xol) + fyv * sy.sin(angle_rad_xol)\nsy.nsimplify(flv) # \u7b80\u5316\uff0c\u4e0d\u7136\u8fd9\u91cc\u53ef\u80fd\u663e\u793a\u6210 -0.5\\sqrt(2)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705115396716.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 137px; --smush-placeholder-aspect-ratio: 137\/131;\" \/><\/p>\n<p>SciPy\u3001NumPy<br \/>\n\u8ba1\u7b97\u89d2\u5ea6<\/p>\n<pre><code class=\"language-py\">import numpy as np\n\nP = np.array([1, 0])\nQ = np.array([2, -1])\nangle_rad_xol = np.arctan2(Q[1] - P[1], Q[0] - P[0])\ndisplay(angle_rad_xol)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705217298511.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 159px; --smush-placeholder-aspect-ratio: 159\/31;\" \/><br \/>\n\u8ba1\u7b97\u65b9\u5411\u5bfc\u6570<\/p>\n<pre><code class=\"language-py\">from scipy.optimize import approx_fprime\n\ndef f(xy):\n    x, y = xy\n    return x * np.exp(2 * y)\n\ngrad = approx_fprime([1, 0], f, 1e-6) # \u8ba1\u7b97\u68af\u5ea6\nnp.dot(grad, np.array([np.cos(angle_rad_xol), np.sin(angle_rad_xol)]))<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705217387832.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 177px; --smush-placeholder-aspect-ratio: 177\/39;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x, y;\n\nP = [1, 0];\nQ = [2, -1];\n\nangle_rad_xol = atan((Q(2) - P(2)) \/ (Q(1) - P(1)));\n\nf = x * exp(2 * y);\n\nfx = diff(f, x);\nfy = diff(f, y);\n\nfxv = subs(fx, [x, y], [1, 0]);\nfyv = subs(fy, [x, y], [1, 0]);\n\nflv = fxv * cos(angle_rad_xol) + fyv * sin(angle_rad_xol);\neval(flv)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705213641617.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 202px; --smush-placeholder-aspect-ratio: 202\/70;\" \/><\/p>\n<h1><span class=\"ez-toc-section\" id=\"6_%E6%A2%AF%E5%BA%A6\"><\/span>6 \u68af\u5ea6<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u68af\u5ea6\u662f\u4e00\u4e2a\u5411\u91cf\uff0c\u6570\u503c\u4e0a\u7531\u504f\u5bfc\u6570\u7ec4\u6210\u3002\u5728\u4e00\u4e2a\u66f2\u9762\u4e0a\u7684\u67d0\u70b9\u5904\uff0c\u68af\u5ea6\u65b9\u5411\u7684\u53d8\u5316\u7387\u6700\u5feb\u3002<\/p>\n<h2><span class=\"ez-toc-section\" id=\"61_%E4%BE%8B_1_%E6%A2%AF%E5%BA%A6%E4%B8%8B%E9%99%8D%E6%B3%95%E6%B1%82%E6%9C%80%E5%B0%8F%E5%80%BC_%E2%80%93_%E7%BB%98%E5%9B%BE\"><\/span>6.1 \u4f8b 1 \u68af\u5ea6\u4e0b\u964d\u6cd5\u6c42\u6700\u5c0f\u503c &#8211; \u7ed8\u56fe<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><code class=\"katex-inline\">f(x,y)=x-y+2x^2+2xy+y^2<\/code>\uff0c\u521d\u503c\u4e3a(2,2)\uff0c\u6c42\u6700\u5c0f\u503c\u70b9<\/p>\n<p>\u6c42\u89e3\u5b9e\u73b0<\/p>\n<pre><code class=\"language-py\">%matplotlib widget\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\n\n# \u539f\u51fd\u6570\ndef f(x, y):\n    return x - y + 2 * x**2 + 2 * x * y + y**2\n\n# \u5173\u4e8e x \u7684\u504f\u5bfc\u6570\ndef fx(x, y):\n    return 1 + 4 * x + 2 * y\n\n# \u5173\u4e8e y \u7684\u504f\u5bfc\u6570\ndef fy(x, y):\n    return -1 + 2 * x + 2 * y\n\n# x = np.linspace(-2, 2, 100)\n# y = np.linspace(-2, 2, 100)\n# X, Y = np.meshgrid(x, y)\nX, Y = np.mgrid[-2:2:100j, -2:2:100j] # \u4e5f\u53ef\u4ee5\u4f7f\u7528\u4e0a\u9762\u7684\u5199\u6cd5\u751f\u6210\u7f51\u683c\u5750\u6807\u77e9\u9635\nZ = f(X, Y)\n\nfig = plt.figure()\nax = fig.add_subplot(111, projection=&#039;3d&#039;)\n\nax.plot_surface(X, Y, Z, cmap=&#039;rainbow&#039;) # \u7ed8\u5236\u539f\u51fd\u6570\n\n# \u5750\u6807\u8f74\u6807\u7b7e\nax.set_xlabel(&#039;x&#039;)\nax.set_ylabel(&#039;y&#039;)\nax.set_zlabel(&#039;z&#039;)\n\n# \u68af\u5ea6\u4e0b\u964d\n###########\n\n# \u6b65\u957f\nstep = 0.0008\n\n# \u8d77\u59cb\u70b9\nnew_x = 2\nnew_y = 2\n\nx = new_x\ny = new_y\n\nxs = [x]\nys = [y]\nzs = [f(x, y)]\n\nOver = False\n\nwhile Over == False:\n    new_x -= step * fx(x, y)\n    new_y -= step * fy(x, y)\n\n    if f(x, y) - f(new_x, new_y) &lt; 7e-9:\n        Over = True\n\n    x = new_x\n    y = new_y\n\n    xs.append(x)\n    ys.append(y)\n    zs.append(f(x, y))\n\nax.plot(xs, ys, zs, color=&#039;k&#039;) # \u7ed8\u5236\u8def\u5f84\nprint(&#039;({},{},{})&#039;.format(xs[-1], ys[-1], zs[-1])) # \u6700\u5c0f\u503c\u70b9<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705555422208.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 680px; --smush-placeholder-aspect-ratio: 680\/661;\" \/><br \/>\n\u56fe\u4e2d\u66f2\u9762\u91cc\u7684\u90a3\u6761\u9ed1\u7ebf\u5c31\u662f\u4ece(2,2)\u6700\u5feb\u8d70\u5230\u66f2\u9762\u6700\u4f4e\u70b9\u7684\u8def\u5f84<br \/>\n\u6c42\u539f\u51fd\u6570\u4e00\u9636\u504f\u5bfc\u4e3a 0 \u7684\u70b9\u53ef\u4ee5\u5f97\u5230(x,y)~(-1,1.5)\uff0c\u5373\u5728\u8fd9\u4e2a\u4f4d\u7f6e\u53ef\u80fd\u662f\u4e00\u4e2a\u6781\u503c\uff0c\u548c\u7ed8\u56fe\u627e\u51fa\u7684\u70b9\u5339\u914d\uff0c\u8bf4\u660e\u8fd9\u4e2a\u70b9\u5c31\u662f\u8981\u627e\u7684\u70b9\uff0c\u53ea\u662f\u56e0\u4e3a\u7ed8\u56fe\u6240\u53d6\u6b65\u957f\u7684\u5f71\u54cd\uff0c\u7ed8\u56fe\u627e\u51fa\u7684\u70b9\u662f\u4e00\u4e2a\u8fd1\u4f3c\u503c\u3002\u5982\u679c\u6b65\u957f\u65e0\u9650\u7684\u5c0f\uff0c\u90a3\u4e48\u6700\u7ec8\u5c31\u662f\u4e00\u4e2a\u65e0\u9650\u903c\u8fd1\u7684\u6700\u5c0f\u503c\u70b9\u3002<\/p>\n<h1><span class=\"ez-toc-section\" id=\"7_%E7%A7%AF%E5%88%86\"><\/span>7 \u79ef\u5206<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u4eba\u5de5\u8ba1\u7b97\u53c2\u8003\uff1a<a href=\"https:\/\/blog.iyatt.com\/?p=11227\">https:\/\/blog.iyatt.com\/?p=11227<\/a><\/p>\n<h2><span class=\"ez-toc-section\" id=\"71_%E4%BE%8B_1\"><\/span>7.1 \u4f8b 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u8ba1\u7b97<code class=\"katex-inline\">\\int_0^3\\cos^2(e^x)dx<\/code><\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx = sy.symbols(&#039;x&#039;)\nf = sy.cos(sy.E**x)**2\n\nindefinite_integral = sy.integrate(f, x) # \u4e0d\u5b9a\u79ef\u5206\ndisplay(indefinite_integral)\n\ndefinite_integral = indefinite_integral.subs(x, 3) - indefinite_integral.subs(x, 0) # \u5b9a\u79ef\u5206\uff08\u725b\u987f-\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f\uff09\ndisplay(definite_integral)\ndisplay(sy.N(definite_integral)) # Ci \u662f\u4f59\u5f26\u79ef\u5206\u51fd\u6570\uff0c\u8fd9\u91cc\u53ef\u4ee5\u8f6c\u4e3a\u6570\u503c\u7684\n\ndisplay(\n    sy.integrate(f, (x, 0, 3)) # \u76f4\u63a5\u8ba1\u7b97\u5b9a\u79ef\u5206\n)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705245819409.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 226px; --smush-placeholder-aspect-ratio: 226\/200;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = cos(exp(x))^2;\nindefinite_integral = int(f, x) % \u8ba1\u7b97\u4e0d\u5b9a\u79ef\u5206\n\ndefinite_integral1 = subs(indefinite_integral, x, 3) - subs(indefinite_integral, x, 0) % \u5b9a\u79ef\u5206\uff08\u725b\u987f-\u83b1\u5e03\u5c3c\u8328\u516c\u5f0f\uff09\n\ndefinite_integral2 = int(f, x, 0, 3) % \u542b\u8868\u8fbe\u5f0f\u7684\u7ed3\u679c\neval(definite_integral2) % \u8f6c\u4e3a\u6570\u503c\u7684\u7ed3\u679c<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705246021557.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 222px; --smush-placeholder-aspect-ratio: 222\/272;\" \/><\/p>\n<p>SciPy<\/p>\n<pre><code class=\"language-py\">from scipy.integrate import quad\nimport numpy as np\n\nf = lambda x: np.cos(np.exp(x))**2\ndisplay(quad(f, 0, 3)) # \u663e\u793a\u7ed3\u679c\u4e3a\u79ef\u5206\u503c\u548c\u8bef\u5dee<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705246311590.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 334px; --smush-placeholder-aspect-ratio: 334\/33;\" \/><\/p>\n<p>\u57fa\u4e8e\u5b9a\u4e49\u6c42\u8fd1\u4f3c\u503c<\/p>\n<pre><code class=\"language-py\">import numpy as np\n\na = 0 # \u4e0a\u9650\nb = 3 # \u4e0b\u9650\n\ndef f(x):\n    return np.cos(np.exp(x))**2\n\ndef trape(n):\n    sum = 0\n    x1 = a\n    step = (b - a) \/ n # \u6b65\u957f\n    for i in range(n):\n        x2 = a + i * step\n        sum += (f(x1) + f(x2)) * step \/ 2 # \u7d2f\u52a0\u5c0f\u68af\u5f62\n        x1 = x2\n    return sum\n\nprint(trape(10))\nprint(trape(100))\nprint(trape(1000))\nprint(trape(10000))\nprint(trape(100000))<\/code><\/pre>\n<p>\u6b65\u957f\u8d8a\u5c0f\uff0c\u5c0f\u68af\u5f62\u8d8a\u5c0f\uff0c\u8ba1\u7b97\u7ed3\u679c\u5c31\u8d8a\u63a5\u8fd1<br \/>\n<img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705286200349.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 155px; --smush-placeholder-aspect-ratio: 155\/100;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"72_%E4%BE%8B_2\"><\/span>7.2 \u4f8b 2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx, y = sy.symbols(&#039;x y&#039;)\nf = sy.E**(-x**2 - y**2)\ndisplay(sy.integrate(f, x, y)) # \u4e0d\u5b9a\u79ef\u5206\ndefinite_integral = sy.integrate(f, (x, 0, 10), (y, 0, 10)) # \u5b9a\u79ef\u5206\ndisplay(definite_integral)\ndisplay(sy.N(definite_integral)) # \u6570\u503c\u8fd1\u4f3c\u503c<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705285238781.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 165px; --smush-placeholder-aspect-ratio: 165\/139;\" \/><\/p>\n<p>SciPy<\/p>\n<pre><code class=\"language-py\">import numpy as np\nfrom scipy.integrate import dblquad\n\ndef f(x, y):\n    return np.exp(-x**2 - y**2)\n\ndblquad(f, 0, 10, lambda y: 0, lambda y: 10) # y \u7684\u4e0a\u4e0b\u9650\u5efa\u8bae\u5b9a\u4e49\u51fd\u6570\u6216\u8005\u4f7f\u7528 lambda \u8868\u8fbe\u5f0f<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705285264382.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 343px; --smush-placeholder-aspect-ratio: 343\/37;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x y;\n\nf = exp(-x^2 - y^2);\nint(f, x, y) % \u4e0d\u5b9a\u79ef\u5206\n\nf = @(x, y) exp(-x.^2 - y.^2);\nintegral2(f, 0, 10, 0, 10) % \u5b9a\u79ef\u5206<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705285289280.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 211px; --smush-placeholder-aspect-ratio: 211\/142;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"73_%E4%BE%8B_3\"><\/span>7.3 \u4f8b 3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><code class=\"katex-inline\">\\int_1^2(x^2+\\frac{1}{x^4})dx<\/code><\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx = sy.symbols(&#039;x&#039;)\nf = x**2 + 1 \/ x**4\ndi = sy.integrate(f, (x, 1, 2))\ndisplay(di)\nsy.N(di)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705290540924.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 62px; --smush-placeholder-aspect-ratio: 62\/85;\" \/><\/p>\n<p>SciPy<\/p>\n<pre><code class=\"language-py\">from scipy.integrate import quad\n\ndef f(x):\n    return x**2 + 1 \/ x**4\n\nquad(f, 1, 2)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705290565518.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 233px; --smush-placeholder-aspect-ratio: 233\/37;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = x^2 + 1 \/ x^4;\ndi = int(f, x, 1, 2);\ndisp(di)\ndisp(vpa(di)) % \u8f6c\u5c0f\u6570<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705290587329.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 86px; --smush-placeholder-aspect-ratio: 86\/102;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"74_%E4%BE%8B_4\"><\/span>7.4 \u4f8b 4<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><code class=\"katex-inline\">\\int_{-1}^0\\frac{3x^4+3x^2+1}{x^1+1}dx<\/code><\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx = sy.symbols(&#039;x&#039;)\nf = (3 * x**4 + 3 * x**2 + 1) \/ (x**2 + 1)\ndefinite_integral = sy.integrate(f, (x, -1, 0))\n\ndisplay(definite_integral)\ndisplay(sy.N(definite_integral))<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705302695988.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 148px; --smush-placeholder-aspect-ratio: 148\/70;\" \/><\/p>\n<p>SciPy<\/p>\n<pre><code class=\"language-py\">from scipy.integrate import quad\n\ndef f(x):\n    return (3 * x**4 + 3 * x**2 + 1) \/ (x**2 + 1)\n\nquad(f, -1, 0)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705304874574.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 340px; --smush-placeholder-aspect-ratio: 340\/33;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = (3 * x^4 + 3 * x^2 + 1) \/ (x^2 + 1);\ndefinite_integral = int(f, x, -1, 0)\nvpa(definite_integral)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705304847078.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 310px; --smush-placeholder-aspect-ratio: 310\/117;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"75_%E4%BE%8B_5_%E5%9F%BA%E4%BA%8E%E7%A7%AF%E5%88%86%E5%AE%9A%E4%B9%89%EF%BC%88%E9%BB%8E%E6%9B%BC%E5%92%8C%EF%BC%89%E8%AE%A1%E7%AE%97\"><\/span>7.5 \u4f8b 5 \u57fa\u4e8e\u79ef\u5206\u5b9a\u4e49\uff08\u9ece\u66fc\u548c\uff09\u8ba1\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><code class=\"katex-inline\">\\lim_{n\\to\\infty}\\frac{1^p+2^p+\\cdots+n^p}{n^{p+1}}\uff0c(p\\gt0)<\/code><\/p>\n<p>\u8fd9\u4e2a\u6211\u7528 SymPy \u548c MATLAB \u628a\u5f0f\u5b50\u8868\u793a\u51fa\u6765\u4e86\uff0c\u4f46\u662f\u90fd\u65e0\u6cd5\u8ba1\u7b97\uff0cSymPy \u62a5\u9519\uff0cMATLAB \u628a\u8868\u8fbe\u5f0f\u4f5c\u4e3a\u7ed3\u679c\u663e\u793a\u51fa\u6765\uff0c\u90fd\u65e0\u6cd5\u8ba1\u7b97\u3002<br \/>\n<img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705334362267.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 389px; --smush-placeholder-aspect-ratio: 389\/320;\" \/><br \/>\n<img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705332724699.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 607px; --smush-placeholder-aspect-ratio: 607\/228;\" \/><\/p>\n<p>\u6240\u4ee5\u5f97\u56de\u5230\u79ef\u5206\u5b9a\u4e49\u6765\u505a\u3002<br \/>\n\u5047\u5982\u8981\u6c42\u4e00\u4e2a\u51fd\u6570\u4ecex=0\u5230x=E\u8303\u56f4\u5185\u548cx\u8f74\u56f4\u8d77\u6765\u7684\u90e8\u5206\u7684\u9762\u79ef\uff0c\u90a3\u4e48\u53ef\u4ee5\u770b\u505a\u65e0\u9650\u4e2a\u5c0f\u957f\u65b9\u5f62\u4e4b\u548c\uff08\u9ece\u66fc\u548c\uff09\u3002\u5206\u4e3an\u4e2a\u5c0f\u957f\u65b9\u5f62\uff0c\u6bcf\u4e2a\u5c0f\u957f\u65b9\u5f62\u7684\u5bbd\u5c31\u662f<code class=\"katex-inline\">\\frac En<\/code>\uff0c\u9ad8\u5c31\u662f<code class=\"katex-inline\">f(\\frac {iE}{n})<\/code>\uff0ci\u5bf9\u5e94\u4ece0\u5f00\u59cb\u7684\u7b2ci\u4e2a\u957f\u65b9\u5f62\u3002<\/p>\n<pre><code class=\"language-katex\">\\begin{aligned}\n\\int_0^Ef(x)dx&amp;=\\lim_{n\\to\\infty}[f(\\frac{E}{n})\\cdot\\frac{E}{n}+f(\\frac{2E}{n})\\cdot\\frac{E}{n}+\\cdots+f(\\frac{nE}{n})\\frac{E}{n}] \\\\\n&amp;=\\lim_{n\\to\\infty}\\frac{E}{n}\\cdot\\sum_{i=1}^n\\frac{iE}{n}\n\\end{aligned}<\/code><\/pre>\n<p>\u82e5\u8303\u56f4\u4e0d\u662f0\u5230E\uff0c\u800c\u662f\u6307\u5b9a\u4eceS\u5f00\u59cb\u5230E\uff0c\u90a3\u4e48\u6709<\/p>\n<pre><code class=\"language-katex\">\\begin{aligned}\n\\int_S^Ef(x)dx&amp;=\\lim_{n\\to\\infty}[f(S+\\frac{E-S}{n})\\cdot\\frac{E-S}{n}+f(\\frac{S+2(E-S)}{n})\\cdot\\frac{E-S}{n}+\\cdots+f(S+\\frac{n(E-S)}{n})\\frac{E-S}{n}] \\\\\n&amp;=\\lim_{n\\to\\infty}\\frac{E-S}{n}\\cdot\\sum_{i=1}^n[S+\\frac{i(E-S)}{n}]\n\\end{aligned}<\/code><\/pre>\n<p>\u518d\u56de\u5230\u4e0a\u9762\u7684\u9898\u76ee<\/p>\n<pre><code class=\"language-katex\">\\begin{aligned}\n\u539f\u5f0f&amp;=\\lim_{n\\to+\\infty}\\sum_{i=1}^n\\frac{i^p}{n^{p+1}}\\\\\n&amp;=\\lim_{n\\to+\\infty}\\frac{1}{n}\\sum_{i=1}^n(\\frac{i}{n})^p\\\\\n&amp;=\\int_0^1x^pdx\\\\\n&amp;=\\frac{1}{p+1}x^{p+1}|_0^1\\\\\n&amp;=\\frac{1}{p+1}\n\\end{aligned}<\/code><\/pre>\n<h1><span class=\"ez-toc-section\" id=\"8_%E6%B3%B0%E5%8B%92%E5%85%AC%E5%BC%8F\"><\/span>8 \u6cf0\u52d2\u516c\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p>\u6cf0\u52d2\u5b9a\u7406\u53c2\u8003\uff1a<a href=\"https:\/\/blog.iyatt.com\/?p=10375\">https:\/\/blog.iyatt.com\/?p=10375<\/a><\/p>\n<h2><span class=\"ez-toc-section\" id=\"81_%E4%BE%8B_1_ex_%E9%BA%A6%E5%85%8B%E5%8A%B3%E6%9E%97%E5%B1%95%E5%BC%80\"><\/span>8.1 \u4f8b 1 e^x \u9ea6\u514b\u52b3\u6797\u5c55\u5f00<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{aligned}\n&amp;\n\\begin{aligned}\ne^x&amp;=f(0)+\\frac{f&#039;(0)}{1!}(x-0)+\\frac{f&#039;&#039;(0)}{2!}(x-0)^2+\\cdots+\\frac{f^{(n)}}{n!}(x-0)^n \\\\\n&amp;=1+x+\\frac{x^2}{2!}+\\cdots+\\frac{x^n}{n!}\\\\\n\\end{aligned}\n\\\\\n&amp;|R_n(x)|=|\\frac{e^{\\theta x}}{(n+1)!}|\\lt\\frac{e^{|x|}}{(n+1)!}\uff0c0\\lt\\theta\\lt x\n\\end{aligned}<\/code><\/pre>\n<p>\u82e5x=1,\u5219\u6709<\/p>\n<pre><code class=\"language-katex\">e=1+1+\\frac{1}{2!}+\\frac{1}{3!}+\\cdots+\\frac{1}{n!}\uff0c|R_n|\\lt\\frac{e}{(n+1)!}\\lt\\frac{3}{(n+1)!}<\/code><\/pre>\n<p>n=10\u65f6\uff0c<code class=\"katex-inline\">e\\approx2.718281801<\/code>\uff0c\u8bef\u5dee\u4e0d\u8d85\u8fc7<code class=\"katex-inline\">10^{-7}<\/code><\/p>\n<p>Python \u6a21\u62df\u8ba1\u7b97\u8fc7\u7a0b<\/p>\n<pre><code class=\"language-py\">def f(n):\n    sum = 1\n    if n == 1:\n        return sum\n    else:\n        for i in range(1, n):\n            factorial = 1\n            factorial_end = i + 1\n            for j in range(1, factorial_end): # \u9636\u4e58\n                factorial *= j\n            sum += (1.0 \/ factorial)\n    return sum\n\nprint(f(1))\nprint(f(2))\nprint(f(3))\nprint(f(4))\nprint(f(5))\nprint(f(6))\nprint(f(7))\nprint(f(8))\nprint(f(9))\nprint(f(10))<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705373034668.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 151px; --smush-placeholder-aspect-ratio: 151\/180;\" \/><\/p>\n<p>SymPy<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx = sy.symbols(&#039;x&#039;)\nf = sy.E**x\ndisplay(sy.series(f)) # \u9ed8\u8ba4\u9ea6\u514b\u52b3\u6797\u5c55\u5f00\ntaylor = sy.series(f, x, 0, 10) # \u6307\u5b9a\u5728 0 \u5904\u5c55\u5f00\u5373\u9ea6\u514b\u52b3\u6797\uff0c\u5e76\u6307\u5b9a\u5c55\u5f00 10 \u9636\ndisplay(taylor)\ndisplay(taylor.removeO().evalf(subs={x:1})) # \u5148\u79fb\u9664\u9ad8\u9636\u65e0\u7a77\u5c0f\u518d\u4ee3\u503c\u8ba1\u7b97<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705370575753.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 591px; --smush-placeholder-aspect-ratio: 591\/133;\" \/><\/p>\n<p>MATLAB<\/p>\n<pre><code class=\"language-matlab\">syms x;\n\nf = exp(x);\ntaylor(f) % \u9ed8\u8ba4\u9ea6\u514b\u52b3\u6797\u5c55\u5f00\nft = taylor(f, x, &#039;Order&#039;, 10) % \u5c55\u5f00 10 \u9636\nvpa(subs(ft, x, 1)) % \u4ee3\u503c\u8ba1\u7b97<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705373923765.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 332px; --smush-placeholder-aspect-ratio: 332\/204;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"82_%E4%BE%8B_2_sin_x_%E9%BA%A6%E5%85%8B%E5%8A%B3%E6%9E%97%E5%B1%95%E5%BC%80\"><\/span>8.2 \u4f8b 2 sin x \u9ea6\u514b\u52b3\u6797\u5c55\u5f00<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l}\nf(x)=\\sin x \\\\\nf&#039;(x)=\\cos x \\\\\nf&#039;&#039;(x)=-\\sin x \\\\\nf^{(3)}(x)=-\\cos x \\\\\nf^{(4)}(x)=\\sin x \\\\\n\\\\\nf(x)=0+x+0-\\frac{1}{3!}x^3+0+\\frac{1}{5}x^5+0-\\frac{1}{7}x^7+\\cdots+(-1)^{m-1}\\frac{x^{2m-1}}{2m-1}+R_{2m}(x)\\\\\nR_{2m}(x)=\\frac{\\sin[\\theta x+(2m+1)\\cdot\\frac{\\pi}{2}]}{(2m+1)!}x^{2m+1}=(-1)^m\n\\frac{\\cos\\theta x}{(2m+1)!}x^{2m+1}\uff0c0\\lt\\theta\\lt1\n\\end{array}<\/code><\/pre>\n<h1><span class=\"ez-toc-section\" id=\"9_%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E4%B9%98%E5%AD%90%E6%B3%95\"><\/span>9 \u62c9\u683c\u6717\u65e5\u4e58\u5b50\u6cd5<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"91_%E4%BE%8B_1\"><\/span>9.1 \u4f8b 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5df2\u77e5\u76ee\u6807\u51fd\u6570V(x,y,z)=xyz\uff0c\u5728\u7ea6\u675f\u6761\u4ef62xy+2xz+2yz=12\u4e0b\uff0c\u6c42\u4f53\u79efV\u7684\u6700\u5927\u503c\u3002<\/p>\n<p>\u4ee4G(x)=2xy+2xz+2yz-12<br \/>\n\u5219\u6709<\/p>\n<pre><code class=\"language-katex\">\\left \\{\n\\begin{array}{l}\n\\frac{\\partial V}{\\partial x}=-\\lambda\\frac{\\partial G}{\\partial x}\\\\ \\\\\n\\frac{\\partial V}{\\partial y}=-\\lambda\\frac{\\partial G}{\\partial y}\\\\ \\\\\n\\frac{\\partial V}{\\partial z}=-\\lambda\\frac{\\partial G}{\\partial z}\\\\ \\\\\nG(x)=0\n\\end{array}\n\\right .<\/code><\/pre>\n<pre><code class=\"language-katex\">\\left \\{\n\\begin{array}{l}\nyz+\\lambda(2y+2z)=0 \\\\\nxz+\\lambda(2x+2z)=0 \\\\\nxy+\\lambda(2x+2y)=0 \\\\\n2xy+2xz+2yz-12 = 0\n\\end{array}\n\\right .<\/code><\/pre>\n<p>SymPy \u89e3\u65b9\u7a0b\u7ec4<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nsy.init_printing(use_unicode=True) # \u4f7f\u7528 Unicode \u663e\u793a\uff0c\u4fdd\u8bc1\u8fd9\u91cc\u7684\u89e3\u80fd\u4ee5\u624b\u5199\u683c\u5f0f\u663e\u793a\n\nx, y, z, t = sy.symbols(&#039;x y z t&#039;)\nf1 = sy.Eq(y * z + t * (2 * y + 2 * z), 0)\ndisplay(f1)\nf2 = sy.Eq(x * z + t * (2 * x + 2 * z), 0)\ndisplay(f2)\nf3 = sy.Eq(x * z + t * (2 * x + 2 * y), 0)\ndisplay(f3)\nf4 = sy.Eq(2 * x * y + 2 * x * z + 2 * y * z, 12)\ndisplay(f4)\n\nsy.solve((f1, f2, f3, f4), (x, y, z, t))<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705476749081.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 869px; --smush-placeholder-aspect-ratio: 869\/77;\" \/><\/p>\n<p>\u56e0\u4e3a\u65b9\u7a0b\u7ec4\u5b58\u5728\u7ebf\u6027\u76f8\u5173\uff0c\u6240\u4ee5\u89e3\u4e0d\u53ea\u4e00\u4e2a\u3002V(x)\u7684\u4e8c\u9636\u504f\u5bfc\u6052\u4e3a0\uff0c\u6240\u4ee5\u6ca1\u6cd5\u7528\u4e8c\u9636\u504f\u5bfc\u786e\u5b9a\u6700\u5927\u503c\uff0c\u8fd8\u5728\u8fd9\u91cc\u65b9\u7a0b\u7b80\u5355\uff0c\u76f4\u63a5\u7528\u6c42\u51fa\u7684\u70b9\u4ee3\u5165\u8bd5\u5c31\u884c\u3002<br \/>\n\u5728<code class=\"katex-inline\">(\\sqrt2,\\sqrt2,\\sqrt2)<\/code>\u5904\uff0c<code class=\"katex-inline\">\\lambda=-\\frac{\\sqrt2}{4}<\/code>\u65f6\uff0cV(x)\u6709\u6700\u5927\u503c<code class=\"katex-inline\">2\\sqrt2<\/code><\/p>\n<p>MATLAB \u6c42\u89e3\u4e0a\u9762\u7684\u65b9\u7a0b\u7ec4<\/p>\n<pre><code class=\"language-matlab\">syms x y z t;\n\nf1 = y * z + t * (2 * y + 2 * z) == 0;\nf2 = x * z + t * (2 * x + 2 * z) == 0;\nf3 = x * y + t * (2 * x + 2 * y) == 0;\nf4 = 2 * x * y + 2 * x * z + 2 * y * z == 12;\n\nsol = solve(f1, f2, f3, f4);\ndisp(sol.x)\ndisp(sol.y)\ndisp(sol.z)\ndisp(sol.t)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705478591981.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 113px; --smush-placeholder-aspect-ratio: 113\/308;\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"92_%E4%BE%8B_2\"><\/span>9.2 \u4f8b 2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5df2\u77e5\u76ee\u6807\u51fd\u6570\u4e3a<code class=\"katex-inline\">u=x^3y^2z<\/code>\uff0c\u7ea6\u675f\u6761\u4ef6x+y+z=12\uff0c\u6c42\u6700\u5927\u503c<\/p>\n<pre><code class=\"language-katex\">\\left \\{\n\\begin{array}{l}\n3x^2y^2z+\\lambda = 0 \\\\\n2x^3yz+\\lambda=0 \\\\\nx^3y^2 + \\lambda=0\\\\\nx+y+z=12\n\\end{array}\n\\right .<\/code><\/pre>\n<p>SymPy \u89e3\u65b9\u7a0b\u7ec4<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx, y, z, t = sy.symbols(&#039;x y z t&#039;)\nf1 = sy.Eq(3 * x**2 * y**2 * z + t, 0)\nf2 = sy.Eq(2 * x**3 * y * z + t, 0)\nf3 = sy.Eq(x**3 * y**2 + t, 0)\nf4 = sy.Eq(x + y + z, 12)\n\nsy.solve((f1, f2, f3, f4))<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705479491388.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 1009px; --smush-placeholder-aspect-ratio: 1009\/46;\" \/><\/p>\n<p>\u5176\u4e2d(6, 4, 2)\uff0c<code class=\"katex-inline\">\\lambda=-3456<\/code>\u65f6\uff0c<code class=\"katex-inline\">u_{max}=6912<\/code><\/p>\n<h2><span class=\"ez-toc-section\" id=\"93_%E4%BE%8B_3\"><\/span>9.3 \u4f8b 3<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u5728\u7b2c\u4e00\u8c61\u9650\u5185\u505a\u692d\u7403\u9762<code class=\"katex-inline\">\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1<\/code>\u7684\u5207\u5e73\u9762\uff0c\u4f7f\u5207\u5e73\u9762\u4e0e 3 \u4e2a\u5750\u6807\u8f74\u9762\u56f4\u57ce\u7684\u56db\u9762\u4f53\u4f53\u79ef\u6700\u5c0f\uff0c\u6c42\u5207\u70b9\u5750\u6807\u3002<\/p>\n<p>\u8bbe<code class=\"katex-inline\">p(x_0,y_0,z_0)<\/code>\u4e3a\u692d\u7403\u9762\u4e0a\u7684\u4e00\u70b9\uff08\u4f5c\u4e3a\u5207\u70b9\uff09\uff0c\u4ee4<code class=\"katex-inline\">F(x,y,z)=\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}-1<\/code><br \/>\np\u70b9\u5904\u7684\u504f\u5bfc\u6570\u4e3a<\/p>\n<pre><code class=\"language-katex\">\\begin{array}{l}\nF_x&#039;|_p=\\frac{2x_0}{a^2} \\\\ \\\\\nF_y&#039;|_p=\\frac{2y_0}{b^2} \\\\ \\\\\nF_z&#039;|_p=\\frac{2z_0}{c^2}\n\\end{array}<\/code><\/pre>\n<p>\u8fc7p\u70b9\u7684\u5207\u5e73\u9762\u65b9\u7a0b\u4e3a<\/p>\n<pre><code class=\"language-katex\">\\begin{aligned}\n\\frac{2x_0}{a^2}(x-x_0)+\\frac{2y_0}{b^2}(y-y_0)+\\frac{2z_0}{c^2}(z-z_0)&amp;=0 \\\\\n\\frac{x_0}{a^2}(x-x_0)+\\frac{y_0}{b^2}(y-y_0)+\\frac{z_0}{c^2}(z-z_0)&amp;=0 \\\\\n\\frac{xx_0}{a^2}+\\frac{yy_0}{b^2}+\\frac{zz_0}{c^2}&amp;=\\frac{x_0^2}{a^2}+\\frac{y_0^2}{b^2}+\\frac{z_0^2}{c^2}\\xlongequal{(x_0,y_0,z_0)\u5728\u692d\u7403\u9762\u4e0a}1 \\\\\n\u5373\u6709\\frac{xx_0}{a^2}+\\frac{yy_0}{b^2}+\\frac{zz_0}{c^2}&amp;=0\n\\end{aligned}<\/code><\/pre>\n<p>\u53ef\u4ee5\u5f97\u5230\u5207\u5e73\u9762\u4e0e 3 \u8f74\u7684\u622a\u8ddd<code class=\"katex-inline\">x=\\frac{a^2}{x_0}<\/code>\u3001<code class=\"katex-inline\">y=\\frac{b^2}{y_0}<\/code>\u3001<code class=\"katex-inline\">z=\\frac{c^2}{z_0}<\/code><br \/>\n\u5207\u5e73\u9762\u548c 3 \u8f74\u56f4\u6210\u7684\u9762\u6784\u6210\u7684\u56db\u9762\u4f53\u4f53\u79ef <code class=\"katex-inline\">V=\\frac{1}{2}\\cdot\\frac{a^2}{x_0}\\cdot\\frac{b^2}{y_0}\\cdot\\frac{1}{3}\\cdot\\frac{c^2}{z_0}=\\frac{a^2b^2c^2}{6x_0y_0z_0}<\/code><\/p>\n<p>\u76ee\u6807\u51fd\u6570\u5c31\u662f<code class=\"katex-inline\">V=\\frac{a^2b^2c^2}{6x_0b_0z_0}<\/code>\uff0c\u7ea6\u675f\u6761\u4ef6\u4e3a<code class=\"katex-inline\">\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1<\/code><\/p>\n<p>\u6c42V\u6700\u5c0f\u65f6\uff0c\u5b9e\u9645\u5c31\u662f\u6c42<code class=\"katex-inline\">x_0y_0z_0<\/code>\u6700\u5927\u65f6\uff0c\u90a3\u4e48\u53ef\u4ee5\u5f53\u505a\u6c42<code class=\"katex-inline\">\\ln(x_0y_0z_0)=\\ln(x_0)+\\ln(y_0)+\\ln(z_0)<\/code>\u6700\u5927\u65f6<br \/>\n\u6784\u9020\u51fd\u6570\uff1a<code class=\"katex-inline\">F(x,y,z,\\lambda)=\\ln x_0+\\ln y_0+\\ln z_0+\\lambda(\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}-1)<\/code><br \/>\n\u5217\u65b9\u7a0b\u7ec4<\/p>\n<pre><code class=\"language-katex\">\\left \\{\n\\begin{array}{l}\n\\frac{1}{x_0}+2\\lambda\\frac{x_0}{a^2}=0\\\\ \\\\\n\\frac{1}{y_0}+2\\lambda\\frac{y_0}{b^2}=0\\\\ \\\\\n\\frac{1}{z_0}+2\\lambda\\frac{z_0}{b^2}=0\\\\ \\\\\n\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1\n\\end{array}\n\\right .<\/code><\/pre>\n<p>SymPy \u6c42\u89e3<\/p>\n<pre><code class=\"language-py\">import sympy as sy\n\nx0, y0, z0, a, b, c = sy.symbols(&#039;x0 y0 z0 a b c&#039;, positive=True) # \u5207\u70b9\u5728\u7b2c\u4e00\u8c61\u9650\u5fc5\u7136\u4e3a\u6b63\nt = sy.symbols(&#039;t&#039;)\n\nf1 = sy.Eq(1 \/ x0 + 2 * t * x0 \/ a**2, 0)\nf2 = sy.Eq(1 \/ y0 + 2 * t * y0 \/ b**2, 0)\nf3 = sy.Eq(1 \/ z0 + 2 * t * z0 \/ c**2, 0)\nf4 = sy.Eq(x0**2 \/ a**2 + y0**2 \/ b**2 + z0**2 \/ c**2, 1)\n\nfor result in sy.solve((f1, f2, f3, f4), (x0, y0, z0, t)):\n    display(result)<\/code><\/pre>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2024\/01\/image-1705507189346.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 237px; --smush-placeholder-aspect-ratio: 237\/69;\" \/><\/p>\n<p>\u5207\u70b9\u4e3a<code class=\"katex-inline\">(\\frac{\\sqrt3a}{3},\\frac{\\sqrt3b}{3},\\frac{\\sqrt3c}{3})<\/code>\u65f6\uff0c<code class=\"katex-inline\">V_{min}=\\frac{\\sqrt3}{2}abc<\/code><\/p>\n","protected":false},"excerpt":{"rendered":"<p>1 \u5de5\u5177 1.1 Python \u57fa\u7840\u5de5\u5177 Python 3.11.2 \u6570\u5b66\u6a21\u5757 SymPy 1.12 SciP [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"zakra_page_container_layout":"customizer","zakra_page_sidebar_layout":"customizer","zakra_remove_content_margin":false,"zakra_sidebar":"customizer","zakra_transparent_header":"customizer","zakra_logo":0,"zakra_main_header_style":"default","zakra_menu_item_color":"","zakra_menu_item_hover_color":"","zakra_menu_item_active_color":"","zakra_menu_active_style":"","zakra_page_header":true,"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[1,606,592,612],"tags":[911,910,915,914,909,913,912,918,917,923,925,924,908,927,919,916,920,922,921,926,907],"class_list":["post-12906","post","type-post","status-publish","format-standard","hentry","category-all","category-jupyter","category-python","category-612","tag-jupyter-notebook","tag-matlab","tag-matplotlib","tag-numpy","tag-python","tag-scipy","tag-sympy","tag-918","tag-917","tag-923","tag-925","tag-924","tag-908","tag-927","tag-919","tag-916","tag-920","tag-922","tag-921","tag-926","tag-907"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/12906","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=12906"}],"version-history":[{"count":1,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/12906\/revisions"}],"predecessor-version":[{"id":21494,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/12906\/revisions\/21494"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=12906"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=12906"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=12906"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}