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\/>\n\u56e0\u4e3ae\u5206\u4e4b1\u5c31\u662f\u6240\u8c13\u7684\u201c\u5012\u9709\u5e38\u6570\u201d\u3002<\/p>\n<p>\u5c31\u597d\u50cf\u4f60\u5728\u6e38\u620f\u91cc\u62bd\u5361\uff0c\u767e\u5206\u4e4b\u4e00\u51fa\u8d27\u7387\u4e0b\u4f60\u62bd\u4e00\u767e\u6b21\u6ca1\u51fa\u8d27\uff0c\u4f60\u4f1a\u8ba4\u4e3a\u662f\u81ea\u5df1\u5012\u9709\uff1b\u4e07\u5206\u4e4b\u4e00\u51fa\u8d27\u7387\u4e0b\u4f60\u62bd\u4e00\u4e07\u6b21\u6ca1\u51fa\u8d27\uff0c\u4f60\u4e5f\u4f1a\u8ba4\u4e3a\u81ea\u5df1\u5012\u9709\u3002<\/p>\n<p>\u4f60\u7528\u8ba1\u7b97\u5668\u6309\u4e00\u4e0b\u8fd9\u4e24\u8005\u7684\u51e0\u7387\u5c31\u4f1a\u53d1\u73b0\u5f88\u63a5\u8fd1\uff08\u56db\u820d\u4e94\u5165\u5747\u4e3a\u7ea637%\uff09\u3002\u90a3\u4e48\u4f60\u7ee7\u7eed\u5e26\u5165\u4e00\u767e\u4e07\u3001\u4e00\u4ebf\u3001\u4e00\u767e\u4ebf\u3002\u3002\u3002\u5c31\u4f1a\u53d1\u73b0\u51e0\u7387\u65e0\u9650\u903c\u8fd1e\u5206\u4e4b\u4e00\u3002<\/p>\n<p>\u6ca1\u9519\uff0ce\u5206\u4e4b\u4e00\u5c31\u662f(1-1\/n)^n\u7684\u6781\u9650\u3002<\/p>\n<p>\u8fd9\u5c31\u662f\u968f\u673a\u7cfb\u7edf\u81ea\u53d1\u7684\u8d8b\u52bf\uff1a\u5f53\u6210\u529f\u51e0\u7387\u548c\u673a\u4f1a\u6b21\u6570\u4e92\u4e3a\u5012\u6570\u65f6\uff0c\u7ea6\u670937%\u7684\u4eba\u4f1a\u5012\u9709\u3002<\/p>\n<p>\u653f\u5e9c\u4e0d\u6709\u610f\u7ba1\u4eba\u6c11\u4e5f\u4e0d\u6545\u610f\u5bb3\u4eba\u6c11\u65f6\uff0c\u8fd9\u5c31\u662f\u81ea\u7136\u7684\u5e38\u6570\uff0c\u8fd9\u5c31\u662f\u5929\u7406\u3002<\/p>\n<p>\u73b0\u5728\u4f60\u53ef\u80fd\u60f3\u95ee\uff1a\u90a3\u4e48\u54b1\u4eec\u6709\u6ca1\u6709\u529e\u6cd5\u4e0d\u8ba9\u90a3\u4e48\u591a\u4eba\u5012\u9709\u554a\uff1f<\/p>\n<p>\u5f53\u7136\u6709\u3002<\/p>\n<p>\u4e0d\u8ba9\u6210\u529f\u51e0\u7387\u548c\u673a\u4f1a\u6b21\u6570\u4e92\u4e3a\u5012\u6570\u5c31\u53ef\u4ee5\u4e86\u3002\u673a\u4f1a\u7684\u6b21\u6570\u662f\u6210\u529f\u7387\u5012\u6570\u76842\u500d\uff0c\u5012\u9709\u7684\u4eba\u5c31\u53ea\u670913%\u4e86\uff1b\u63d0\u9ad8\u52303\u500d\uff0c\u5012\u9709\u7684\u4eba\u5c31\u53ea\u6709\u4e0d\u52305%\u4e86\uff1b\u63d0\u9ad8\u52304\u500d\uff0c\u5012\u9709\u7684\u4eba\u5c31\u4e0d\u52302%\u4e86\uff1b\u63d0\u9ad8\u52305\u500d\uff0c\u5012\u9709\u7684\u4eba\u5c31\u4e0d\u52301%\u4e86\u3002<\/p>\n<p>\u5728\u4f60\u5012\u9709\uff08\u7528\u5b8c\u5929\u7ecf\u5730\u4e49\u7684\u673a\u4f1a\u6b21\u6570\u4e4b\u540e\u8fd8\u6ca1\u80fd\u6210\u529f\u7ef4\u6301\u751f\u5b58\uff09\u65f6\u591a\u7ed9\u4f60\u51e0\u6b21\u673a\u4f1a\u2014\u2014\u8fd9\u5c31\u662f\u6211\u56fd\u653f\u5e9c\u5728\u505a\u7684\u4e8b\u3002<\/p>\n<p>\u800c\u5728\u4f60\u5012\u9709\uff08\u7528\u5b8c\u5929\u7ecf\u5730\u4e49\u7684\u673a\u4f1a\u6b21\u6570\u4e4b\u540e\u8fd8\u6ca1\u80fd\u6210\u529f\u7ef4\u6301\u751f\u5b58\uff09\u65f6\u76f4\u63a5\u7ed3\u675f\u6e38\u620f\u2014\u2014\u8fd9\u5c31\u662f\u7f8e\u56fd\u653f\u5e9c\u505a\u7684\u4e8b\uff0c\u4e5f\u5c31\u662f\u6240\u8c13\u7684\u65a9\u6740\u7ebf\u3002<\/p>\n<p>\u5929\u7406\u5c31\u89c4\u5b9a\u4e8637%\u7684\u4eba\u8981\u5012\u9709\uff0c\u800c\u6211\u4eec\u7684\u653f\u5e9c\u5c31\u662f\u4e3a\u4e86\u4eba\u6c11\u5728\u9006\u5929\u800c\u884c\u3002<\/p>\n<p>\u56de\u7b543:<br \/>\n\u5176\u5b9e\u662f\u4e00\u4e2a\u975e\u5e38\u7ecf\u5178\u7684\u95ee\u9898\uff0c\u5728quant\u6c42\u804c\u65f6\u5e38\u89c1\u7684brain teaser\uff0c\u800c\u4e14\u53ea\u9700\u8981\u57fa\u672c\u7684\u9ad8\u7b49\u6570\u5b66\u77e5\u8bc6\u5373\u53ef\u5b8c\u6210\u3002<\/p>\n<p>\u8fd9\u4e2a\u95ee\u9898\u7684\u57fa\u7840\u7248\u672c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\uff1a\u5047\u8bbe\u5341\u5e74\u65f6\u95f4\u91cc\u6bcf\u5e74\u90fd\u4e0d\u91cd\u590d\u5730\u8c08\u4e00\u6b21\u604b\u7231\uff0c\u90a3\u4e48\u9009\u62e9\u5728\u7b2c\u51e0\u6b21\u7ed3\u5a5a\uff1f<\/p>\n<p>\u53bb\u9ea6\u7530\u6458\u9ea6\u7a57\uff0c\u5047\u8bbe\u6bcf\u68f5\u9ea6\u7a57\u53ea\u80fd\u7ecf\u8fc7\u4e00\u6b21\uff0c\u600e\u4e48\u627e\u51fa\u6700\u5927\u7684\u9ea6\u7a57\uff1f<\/p>\n<p>\u8fd9\u7c7b\u95ee\u9898\u90fd\u6709\u4e00\u4e2a\u5171\u540c\u70b9\uff0c\u5373\uff1a\u7ed9\u5b9a\u6570\u91cf\u7684\u6837\u672c\u8303\u56f4\uff0c\u5728\u6bcf\u4e2a\u6837\u672c\u53ea\u6709\u4e00\u6b21\u89c2\u5bdf\u7684\u673a\u4f1a\uff08\u5373\u8981\u4e48\u9009\u62e9\u8981\u4e48\u653e\u5f03\uff09\u7684\u524d\u63d0\u4e0b\uff0c\u5982\u4f55\u64cd\u4f5c\uff0c\u4f7f\u5f97\u627e\u51fa\u6700\u4f18\u6837\u672c\u7684\u6982\u7387\u6700\u5927\uff1f<\/p>\n<p>\u8fd9\u91cc\u627e\u51fa\u6700\u4f18\u89e3\u9700\u8981\u8003\u8651\u4e09\u4e2a\u70b9\uff0c\u9996\u5148\uff0c\u6211\u4eec\u5e0c\u671b\u8003\u5bdf\u8fd9\u4e2a\u6837\u672c\u96c6\u5408\u7684\u5927\u81f4\u6c34\u5e73\uff0c\u4ee5\u4fbf\u5c3d\u53ef\u80fd\u4f30\u8ba1\u6700\u4f18\u6837\u672c\u7684\u6c34\u5e73\uff0c\u4e3a\u5b9e\u73b0\u8fd9\u4e2a\u76ee\u7684\uff0c\u9700\u8981\u9884\u5148\u89c2\u5bdf\u82e5\u5e72\u4e2a\u6837\u672c\uff0c\u89c2\u5bdf\u540e\u518d\u8fdb\u884c\u9009\u62e9\uff1b\u5176\u6b21\uff0c\u5bf9\u6bcf\u4e2a\u6837\u672c\uff0c\u89c2\u5bdf\u673a\u4f1a\u53ea\u6709\u4e00\u6b21\uff0c\u81ea\u7136\u5e0c\u671b\u6700\u4f18\u89e3\u6ca1\u6709\u51fa\u73b0\u5728\u9884\u5148\u89c2\u5bdf\u7684\u6837\u672c\u96c6\u5408\u4e2d\uff1b\u6700\u540e\uff0c\u5728\u5b8c\u6210\u5bf9\u6837\u672c\u7684\u9884\u5148\u89c2\u5bdf\u540e\uff0c\u53ea\u8981\u65b0\u89c2\u5bdf\u7684\u6837\u672c\u6bd4\u9884\u5148\u89c2\u5bdf\u96c6\u5408\u4e2d\u7684\u5c40\u90e8\u6700\u4f18\u6837\u672c\u66f4\u597d\uff0c\u5c31\u89c6\u4e3a\u5168\u90e8\u6837\u672c\u4e2d\u7684\u6700\u4f18\u89e3\uff0c\u7ed3\u675f\u89c2\u5bdf\uff0c\u90a3\u4e48\u81ea\u7136\u5e0c\u671b\u6b21\u4f18\u89e3\u51fa\u73b0\u5728\u9884\u5148\u89c2\u5bdf\u7684\u6837\u672c\u96c6\u5408\u4e2d\uff0c\u5e76\u5047\u8bbe\u6b21\u4f18\u89e3\u51fa\u73b0\u5728\u6700\u4f18\u89e3\u4e4b\u524d\u3002<\/p>\n<p>\u660e\u786e\u4e86\u8fd9\u4e09\u70b9\uff0c\u5c31\u53ef\u4ee5\u5f00\u59cb\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u4e86\uff0c\u95ee\u9898\u6c42\u89e3\u5982\u4e0b:<\/p>\n<p>\u6837\u672c\u96c6\u5408\u4e3a \\left\\{ X_{i} \\right\\}(1\\leq i\\leq n) \uff0c\u9884\u5148\u89c2\u5bdf\u7684\u6837\u672c\u96c6\u5408\u4e3a {X_{i}}(1\\leq i \\leq k) \uff0c\u8bb0<\/p>\n<p>X_{j}=max{X_{i}} (1\\leq i \\leq n) \uff0c X_{l}=X_{(n-1)} \uff0c\u89c2\u5bdfk\u4e2a\u6837\u672c\u540e\u627e\u51fa\u6700\u4f18\u6837\u672c\u7684\u6982\u7387\u4e3a P_{k} .<\/p>\n<p>\u7531\u524d\u8ff0\u7b2c\u4e8c\u4e2a\u70b9\uff0c k<j\\leq n \uff1b\u7531\u524d\u8ff0\u7b2c\u4e09\u4e2a\u70b9\uff0c 1<l\\leq k \uff0c\u82e5\u4e0d\u7136\uff0c P_{k}=0 .\n\n\u5bf9j\u6c42\u548c\uff0cj\u6709n\u4e2a\u53d6\u503c\uff0c\u53d6\u540en-k\u4e2a\u6709\u6548\uff1bl\u6709j-1\u4e2a\u53d6\u503c\uff0c\u53d6\u524dk\u4e2a\u6709\u6548 \uff0c\u6240\u4ee5\n\nP_{k}=\\sum_{k+1}^{n}{\\frac{k}{j-1}\\cdot\\frac{1}{n}}=\\frac{k}{n}\\sum_{k+1}^{n}{\\frac{1}{j-1}} .\n\n\u5f53 n\\rightarrow \\infty \u65f6\uff0c\u7531Riemann\u79ef\u5206\u5b9a\u4e49\uff0c P_{k}=\\frac{k}{n}\\int_{\\frac{k}{n}}^{1}\\frac{1}{t}dt=-\\frac{k}{n}ln\\frac{k}{n} .\n\n\u8fd9\u662f\u5173\u4e8ek\u7684\u51fd\u6570\uff0c\u5bf9k\u6c42\u5bfc\uff0c\u4ee4\u5bfc\u6570\u4e3a0\uff0c\u5373\u53ef\u5f97\u5230\u6700\u5927\u503c\u70b9 k=\\frac{n}{e} .\n\n\u4e5f\u5c31\u662f\u5728e\u5206\u4e4b\u4e00\u7684\u4f4d\u7f6e\u505a\u9884\u5148\u89c2\u5bdf\u540e\uff0c\u540e\u7eed\u53ea\u8981\u65b0\u89c2\u5bdf\u7684\u6837\u672c\u6bd4\u9884\u5148\u89c2\u5bdf\u96c6\u5408\u4e2d\u7684\u5c40\u90e8\u6700\u4f18\u6837\u672c\u66f4\u597d\uff0c\u5c31\u53ef\u4ee5\u6700\u5927\u6982\u7387\u5f97\u5230\u6700\u4f18\u6837\u672c\u3002\n\n\u8865\u5145\u8bf4\u660e\u4e00\u4e0b\uff0c\u8fd9\u4e2a\u7406\u8bba\u662f\u9700\u8981\u6ee1\u8db3\u4e00\u5b9a\u6761\u4ef6\u624d\u9002\u7528\u7684\uff0c\u6700\u91cd\u8981\u7684\u5c31\u662f\uff1a\u5fc5\u987b\u5148\u7ecf\u8fc7\u6b21\u4f18\u89e3\u518d\u7ecf\u8fc7\u6700\u4f18\u89e3\uff1b\u6837\u672c\u6570\u91cf\u8981\u8db3\u591f\u5927\u3002\n\n\u6b64\u5916\uff0c\u6bcf\u4e2a\u6837\u672c\u53ef\u80fd\u5e76\u975e\u53ea\u6709\u4e00\u6b21\u89c2\u5bdf\u673a\u4f1a\u3002\n\n\u56e0\u4e3ae\u5206\u4e4b1\u5c31\u662f\u6240\u8c13\u7684\u201c\u5012\u9709\u5e38\u6570\u201d\u3002\n\n\u5c31\u597d\u50cf\u4f60\u5728\u6e38\u620f\u91cc\u62bd\u5361\uff0c\u767e\u5206\u4e4b\u4e00\u51fa\u8d27\u7387\u4e0b\u4f60\u62bd\u4e00\u767e\u6b21\u6ca1\u51fa\u8d27\uff0c\u4f60\u4f1a\u8ba4\u4e3a\u662f\u81ea\u5df1\u5012\u9709\uff1b\u4e07\u5206\u4e4b\u4e00\u51fa\u8d27\u7387\u4e0b\u4f60\u62bd\u4e00\u4e07\u6b21\u6ca1\u51fa\u8d27\uff0c\u4f60\u4e5f\u4f1a\u8ba4\u4e3a\u81ea\u5df1\u5012\u9709\u3002\n\n\u4f60\u7528\u8ba1\u7b97\u5668\u6309\u4e00\u4e0b\u8fd9\u4e24\u8005\u7684\u51e0\u7387\u5c31\u4f1a\u53d1\u73b0\u5f88\u63a5\u8fd1\uff08\u56db\u820d\u4e94\u5165\u5747\u4e3a\u7ea637%\uff09\u3002\u90a3\u4e48\u4f60\u7ee7\u7eed\u5e26\u5165\u4e00\u767e\u4e07\u3001\u4e00\u4ebf\u3001\u4e00\u767e\u4ebf\u3002\u3002\u3002\u5c31\u4f1a\u53d1\u73b0\u51e0\u7387\u65e0\u9650\u903c\u8fd1e\u5206\u4e4b\u4e00\u3002\n\n\u6ca1\u9519\uff0ce\u5206\u4e4b\u4e00\u5c31\u662f(1-1\/n)^n\u7684\u6781\u9650\u3002\n\n\u8fd9\u5c31\u662f\u968f\u673a\u7cfb\u7edf\u81ea\u53d1\u7684\u8d8b\u52bf\uff1a\u5f53\u6210\u529f\u51e0\u7387\u548c\u673a\u4f1a\u6b21\u6570\u4e92\u4e3a\u5012\u6570\u65f6\uff0c\u7ea6\u670937%\u7684\u4eba\u4f1a\u5012\u9709\u3002\n\n\u653f\u5e9c\u4e0d\u6709\u610f\u7ba1\u4eba\u6c11\u4e5f\u4e0d\u6545\u610f\u5bb3\u4eba\u6c11\u65f6\uff0c\u8fd9\u5c31\u662f\u81ea\u7136\u7684\u5e38\u6570\uff0c\u8fd9\u5c31\u662f\u5929\u7406\u3002\n\n\u73b0\u5728\u4f60\u53ef\u80fd\u60f3\u95ee\uff1a\u90a3\u4e48\u54b1\u4eec\u6709\u6ca1\u6709\u529e\u6cd5\u4e0d\u8ba9\u90a3\u4e48\u591a\u4eba\u5012\u9709\u554a\uff1f\n\n\u5f53\u7136\u6709\u3002\n\n\u4e0d\u8ba9\u6210\u529f\u51e0\u7387\u548c\u673a\u4f1a\u6b21\u6570\u4e92\u4e3a\u5012\u6570\u5c31\u53ef\u4ee5\u4e86\u3002\u673a\u4f1a\u7684\u6b21\u6570\u662f\u6210\u529f\u7387\u5012\u6570\u76842\u500d\uff0c\u5012\u9709\u7684\u4eba\u5c31\u53ea\u670913%\u4e86\uff1b\u63d0\u9ad8\u52303\u500d\uff0c\u5012\u9709\u7684\u4eba\u5c31\u53ea\u6709\u4e0d\u52305%\u4e86\uff1b\u63d0\u9ad8\u52304\u500d\uff0c\u5012\u9709\u7684\u4eba\u5c31\u4e0d\u52302%\u4e86\uff1b\u63d0\u9ad8\u52305\u500d\uff0c\u5012\u9709\u7684\u4eba\u5c31\u4e0d\u52301%\u4e86\u3002\n\n\u5728\u4f60\u5012\u9709\uff08\u7528\u5b8c\u5929\u7ecf\u5730\u4e49\u7684\u673a\u4f1a\u6b21\u6570\u4e4b\u540e\u8fd8\u6ca1\u80fd\u6210\u529f\u7ef4\u6301\u751f\u5b58\uff09\u65f6\u591a\u7ed9\u4f60\u51e0\u6b21\u673a\u4f1a\u2014\u2014\u8fd9\u5c31\u662f\u6211\u56fd\u653f\u5e9c\u5728\u505a\u7684\u4e8b\u3002\n\n\u800c\u5728\u4f60\u5012\u9709\uff08\u7528\u5b8c\u5929\u7ecf\u5730\u4e49\u7684\u673a\u4f1a\u6b21\u6570\u4e4b\u540e\u8fd8\u6ca1\u80fd\u6210\u529f\u7ef4\u6301\u751f\u5b58\uff09\u65f6\u76f4\u63a5\u7ed3\u675f\u6e38\u620f\u2014\u2014\u8fd9\u5c31\u662f\u7f8e\u56fd\u653f\u5e9c\u505a\u7684\u4e8b\uff0c\u4e5f\u5c31\u662f\u6240\u8c13\u7684\u65a9\u6740\u7ebf\u3002\n\n\u5929\u7406\u5c31\u89c4\u5b9a\u4e8637%\u7684\u4eba\u8981\u5012\u9709\uff0c\u800c\u6211\u4eec\u7684\u653f\u5e9c\u5c31\u662f\u4e3a\u4e86\u4eba\u6c11\u5728\u9006\u5929\u800c\u884c\u3002\n\n\u6ca1\u6709\u4efb\u4f55\u7ba1\u5236\u7684\u590d\u6742\u7cfb\u7edf\uff0c\u5728\u6781\u5ea6\u8ffd\u6c42\u6548\u7387\u548c\u9c81\u68d2\u6027\u65f6\uff0c\u81ea\u7136\u574d\u7f29\u5230\u7684\u4e00\u4e2a\u6570\u5b66\u7a33\u6001\u3002\n\n\u6240\u8c13\u76841\/e\uff08\u7ea6\u7b49\u4e8e36.8%\uff09\uff0c\u5728\u6570\u5b66\u4e0a\u6700\u8457\u540d\u7684\u5e94\u7528\u662f\u4ec0\u4e48\uff1f\n\u6700\u4f18\u505c\u6b62\u7406\u8bba\uff08Optimal Stopping Theory\uff09\uff0c\u4fd7\u79f0\u201c\u79d8\u4e66\u95ee\u9898\u201d\u6216\u8005\u201c\u82cf\u683c\u62c9\u5e95\u9ea6\u7530\u95ee\u9898\u201d\u3002\n\n\u7b80\u5355\u8bf4\uff1a\u5982\u679c\u4f60\u8981\u62db\u4e00\u4e2a\u79d8\u4e66\uff0c\u60f3\u62db\u5230\u6700\u597d\u7684\uff0c\u4f46\u4f60\u53ea\u80fd\u4e00\u4e2a\u4e00\u4e2a\u9762\u8bd5\uff0c\u9762\u5b8c\u5c31\u5f97\u51b3\u5b9a\u5f55\u4e0d\u5f55\uff0c\u4e0d\u80fd\u56de\u5934\u3002\u6570\u5b66\u544a\u8bc9\u4f60\u7684\u6700\u4f18\u7b56\u7565\u662f\uff1a\u65e0\u8111\u62d2\u7edd\u6389\u524d37%\uff081\/e\uff09\u7684\u4eba\uff0c\u628a\u4ed6\u4eec\u5f53\u4f5c\u6837\u672c\u6765\u5efa\u7acb\u6807\u51c6\uff0c\u7136\u540e\u4ece\u540e\u9762\u7684\u4eba\u91cc\uff0c\u9009\u7b2c\u4e00\u4e2a\u6bd4\u524d\u9762\u90fd\u5f3a\u7684\u4eba\u3002\n\n\u542c\u61c2\u4e86\u5417\uff1f\n\u90a337%\u7684\u4eba\uff0c\u5728\u6570\u5b66\u4e0a\u7684\u5b9a\u4e49\u5c31\u662f\u2014\u2014\u6837\u672c\u3002\n\u4ed6\u4eec\u7684\u5b58\u5728\u610f\u4e49\uff0c\u5c31\u662f\u88ab\u7cfb\u7edf\u89c2\u5bdf\u548c\u629b\u5f03\uff0c\u7528\u6765\u7ed9\u5269\u4e0b\u768463%\u786e\u7acb\u4e00\u4e2a\u57fa\u51c6\u7ebf\uff01\n\n\u628a\u8fd9\u4e2a\u903b\u8f91\u653e\u5230\u7f8e\u56fd\u793e\u4f1a\u8fd9\u4e2a\u5de8\u5927\u7684\u8d44\u672c\u4e3b\u4e49\u673a\u5668\u91cc\uff0c\u662f\u4e0d\u662f\u77ac\u95f4\u5c31\u9ed1\u6697\u4e86\u8d77\u6765\uff1f\n\n\u90a337%\u7684\u5e95\u5c42\uff08\u5927\u6982\u5c31\u662f\u5e74\u85aa\u51e0\u4e07\u5200\u4ee5\u4e0b\uff0c\u4e00\u65e6\u5931\u4e1a\u7acb\u523b\u6d41\u843d\u8857\u5934\u7684\u7fa4\u4f53\uff09\uff0c\u5c31\u662f\u8d44\u672c\u4e3b\u4e49\u793e\u4f1a\u7684\u8017\u6750\u6837\u672c\u3002\n\u7cfb\u7edf\u5e76\u4e0d\u60f3\u628a\u6240\u6709\u4eba\u90fd\u5f04\u6b7b\uff0c\u90a3\u6837\u6ca1\u4eba\u5e72\u6d3b\uff1b\u7cfb\u7edf\u4e5f\u4e0d\u60f3\u8ba9\u6240\u6709\u4eba\u90fd\u8fc7\u5f97\u597d\uff0c\u90a3\u6837\u52b3\u52a8\u529b\u6210\u672c\u592a\u9ad8\uff0c\u5229\u6da6\u4f1a\u5f52\u96f6\u3002\n\n\u5047\u8bbe S \u662f\u793e\u4f1a\u7684\u7a33\u5b9a\u5ea6\uff0cP \u662f\u88ab\u5265\u524a\u7684\u7a0b\u5ea6\uff08\u6216\u8005\u8bf4\u8d2b\u56f0\u4eba\u53e3\u6bd4\u4f8b\uff09\u3002\n\n    \u6050\u60e7\u9a71\u52a8\u6548\u7387\uff1a \u5982\u679c\u6ca1\u4eba\u997f\u6b7b\uff0c\u6ca1\u4eba\u7761\u5927\u8857\uff0c\u90a3\u4e2d\u95f4\u90a360%\u7684\u6253\u5de5\u4eba\u5c31\u4f1a\u53d8\u61d2\uff0c\u4f1a\u8981\u6c42\u52a0\u85aa\uff0c\u4f1a\u7f62\u5de5\u3002\u6240\u4ee5\u5fc5\u987b\u6709\u4e00\u90e8\u5206\u4eba\u6d3b\u5728\u5730\u72f1\u91cc\uff0c\u4f5c\u4e3a\u6d3b\u4f53\u5e7f\u544a\u5c55\u793a\u7ed9\u4e2d\u95f4\u5c42\u770b\u3002\u8fd9\u53eb\u7ef4\u6301\u52b3\u52a8\u7eaa\u5f8b\u7684\u5fc5\u8981\u4ee3\u4ef7\u3002\u9a6c\u514b\u601d\u7ba1\u8fd9\u53eb\u4ea7\u4e1a\u540e\u5907\u519b\u3002\n    \u6210\u672c\u5236\u7ea6\u538b\u69a8\uff1a \u5982\u679c\u997f\u6b7b\u7684\u4eba\u592a\u591a\u8d85\u8fc7\u67d0\u4e2a\u9608\u503c\uff0c\u6ee1\u5927\u8857\u90fd\u662f\u4e27\u5c38\uff0c\u6cbb\u5b89\u7ef4\u62a4\u6210\u672c\u2014\u2014\u8b66\u5bdf\u3001\u76d1\u72f1\u3001\u9632\u5f39\u73bb\u7483\u5c31\u4f1a\u6307\u6570\u7ea7\u4e0a\u5347\uff0c\u751a\u81f3\u7206\u53d1\u9769\u547d\uff0c\u5bfc\u81f4\u7cfb\u7edf\u5d29\u6e83\u3002\n\n\u8d44\u672c\u4e3b\u4e49\u7684\u7b97\u6cd5\uff0c\u5b9e\u9645\u4e0a\u662f\u5728\u6c42\u89e3\u4e00\u4e2a\u6cdb\u51fd\u6781\u503c\uff1a \u5982\u4f55\u5728\u6700\u5c0f\u5316\u7ef4\u7a33\u6210\u672c\u7684\u540c\u65f6\uff0c\u6700\u5927\u5316\u6253\u5de5\u4eba\u7684\u6050\u60e7\u611f\uff08\u5373\u52b3\u52a8\u6548\u7387\uff09\uff1f\n\n\u5728\u8fd9\u4e2a\u65b9\u7a0b\u91cc\uff0c\u90a3\u4e2a37%\uff0c\u5c31\u662f\u7cfb\u7edf\u9c81\u68d2\u6027\u7684\u4e34\u754c\u70b9\u3002\n\n\u8fd9\u5f88\u50cf\u4fe1\u606f\u8bba\u91cc\u7684\u71b5\u3002\u51fd\u6570 -x \\ln x \u7684\u6781\u503c\u70b9\u5728\u54ea\u91cc\uff1f\u5c31\u5728 1\/e \u9644\u8fd1\u3002 \u8fd9\u8bf4\u660e\u4ec0\u4e48\uff1f\u8bf4\u660e\u7ef4\u630137%\u7684\u4eba\u5904\u4e8e\u201c\u859b\u5b9a\u8c14\u7684\u6b7b\u6d3b\u72b6\u6001\u201d\u2014\u2014\u65e2\u5728\u8fd9\u4e2a\u793e\u4f1a\u91cc\u6d3b\u7740\uff0c\u53c8\u597d\u50cf\u5df2\u7ecf\u6b7b\u4e86\uff1b\u65e2\u662f\u52b3\u52a8\u529b\uff0c\u53c8\u662f\u5e9f\u6599\u2014\u2014\u8fd9\u79cd\u72b6\u6001\u4e0b\uff0c\u7cfb\u7edf\u80fd\u69a8\u53d6\u7684\u4fe1\u606f\u91cf\uff0c\u6216\u8005\u8bf4\u4ece\u6df7\u4e71\u4e2d\u63d0\u53d6\u7684\u79e9\u5e8f\u662f\u6700\u5927\u7684\u3002\n\n\u4e3a\u4ec0\u4e48\u662f1\/e\u7a33\u5b9a\uff1f\n\n\u56e0\u4e3a\u5982\u679c\u8fd9\u4e2a\u6bd4\u4f8b\u5c0f\u4e8e37%\uff0c\u6bd4\u5982\u53ea\u670910%\u7684\u4eba\u7a77\uff0c\u90a3\u5269\u4e0b90%\u7684\u4eba\u4f1a\u89c9\u5f97\u81ea\u5df1\u5f88\u5b89\u5168\uff0c\u4ed6\u4eec\u4f1a\u8054\u5408\u8d77\u6765\u5bf9\u6297\u8d44\u672c\uff0c\u8d44\u672c\u7684\u5229\u6da6\u7387\u4f1a\u4e0b\u964d\u3002\n\u5982\u679c\u8fd9\u4e2a\u6bd4\u4f8b\u5927\u4e8e37%\uff0c\u6bd4\u598250%\u7684\u4eba\u90fd\u6d3b\u4e0d\u4e0b\u53bb\u4e86\uff0c\u90a3\u793e\u4f1a\u7684\u6df7\u4e71\u7a0b\u5ea6\uff08\u71b5\uff09\u4f1a\u8d85\u8fc7\u8b66\u5bdf\u7cfb\u7edf\u7684\u63a7\u5236\u80fd\u529b\uff0c\u5bcc\u4eba\u7684\u8d44\u4ea7\u5b89\u5168\u5c31\u6ca1\u6709\u4fdd\u969c\u3002\n\n\u53ea\u6709\u57281\/e\u8fd9\u4e2a\u4f4d\u7f6e\uff1a\n\n    \u8db3\u591f\u591a\u7684\u4eba\u5728\u53d7\u82e6\uff0c\u591a\u5230\u8ba9\u4f60\u4e0b\u697c\u4e70\u5496\u5561\u65f6\u4e0d\u5f97\u4e0d\u8de8\u8fc7\u6d41\u6d6a\u6c49\u7684\u8eab\u4f53\uff0c\u8ba9\u4f60\u65f6\u523b\u5728\u8fd9\u4e2a\u5de8\u5927\u7684\u201c\u6837\u672c\u5e93\u201d\u9762\u524d\u745f\u745f\u53d1\u6296\uff0c\u4e0d\u6562\u8fdf\u5230\u65e9\u9000\u3002\n    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   \u798f\u5229\u53d1\u591a\u4e86\uff0c\u901a\u80c0\u5c31\u628a\u5e95\u5c42\u6253\u56de\u539f\u5f62\u3002\n    \u538b\u69a8\u592a\u72e0\u4e86\uff0c\u96f6\u5143\u8d2d\u548c\u9a9a\u4e71\u5c31\u4f1a\u903c\u7740\u653f\u5e9c\u53d1\u70b9\u7cae\u7968\u3002\n\n\u6700\u7ec8\uff0c\u793e\u4f1a\u5c31\u50cf\u4e00\u4e2a\u88ab\u8bbe\u5b9a\u597d\u7a0b\u5e8f\u7684\u5404\u79cd\u7269\u7406\u7c92\u5b50\uff0c\u81ea\u52a8\u505c\u5728\u4e86\u80fd\u91cf\u6700\u4f4e\u3001\u7ed3\u6784\u6700\u7a33\u5b9a\u7684\u5751\u91cc\u3002\u800c\u8fd9\u4e2a\u5751\u7684\u6df1\u5ea6\uff0c\u6070\u597d\u5c31\u662f\u81ea\u7136\u5bf9\u6570\u7684\u5e95\u3002\n\n\u8fd9\u624d\u662f\u6700\u6050\u6016\u7684\u3002\n\u5982\u679c\u662f\u56e0\u4e3a\u6709\u4eba\u574f\uff0c\u90a3\u4f60\u8fd8\u53ef\u4ee5\u628a\u574f\u4eba\u6302\u8def\u706f\u3002\n\u4f46\u5982\u679c\u8fd9\u662f\u6570\u5b66\u89c4\u5f8b\uff0c\u662f\u79c1\u6709\u5236+\u81ea\u7531\u5e02\u573a+\u6781\u5ea6\u5f02\u5316\u540e\u7684\u5fc5\u7136\u6570\u5b66\u6536\u655b\u2026\u2026\n\n\u90a3\u4f60\u9762\u5bf9\u7684\u4e0d\u662f\u4e00\u4e2a\u66b4\u541b\uff0c\u800c\u662f\u4e00\u4e2a\u51b0\u51b7\u7684\u3001\u4e0d\u53ef\u8fdd\u6297\u7684\u7269\u7406\u5b9a\u5f8b\u3002\u90a337%\u7684\u4eba\uff0c\u4e0d\u662f\u4e0d\u5e78\uff0c\u4ed6\u4eec\u662f\u8fd9\u4e2a\u65b9\u7a0b\u91cc\u5fc5\u987b\u88ab\u727a\u7272\u6389\u7684\u90a3\u4e2a\u5206\u6bcd\uff0c\u4e3a\u4e86\u8ba9\u5206\u5b50\u663e\u5f97\u66f4\u6709\u610f\u4e49\u3002\n\n\u8fd9\u5c31\u662f\u73b0\u5b9e\u4e2d\u7684\u5b8c\u7f8e\u7684\u8d44\u672c\u4e3b\u4e49\u5730\u72f1\uff1a\u8fd9\u91cc\u6ca1\u6709\u9b3c\u795e\uff0c\u53ea\u6709\u7cbe\u51c6\u5f97\u4ee4\u4eba\u4f5c\u5455\u7684\u7b97\u672f\u3002\n\n\u56de\u7b544\uff1a\n\u6ce8\u610f\u529b\u975e\u5e38\u60ca\u4eba\uff0c\u582a\u6bd4\u62c9\u9a6c\u52aa\u91d1\u3002\n\n\u591a\u8bf4\u4e00\u53e5\uff0c\u8fd9\u4e2a\u7edf\u8ba1\u6570\u636e\u63a5\u8fd137%\u591a\u534a\u662f\u4e00\u79cd\u5de7\u5408\u3002\n\n\u4f46\u662f\u5f88\u591a\u60c5\u51b5\u4e0b\u51fa\u73b037%\u8ddf1\/e\u5012\u771f\u662f\u6709\u5173\u7cfb\u3002\n\n\u8981\u4e86\u89e3\u8fd9\u4e2a\uff0c\u6211\u4eec\u5148\u8981\u6765\u4e86\u89e3\u4e00\u4e0b\uff0c\u4ec0\u4e48\u662f\u81ea\u7136\u5e95\u6570e\uff1f\n\ne\u22482.71828182845904523536...\n\ne\u662f\u4e00\u4e2a\u8d85\u8d8a\u6570\uff0c\u662f\u6307\u4e00\u79cd\u4e0d\u53ef\u4ee5\u4f5c\u4e3a\u67d0\u4e2a\u4ee3\u6570\u65b9\u7a0b\u7684\u6839\u7684\u5b9e\u6570\u6216\u590d\u6570\u3002\n\ne\u8fd9\u73a9\u610f\u770b\u8d77\u6765\u4e00\u70b9\u90fd\u4e0d\u201c\u81ea\u7136\u201d\uff0c\u5b83\u5230\u5e95\u662f\u4ec0\u4e48\u81ea\u7136\u7684\u5e95\u6570\uff1f\n\n\u522b\u614c\uff0c\u6211\u4eec\u753b\u4e2a\u65b9\u7a0b\uff1a\n\ny= e^x\n\n\u55ef\uff0c\u5e73\u5e73\u65e0\u5947\u561b\uff0c\u4e0d\u7740\u6025\uff0c\u6211\u4eec\u628a\u5b83\u6362\u6210\u6781\u5750\u6807\uff1a\n\n\\\\\u5728\u6781\u5750\u6807\u4e2d\uff0cx = \\cos(\\theta) \u548c y=sin\u2061(\u03b8)\uff0c\u56e0\u6b64\uff1a sin\u2061(\u03b8) = e^{\\cos(\\theta)}\n\n<img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"15889\" data-permalink=\"https:\/\/opgogo.com\/blog2\/?attachment_id=15889\" data-orig-file=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-c81db1ec71d0c5eca4127e21e314ee46_1440w.jpg\" data-orig-size=\"330,330\" data-comments-opened=\"0\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"v2-c81db1ec71d0c5eca4127e21e314ee46_1440w\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-c81db1ec71d0c5eca4127e21e314ee46_1440w.jpg\" src=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-c81db1ec71d0c5eca4127e21e314ee46_1440w-300x300.jpg\" alt=\"\" width=\"300\" height=\"300\" class=\"alignnone size-medium wp-image-15889\" srcset=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-c81db1ec71d0c5eca4127e21e314ee46_1440w-300x300.jpg 300w, https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-c81db1ec71d0c5eca4127e21e314ee46_1440w-150x150.jpg 150w, https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-c81db1ec71d0c5eca4127e21e314ee46_1440w.jpg 330w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>\u8fd9\u4e2a\u56fe\u50cf\u773c\u719f\u4e86\u4e00\u4e9b\u5417\uff1f<\/p>\n<p>\u662f\u4e0d\u662f\u5f88\u63a5\u8fd1\u81ea\u7136\u754c\u7684\u5404\u79cd\u87ba\u7ebf\uff1f<br \/>\n\u5927\u81ea\u7136\u7684\u5bf9\u6570\u87ba\u65cb<\/p>\n<p>\u5728\u51e0\u79cd\u81ea\u7136\u73b0\u8c61\u4e2d\uff0c\u4f60\u4f1a\u53d1\u73b0\u51e0\u4e4e\u662f\u5bf9\u6570\u87ba\u65cb\u7684\u66f2\u7ebf\u3002\u6bd4\u5982\uff1a<\/p>\n<p>    \u9e70\u627e\u5230\u730e\u7269\uff1a\u4ed6\u4eec\u6700\u6e05\u6670\u7684\u89c6\u91ce\u4e0e\u4ed6\u4eec\u7684\u98de\u884c\u65b9\u5411\u6709\u4e00\u5b9a\u7684\u89d2\u5ea6\u3002\u8fd9\u4e2a\u89d2\u5ea6\u4e0e\u5bf9\u6570\u87ba\u65cb\u7684\u5939\u89d2\u76f8\u540c\u3002<br \/>\n<img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"15890\" data-permalink=\"https:\/\/opgogo.com\/blog2\/?attachment_id=15890\" data-orig-file=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-9fb2657e16176808e711668805c9ecb8_1440w.webp\" data-orig-size=\"348,408\" data-comments-opened=\"0\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"v2-9fb2657e16176808e711668805c9ecb8_1440w\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-9fb2657e16176808e711668805c9ecb8_1440w.webp\" src=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-9fb2657e16176808e711668805c9ecb8_1440w-256x300.webp\" alt=\"\" width=\"256\" height=\"300\" class=\"alignnone size-medium wp-image-15890\" srcset=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-9fb2657e16176808e711668805c9ecb8_1440w-256x300.webp 256w, https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-9fb2657e16176808e711668805c9ecb8_1440w.webp 348w\" sizes=\"auto, (max-width: 256px) 100vw, 256px\" \/><\/p>\n<p>    \u6606\u866b\u5230\u5149\u6e90\uff1a\u5b83\u4eec\u4e60\u60ef\u4e8e\u5c06\u5149\u6e90\u4e0e\u98de\u884c\u8def\u5f84\u4fdd\u6301\u6052\u5b9a\u89d2\u5ea6\u3002\u901a\u5e38\uff0c\u592a\u9633\uff08\u6216\u591c\u95f4\u7269\u79cd\u7684\u6708\u4eae\uff09\u662f\u552f\u4e00\u7684\u5149\u6e90\uff0c\u4ee5\u8fd9\u79cd\u65b9\u5f0f\u98de\u884c\u5c06\u5bfc\u81f4\u51e0\u4e4e\u76f4\u7ebf\u3002 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data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"v2-6c48819b0e9caeae38ed21e7dbea95aa_r\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-6c48819b0e9caeae38ed21e7dbea95aa_r.jpg\" src=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-6c48819b0e9caeae38ed21e7dbea95aa_r-300x89.jpg\" alt=\"\" width=\"300\" height=\"89\" class=\"alignnone size-medium wp-image-15892\" srcset=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-6c48819b0e9caeae38ed21e7dbea95aa_r-300x89.jpg 300w, https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-6c48819b0e9caeae38ed21e7dbea95aa_r-768x229.jpg 768w, https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-6c48819b0e9caeae38ed21e7dbea95aa_r.jpg 949w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>    \u89d2\u819c\u795e\u7ecf \uff08\u8fd9\u662f\u51e0\u79cd\u4e0d\u540c\u52a8\u7269\u7684\u89d2\u819c\u795e\u7ecf\uff0c\u4ee5\u5bf9\u87ba\u65cb\u6a21\u5f0f\u5728\u89d2\u819c\u7684\u6d45\u8868\u76ae\u5c42\u9644\u8fd1\u7ec8\u6b62\uff09\u3002<br \/>\n<img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"15891\" data-permalink=\"https:\/\/opgogo.com\/blog2\/?attachment_id=15891\" data-orig-file=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-97cde3deb99b36cd5bb497fda40c8fd3_r.jpg\" data-orig-size=\"736,670\" data-comments-opened=\"0\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"v2-97cde3deb99b36cd5bb497fda40c8fd3_r\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-97cde3deb99b36cd5bb497fda40c8fd3_r.jpg\" src=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-97cde3deb99b36cd5bb497fda40c8fd3_r-300x273.jpg\" alt=\"\" width=\"300\" height=\"273\" class=\"alignnone size-medium wp-image-15891\" srcset=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-97cde3deb99b36cd5bb497fda40c8fd3_r-300x273.jpg 300w, https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-97cde3deb99b36cd5bb497fda40c8fd3_r.jpg 736w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>    \u70ed\u5e26\u6c14\u65cb\uff0c\u4f8b\u5982\u98d3\u98ce\u3002<br \/>\n    \u8bb8\u591a\u751f\u7269\u7ed3\u6784\uff0c\u5305\u62ec\u8f6f\u4f53\u52a8\u7269\u7684\u58f3\u3002<\/p>\n<p>\u7531\u6b64\u53ef\u89c1e\u5728\u81ea\u7136\u754c\u5b58\u5728\u975e\u5e38\u5e7f\u6cdb\uff0c\u4f46\u662f\u8fd9\u8fd8\u4e0d\u662f\u5b83\u53eb\u505a\u81ea\u7136\u5e95\u6570\u7684\u539f\u56e0\u3002<\/p>\n<p>e\u662f\u8fd9\u6837\u5b9a\u4e49\u7684\uff1a<\/p>\n<p>\u5f53n\u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927\u65f6,(1 + 1\/n)^n\u7684\u6781\u9650,\u5373e = lim_{n\u2192\u221e} (1 + 1\/n)^n<\/p>\n<p>\u5b83\u4e4b\u6240\u4ee5\u53eb\u81ea\u7136\u5e95\u6570\uff0c\u6e90\u81ea\u5b83\u90a3\u53e4\u5e0c\u814a\u54f2\u5b66\u4e00\u822c\u7684\u5f62\u5f0f\u7f8e\u611f\uff1ae \u7684\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\u662f\u5b83\u662f\u552f\u4e00\u4e00\u4e2a\u4f7f\u5f97\u5176\u5bfc\u6570\u7b49\u4e8e\u81ea\u8eab\u7684\u6570\u3002\u4e5f\u5c31\u662f\u8bf4\uff1a<\/p>\n<p>\u51fd\u6570 f(x)= e^x\u7684\u5bfc\u6570 f\u2032(x)\u4e5f\u7b49\u4e8e e^x<\/p>\n<p>\u8fd9\u4f7f\u5f97\u5b83\u5728\u5fae\u79ef\u5206\u548c\u6570\u5b66\u5206\u6790\u4e2d\u975e\u5e38\u91cd\u8981\u3002<br \/>\n\u7b80\u5355\u6765\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u8ba4\u4e3ae\u7684\u672c\u8d28\u662f\u8fde\u7eed\u4e0d\u65ad\u7684\u589e\u957f\uff08\u6216\u8870\u51cf\uff09\u3002<\/p>\n<p>\u73b0\u5b9e\u4e16\u754c\u4e2d\uff0c\u6781\u5c11\u6709\u4e8b\u60c5\u662f\u50cf\u5149\u7535\u6548\u5e94\u4e00\u6837\u5230\u4e00\u5b9a\u6570\u503c\u624d\u6fc0\u53d1\uff0c\u5927\u591a\u6570\u4e8b\u60c5\uff08\u7ec6\u83cc\u5206\u88c2\u3001\u8c23\u8a00\u4f20\u64ad\u3001\u8d44\u4ea7\u589e\u503c\uff09\u662f\u6bcf\u5206\u6bcf\u79d2\u90fd\u5728\u53d1\u751f\u7684\u3002\u6240\u4ee5\u7528\u5230e\u7684\u5730\u65b9\u4f1a\u5f88\u591a\u3002<\/p>\n<p>\u6bd4\u5982\u5047\u8bbe\u4f60\u6709 1 \u5757\u94b1\uff0c\u94f6\u884c\u5f88\u5927\u65b9\uff0c\u5e74\u5229\u7387 100%\u3002<\/p>\n<p>\u4e00\u5e74\u7ed3\u4e00\u6b21\u606f\uff1a \u5e74\u672b\u4f60\u6709 2 \u5757\u94b1\u3002<\/p>\n<p>\u534a\u5e74\u7ed3\u4e00\u6b21\u606f\uff0c\u5230\u4e86\u5e74\u5e95\u4f60\u5c31\u6709 \uff1a<br \/>\n(1+1\/2)2=2.25(1+1\/2)2=2.25<br \/>\n\u6bcf\u5929\u7ed3\u4e00\u6b21\u606f\uff0c\u5230\u4e86\u5e74\u5e95\u4f60\u5c31\u6709 \uff1a<br \/>\n(1+1\/365)365\u22482.714(1+1\/365)365\u22482.714<\/p>\n<p>\u8981\u662f\u94f6\u884c\u7528\u7535\u8111\u7ed9\u4f60\u6bcf\u79d2\u3001\u6bcf\u6beb\u79d2\u3001\u8fde\u7eed\u4e0d\u65ad\u5730\u7ed3\u606f\u5462\uff1f<\/p>\n<p>\u90a3\u5230\u4e86\u5e74\u5e95\u4f60\u5c31\u6709 \\lim_{n \\to \\infty} (1 + \\frac{1}{n})^n = e \\approx 2.71828&#8230;<\/p>\n<p>\u8fd9\u5c31\u662f\u8fd9\u5c31\u662f\u8fde\u7eed\u590d\u5229\u516c\u5f0f\u7684\u7531\u6765 :<\/p>\n<p>A = P \u22c5 e ^r t<\/p>\n<p>\u5b83\u544a\u8bc9\u6211\u4eec\uff0c\u9760\u201c\u5229\u6eda\u5229\u201d\u7684\u9891\u7387\u6765\u589e\u52a0\u8d22\u5bcc\u662f\u6709\u5929\u82b1\u677f\u7684\u3002<\/p>\n<p>\u4f60\u5728\u4e00\u5e74\u7684\u65f6\u95f4\u91cc\u4e0d\u80fd\u901a\u8fc7\u65e0\u9650\u6b21\u5730\u7ed3\u7b97\u5229\u606f\u6765\u628a 1 \u5757\u94b1\u53d8\u6210 100 \u4e07\uff0c\u5b87\u5b99\u7684\u6781\u9650\u9501\u6b7b\u5728\u4e86 2.718 \u500d\u5de6\u53f3\uff08\u5728 100% \u5229\u7387\u4e0b\uff09\u3002<\/p>\n<p>\u90a3\u4e48\uff0c\u590d\u5229\u589e\u957f\u53ef\u4ee5\u7528e\uff0c\u4ef7\u503c\u8870\u51cf\u80fd\u4e0d\u80fd\u7528e\u5462\uff1f<\/p>\n<p>\u4e5f\u662f\u53ef\u4ee5\u7684\u3002<br \/>\n\u6211\u7ecf\u5e38\u5728\u7f51\u4e0a\u4e70\u4e8c\u624b\u7684\u4e1c\u897f\u7535\u5b50\u4ea7\u54c1\uff0c\u4f1a\u53d1\u73b0\u4e00\u4e2a\u6709\u8da3\u7684\u73b0\u8c61\u3002<\/p>\n<p>\u6240\u6709\u65b0\u7684\u7535\u5b50\u4ea7\u54c1\uff0c\u53ea\u8981\u62c6\u5f00\u6fc0\u6d3b\u4f7f\u7528\u5c31\u5f00\u59cb\u8d2c\u503c\uff0c\u4e0d\u7ba1\u4f60\u662f\u7528\u4e00\u6b21\u8fd8\u662f\u5341\u51e0\u6b21\uff0c\u90fd\u4f1a\u8dcc\u4e00\u4e2a\u8ba9\u4f60\u8089\u75bc\u7684\u4ef7\u683c\uff0c\u5927\u51e0\u5343\u7684\u65b0\u673a\u5668\uff0c\u4f7f\u7528\u51e0\u6b21\u540e\u4e0d\u662f\u4f60\u964d\u4ef7\u51e0\u5341\u6216\u8005100\u5757\u94b1\u5c31\u80fd\u5356\u51fa\u53bb\u7684\uff0c\u6bd4\u5982\u6211\u4e70\u4e86\u4e00\u4e2a12\u67081\u65e5\u624d\u6fc0\u6d3b\u4ec5\u4f7f\u7528\u8fc7\u4e00\u6b21\u7684\u65e0\u4eba\u673a\uff0c\u539f\u4ef71500\uff0c\u5356\u5bb6\u6700\u7ec81200\u51fa\u552e\u3002<\/p>\n<p>\u5927\u591a\u6570\u7535\u5b50\u5546\u54c1\u4f7f\u7528\u4e00\u5e74\u5de6\u53f3\uff0c\u4ef7\u683c\u4f1a\u8dcc\u5230\u539f\u4ef7\u7684\u4e00\u534a\u4ee5\u4e0a\u3002<\/p>\n<p>\u4f46\u662f\u4f60\u8fd9\u4e2a\u65f6\u5019\u4e70\u4e8c\u624b\u7684\u518d\u7528\u4e24\u4e2a\u6708\uff0c\u4ef7\u683c\u4e0d\u4f1a\u53d8\u6210\u539f\u4ef7\u76841\/4\uff0c\u800c\u662f\u4f9d\u7136\u80fd\u4ee5\u5dee\u4e0d\u591a\u59cb\u53d1\u4ef7\u534a\u4ef7\u5356\u51fa\u53bb\u3002<\/p>\n<p>\u5176\u5b9e\u4ed4\u7ec6\u7b97\u7b97\u5e76\u975e\u4e00\u534a\uff0c\u800c\u662f\u63a5\u8fd136.8%\uff0c\u4e5f\u5c31\u662fe\u7684\u5012\u65701\/e\u3002<\/p>\n<p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u4f60\u4eca\u5929\u4e70\u4e86\u4e00\u4e2a\u4e00\u4e07\u7684\u65b0\u82f9\u679c\u624b\u673a\uff0c\u4f46\u662f\u4f60\u6253\u5f00\u6fc0\u6d3b\u7684\u4e00\u77ac\u95f4\u5c31\u5f00\u59cb\u8d2c\u503c\u4e86\u3002<\/p>\n<p>\u4f60\u652f\u4ed8\u7684\u8d2d\u7f6e\u7a0e\u3001\u4fdd\u4fee\u3001\u65b0\u624b\u673a\u6ea2\u4ef7\u7b49\u8d39\u7528\uff0c\u5728\u8f6c\u624b\u65f6\u6ca1\u4eba\u4f1a\u8003\u8651\u3002<\/p>\n<p>\u8fd9\u90e8\u5206\u8d2c\u503c\u662f\u5373\u65f6\u4e14\u5267\u70c8\u7684\uff0c\u901a\u5e38\u5728\u7b2c\u4e00\u4e2a\u6708\u5c31\u80fd\u635f\u5931\u539f\u4ef7\u7684 10% \u5230 20% \u751a\u81f3\u66f4\u9ad8\u3002<\/p>\n<p>\u4e5f\u5c31\u662f\u8bf4\u4e0d\u7ba1\u4f60\u662f\u5f00\u4e86\u4e00\u5468\u8fd8\u662f\u4e00\u4e2a\u6708\uff0c\u8981\u5356\u7684\u8bdd\u4f30\u8ba1\u90fd\u8981\u670910%\u5de6\u53f3\u7684\u8dcc\u4ef7\u635f\u8017\u3002<\/p>\n<p>\u8fc7\u4e86\u8fd9\u4e2a\u5267\u70c8\u8870\u51cf\u671f\u95f4\uff0c\u540e\u9762\u4ef7\u683c\u7684\u53d8\u5316\u5c31\u6309\u7167\u6307\u6570\u8870\u51cf\uff1a<\/p>\n<p>V(t) = V_0 \\cdot e^{-kt}<\/p>\n<p>V\u6307\u7684\u662f\u5546\u54c1\u7684\u521d\u59cb\u4ef7\u503c\uff08\u6bd4\u5982\u65b0\u8f66\u7684\u8d2d\u4e70\u4ef7\uff09\u3002<\/p>\n<p>k\u8870\u51cf\u7387\u5e38\u6570\u3002\u8fd9\u4e2a\u503c\u8d8a\u5927\uff0c\u5546\u54c1\u8d2c\u503c\u8d8a\u5feb\u3002<\/p>\n<p>\u8fd9\u4e2a\u516c\u5f0f\u610f\u5473\u7740\u4f60\u4e70\u7684\u5546\u54c1\u7684\u4ef7\u503c\u4e0d\u662f\u56fa\u5b9a\u6570\u91cf\u7684\u51cf\u5c11\uff0c\u4e5f\u4e0d\u662f\u56fa\u5b9a\u767e\u5206\u6bd4\u7684\u51cf\u5c11\uff08\u6bd4\u5982\u6bcf\u5e74\u90fd\u51cf\u5c11\u51e0\u5343\u5757\u94b1\uff09\uff0c\u800c\u662f\u6bcf\u5355\u4f4d\u65f6\u95f4\u51cf\u5c11\u4e00\u4e2a\u56fa\u5b9a\u767e\u5206\u6bd4\u3002\u800c\u6b63\u662f\u8fd9\u79cd\u201c\u8fde\u7eed\u590d\u5229\u5f0f\u201d\u8870\u51cf\u7684\u6570\u5b66\u57fa\u7840\u3002<\/p>\n<p>\u5047\u5982\u6211\u4eec\u53d6k=1\uff0c\u5355\u4f4d\u65f6\u95f4\u4e00\u5e74\uff0c\u4e5f\u53d6\u505a1\u3002<\/p>\n<p>\u90a3\u4e48\u6211\u4eec\u7684\u82f9\u679c\u624b\u673a\u4e00\u5e74\u540e\u4f1a\u8d2c\u503c\u5230\u539f\u4ef7\u683c\u76841\/e\uff0c\u5c31\u7ea6\u7b49\u4e8e36.8%<\/p>\n<p>\u5f53\u7136\uff0c\u73b0\u5b9e\u751f\u6d3b\u4e2d\u82f9\u679c\u8d2c\u503c\u6ca1\u8fd9\u4e48\u5feb\uff0c\u8fd9\u91cc\u5355\u4f4d\u65f6\u95f4\u8981\u4e45\u4e00\u70b9\u3002<\/p>\n<p>\u6211\u8fd9\u91cc\u627e\u4e86\u4e00\u4e2a\u62112021\u5e74\u4e70\u7684\u82f9\u679c12\uff1a<br \/>\n<img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"15893\" data-permalink=\"https:\/\/opgogo.com\/blog2\/?attachment_id=15893\" data-orig-file=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-4ee37c47092d23e21cb4fbb4768b14ac_1440w.webp\" data-orig-size=\"1206,1769\" data-comments-opened=\"0\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"v2-4ee37c47092d23e21cb4fbb4768b14ac_1440w\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/opgogo.com\/blog2\/wp-content\/uploads\/2025\/12\/v2-4ee37c47092d23e21cb4fbb4768b14ac_1440w-698x1024.webp\" 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