41 lines
12 KiB
JSON
41 lines
12 KiB
JSON
[
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{
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"username": "abigwc",
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"password": "$2b$12$OVddteRc.blsLo6PFeun7.oZWOJjcCRYWW7/PH6irlDVfXpwp.p8O",
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"id": 1778114304058,
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"createdAt": "2026-05-07T08:38:24.058955",
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"history": [
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{
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"id": 1778116342815,
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"timestamp": "2026-05-07T09:12:22.815205",
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"summary": "\u8981\u901a\u4fd7\u6613\u61c2\u5730\u7406\u89e3**\u6cf0\u52d2\u516c\u5f0f\uff08Taylor's Formula\uff09**\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u629b\u5f00\u590d\u6742\u7684\u6570\u5b66\u7b26\u53f7\uff0c...",
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"markdown": "\u8981\u901a\u4fd7\u6613\u61c2\u5730\u7406\u89e3**\u6cf0\u52d2\u516c\u5f0f\uff08Taylor's Formula\uff09**\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u629b\u5f00\u590d\u6742\u7684\u6570\u5b66\u7b26\u53f7\uff0c\u4ece\u5b83\u7684**\u6838\u5fc3\u601d\u60f3**\u8bf4\u8d77\u3002\n\n### \u4e00\u3001 \u6cf0\u52d2\u516c\u5f0f\u7684\u6838\u5fc3\u601d\u60f3\uff1a\u7528\u201c\u7b80\u5355\u201d\u66ff\u6362\u201c\u590d\u6742\u201d\n\n\u5728\u6570\u5b66\u4e2d\uff0c\u50cf\u52a0\u3001\u51cf\u3001\u4e58\u3001\u9664\u8fd9\u6837\u7684\u8fd0\u7b97\u662f\u975e\u5e38\u7b80\u5355\u7684\uff08\u8ba1\u7b97\u673a\u548c\u8ba1\u7b97\u5668\u6700\u64c5\u957f\u505a\u8fd9\u4e9b\uff09\uff0c\u7531\u8fd9\u4e9b\u8fd0\u7b97\u7ec4\u6210\u7684\u5f0f\u5b50\u53eb\u505a**\u591a\u9879\u5f0f**\uff08\u6bd4\u5982 $2x^2 + 3x + 1$\uff09\u3002\n\n\u4f46\u662f\uff0c\u50cf $\\sin(x)$\u3001$\\cos(x)$\u3001$e^x$\u3001$\\ln(x)$ \u8fd9\u4e9b**\u8d85\u8d8a\u51fd\u6570**\uff0c\u8ba1\u7b97\u8d77\u6765\u975e\u5e38\u590d\u6742\u3002\u4f60\u6709\u6ca1\u6709\u60f3\u8fc7\uff0c\u5f53\u4f60\u5728\u8ba1\u7b97\u5668\u4e0a\u6309\u4e0b $\\sin(37^\\circ)$ \u65f6\uff0c\u8ba1\u7b97\u5668\u5185\u90e8\u662f\u600e\u4e48\u7b97\u51fa\u6765\u7684\uff1f\u8ba1\u7b97\u5668\u53ef\u4e0d\u4f1a\u753b\u4e00\u4e2a\u76f4\u89d2\u4e09\u89d2\u5f62\u53bb\u91cf\u8fb9\u957f\uff0c\u5b83\u5176\u5b9e\u662f\u7528**\u52a0\u51cf\u4e58\u9664**\u7b97\u51fa\u6765\u7684\u4e00\u4e2a\u8fd1\u4f3c\u503c\u3002\n\n**\u6cf0\u52d2\u516c\u5f0f\u7684\u4f5c\u7528\uff0c\u5c31\u662f\u642d\u5efa\u4e86\u4e00\u5ea7\u6865\u6881\uff1a\u5b83\u80fd\u628a\u4e00\u5207\u590d\u6742\u7684\u51fd\u6570\uff0c\u8f6c\u5316\u4e3a\u7b80\u5355\u7684\u201c\u591a\u9879\u5f0f\u201d\u3002**\n\n\u7b80\u5355\u6765\u8bf4\uff1a**\u6cf0\u52d2\u516c\u5f0f\u662f\u7528\u591a\u9879\u5f0f\u53bb\u201c\u6a21\u4eff\u201d\u548c\u201c\u903c\u8fd1\u201d\u4e00\u4e2a\u590d\u6742\u66f2\u7ebf\u7684\u65b9\u6cd5\u3002**\n\n---\n\n### \u4e8c\u3001 \u5b83\u662f\u600e\u4e48\u201c\u6a21\u4eff\u201d\u7684\uff1f\uff08\u76f4\u89c2\u7406\u89e3\uff09\n\n\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u590d\u6742\u7684\u66f2\u7ebf $f(x)$\uff0c\u6211\u4eec\u60f3\u5728\u67d0\u4e00\u4e2a\u70b9 $x_0$ \u9644\u8fd1\uff0c\u7528\u4e00\u4e2a\u7b80\u5355\u7684\u591a\u9879\u5f0f\u53bb\u6a21\u4eff\u5b83\u3002\n\n1. **\u7b2c\u4e00\u6b65\uff08\u53ea\u6a21\u4eff\u4f4d\u7f6e\uff09\uff1a**\n \u6211\u4eec\u5728 $x_0$ \u70b9\u53d6\u548c\u539f\u51fd\u6570\u4e00\u6837\u7684\u503c\u3002\u8fd9\u76f8\u5f53\u4e8e\u753b\u4e86\u4e00\u6761\u6c34\u5e73\u7ebf\u3002\n $P(x) = f(x_0)$\n *\u6548\u679c\uff1a\u53ea\u6709\u5728\u8fd9\u4e00\u4e2a\u70b9\u662f\u51c6\u7684\uff0c\u7a0d\u5fae\u504f\u79bb\u4e00\u70b9\u5c31\u9519\u5f97\u79bb\u8c31\u3002*\n\n2. **\u7b2c\u4e8c\u6b65\uff08\u6a21\u4eff\u4f4d\u7f6e + \u503e\u659c\u5ea6\uff09\uff1a**\n \u4e0d\u4ec5\u503c\u8981\u4e00\u6837\uff0c\u8fd9\u6761\u7ebf\u7684\u201c\u503e\u659c\u5ea6\u201d\uff08\u5373**\u4e00\u9636\u5bfc\u6570**\uff09\u4e5f\u8981\u4e00\u6837\u3002\u8fd9\u5c31\u53d8\u6210\u4e86\u539f\u51fd\u6570\u5728 $x_0$ \u70b9\u7684**\u5207\u7ebf**\u3002\n $P(x) = f(x_0) + f'(x_0)(x - x_0)$\n *\u6548\u679c\uff1a\u5728 $x_0$ \u9644\u8fd1\u7684\u4e00\u5c0f\u6bb5\u8ddd\u79bb\u5185\uff0c\u8fd9\u6761\u76f4\u7ebf\u548c\u66f2\u7ebf\u975e\u5e38\u8d34\u5408\u3002*\n\n3. **\u7b2c\u4e09\u6b65\uff08\u6a21\u4eff\u4f4d\u7f6e + \u503e\u659c\u5ea6 + \u5f2f\u66f2\u7a0b\u5ea6\uff09\uff1a**\n \u4e3a\u4e86\u8d34\u5408\u5f97\u66f4\u4e45\u4e00\u70b9\uff0c\u6211\u4eec\u8ba9\u591a\u9879\u5f0f\u7684\u201c\u5f2f\u66f2\u7a0b\u5ea6\u201d\uff08\u5373**\u4e8c\u9636\u5bfc\u6570**\uff09\u4e5f\u548c\u539f\u51fd\u6570\u4e00\u6837\u3002\u8fd9\u5c31\u5f15\u5165\u4e86\u629b\u7269\u7ebf\u3002\n $P(x) = f(x_0) + f'(x_0)(x - x_0) + \\frac{f''(x_0)}{2}(x - x_0)^2$\n *\u6548\u679c\uff1a\u543b\u5408\u7684\u8303\u56f4\u53d8\u5f97\u66f4\u5927\u4e86\u3002*\n\n**\u7ed3\u8bba\uff1a** \u6211\u4eec\u53ea\u8981\u4e0d\u65ad\u5730\u8ba9\u591a\u9879\u5f0f\u7684 $3$\u9636\u3001$4$\u9636...$n$\u9636\u5bfc\u6570\u4e0e\u539f\u51fd\u6570\u76f8\u7b49\uff0c\u8fd9\u4e2a\u591a\u9879\u5f0f\u5c31\u4f1a\u548c\u539f\u66f2\u7ebf**\u8d8a\u6765\u8d8a\u50cf\uff0c\u91cd\u5408\u7684\u8303\u56f4\u8d8a\u6765\u8d8a\u5927**\u3002\u8fd9\u5c31\u662f\u6cf0\u52d2\u516c\u5f0f\u7684\u672c\u8d28\uff01\n\n---\n\n### \u4e09\u3001 \u6cf0\u52d2\u516c\u5f0f\u7684\u6570\u5b66\u8868\u8fbe\n\n\u6839\u636e\u4e0a\u9762\u7684\u63a8\u5bfc\uff0c\u628a\u4e00\u4e2a\u5149\u6ed1\u7684\u51fd\u6570 $f(x)$ \u5728\u70b9 $x_0$ \u5904\u5c55\u5f00\uff0c\u53ef\u4ee5\u5f97\u5230\u6807\u51c6\u7684\u6cf0\u52d2\u516c\u5f0f\uff1a\n\n$$f(x) = f(x_0) + f'(x_0)(x - x_0) + \\frac{f''(x_0)}{2!}(x - x_0)^2 + \\frac{f'''(x_0)}{3!}(x - x_0)^3 + \\dots + \\frac{f^{(n)}(x_0)}{n!}(x - x_0)^n + R_n(x)$$\n\n**\u516c\u5f0f\u62c6\u89e3\uff1a**\n* $f^{(n)}(x_0)$\uff1a\u8868\u793a\u51fd\u6570\u5728 $x_0$ \u70b9\u7684 $n$ \u9636\u5bfc\u6570\u3002\n* $n!$\uff1a\u8868\u793a\u9636\u4e58\uff08\u5982 $3! = 3 \\times 2 \\times 1$\uff09\u3002\u4e3a\u4ec0\u4e48\u9700\u8981\u9664\u4ee5\u9636\u4e58\uff1f\u8fd9\u662f\u4e3a\u4e86\u5728\u6c42\u5bfc\u65f6\uff0c\u628a\u591a\u9879\u5f0f\u6389\u4e0b\u6765\u7684\u6307\u6570\u7ed9\u7ea6\u5206\u6389\uff0c\u4fdd\u8bc1\u6bcf\u4e00\u9636\u5bfc\u6570\u90fd\u80fd\u5b8c\u7f8e\u5bf9\u9f50\u3002\n* $R_n(x)$\uff1a**\u4f59\u9879\uff08\u8bef\u5dee\u9879\uff09**\u3002\u56e0\u4e3a\u9664\u975e\u4f60\u5c55\u5f00\u5230\u65e0\u7a77\u9879\uff0c\u5426\u5219\u591a\u9879\u5f0f\u548c\u539f\u51fd\u6570\u603b\u4f1a\u6709\u4e00\u70b9\u70b9\u8bef\u5dee\u3002\u8fd9\u4e2a\u4f59\u9879\u5c31\u662f\u7528\u6765\u8861\u91cf\u201c\u8bef\u5dee\u5230\u5e95\u6709\u591a\u5927\u201d\u7684\uff08\u5e38\u89c1\u7684\u6709\u76ae\u4e9a\u8bfa\u4f59\u9879\u548c\u62c9\u683c\u6717\u65e5\u4f59\u9879\uff09\u3002\n\n#### \ud83d\udca1 \u7279\u4f8b\uff1a\u9ea6\u514b\u52b3\u6797\u516c\u5f0f (Maclaurin Series)\n\u5982\u679c\u6211\u4eec\u9009\u62e9\u628a\u57fa\u51c6\u70b9\u5b9a\u5728\u5750\u6807\u539f\u70b9\uff0c\u4e5f\u5c31\u662f **$x_0 = 0$**\uff0c\u6cf0\u52d2\u516c\u5f0f\u5c31\u4f1a\u53d8\u5f97\u975e\u5e38\u7b80\u6d01\uff0c\u8fd9\u88ab\u79f0\u4e3a**\u9ea6\u514b\u52b3\u6797\u516c\u5f0f**\u3002\u8fd9\u4e5f\u662f\u8003\u8bd5\u548c\u5b9e\u9645\u8ba1\u7b97\u4e2d\u6700\u5e38\u7528\u7684\u5f62\u5f0f\uff1a\n\n$$f(x) \\approx f(0) + f'(0)x + \\frac{f''(0)}{2!}x^2 + \\frac{f'''(0)}{3!}x^3 + \\dots + \\frac{f^{(n)}(0)}{n!}x^n$$\n\n---\n\n### \u56db\u3001 \u7ecf\u5178\u6848\u4f8b\uff1a\u628a $e^x$ \u53d8\u6210\u591a\u9879\u5f0f\n\n\u6211\u4eec\u7528\u9ea6\u514b\u52b3\u6797\u516c\u5f0f\u6765\u5c55\u5f00\u81ea\u7136\u6307\u6570\u51fd\u6570 $f(x) = e^x$\u3002\n\n* $f(x) = e^x$\uff0c\u5f53 $x=0$ \u65f6\uff0c$f(0) = 1$\n* $e^x$ \u65e0\u8bba\u6c42\u591a\u5c11\u6b21\u5bfc\u6570\uff0c\u7ed3\u679c\u90fd\u662f $e^x$\u3002\u6240\u4ee5\u5b83\u5728 $x=0$ \u5904\u7684\u6240\u6709\u5bfc\u6570\u90fd\u662f $1$\u3002\n\n\u628a\u8fd9\u4e9b $1$ \u4ee3\u5165\u9ea6\u514b\u52b3\u6797\u516c\u5f0f\uff0c\u5947\u8ff9\u51fa\u73b0\u4e86\uff1a\n\n$$e^x = 1 + x + \\frac{1}{2!}x^2 + \\frac{1}{3!}x^3 + \\frac{1}{4!}x^4 + \\dots$$\n\n**\u8fd9\u5c31\u662f $e^x$ \u7684\u6cf0\u52d2\u5c55\u5f00\uff01**\n\u4f60\u53ef\u4ee5\u7528\u5b83\u6765\u624b\u5de5\u8ba1\u7b97\u81ea\u7136\u5e38\u6570 $e$ \u7684\u503c\uff08\u4ee4 $x=1$\uff09\uff1a\n$$e^1 \\approx 1 + 1 + \\frac{1}{2} + \\frac{1}{6} + \\frac{1}{24} = 2.7083...$$\n\u4f60\u770b\uff0c\u4ec5\u4ec5\u7b97\u4e86\u524d5\u9879\uff0c\u5c31\u5df2\u7ecf\u975e\u5e38\u63a5\u8fd1 $e$ \u7684\u771f\u5b9e\u503c\uff08$2.71828...$\uff09\u4e86\uff01\n\n---\n\n### \u4e94\u3001 \u6cf0\u52d2\u516c\u5f0f\u5728\u73b0\u5b9e\u4e2d\u6709\u5565\u7528\uff1f\n\n1. **\u8ba1\u7b97\u673a/\u8ba1\u7b97\u5668\u5e95\u5c42\u7b97\u6cd5**\uff1a\u521a\u624d\u63d0\u5230\u7684\u7b97 $\\sin(x)$\u3001$e^x$\u3001\u5bf9\u6570\u7b49\uff0c\u8ba1\u7b97\u673a\u5e95\u5c42\u5f88\u591a\u662f\u7528\u6cf0\u52d2\u5c55\u5f00\uff08\u6216\u5176\u4ed6\u7c7b\u4f3c\u7684\u591a\u9879\u5f0f\u903c\u8fd1\u7b97\u6cd5\uff09\u628a\u590d\u6742\u7684\u51fd\u6570\u8f6c\u5316\u4e3a\u5b83\u552f\u4e00\u8ba4\u8bc6\u7684\u52a0\u51cf\u4e58\u9664\u8fd0\u7b97\u3002\n2. **\u7269\u7406\u5b66\u4e2d\u7684\u8fd1\u4f3c\u8ba1\u7b97**\uff1a\u4e2d\u5b66\u7269\u7406\u5b66\u8fc7\u5355\u6446\uff0c\u6709\u4e00\u4e2a\u7ed3\u8bba\uff1a\u201c\u5f53\u6446\u89d2 $\\theta$ \u5f88\u5c0f\u65f6\uff0c$\\sin(\\theta) \\approx \\theta$\u201d\u3002\u4e3a\u4ec0\u4e48\uff1f\u4f60\u770b\u4e00\u4e0b $\\sin(x)$ \u7684\u6cf0\u52d2\u5c55\u5f00\u5c31\u77e5\u9053\u4e86\uff1a\n $$\\sin(x) = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\dots$$\n \u5f53 $x$ \u5f88\u5c0f\uff08\u63a5\u8fd10\uff09\u65f6\uff0c$x^3$\u3001$x^5$ \u8fd9\u4e9b\u540e\u9762\u7684\u9879\u592a\u5c0f\u4e86\uff0c\u53ef\u4ee5\u76f4\u63a5\u5ffd\u7565\u4e0d\u8ba1\uff0c\u6240\u4ee5\u5c31\u5269\u4e0b\u4e86\u7b2c\u4e00\u9879 $\\sin(x) \\approx x$\u3002\u6cf0\u52d2\u516c\u5f0f\u4e3a\u7269\u7406\u5b66\u4e2d\u7684\u201c\u8fd1\u4f3c\u201d\u63d0\u4f9b\u4e86\u4e25\u5bc6\u7684\u6570\u5b66\u4f9d\u636e\u3002\n3. **\u6c42\u6781\u9650\uff08\u9ad8\u6570\u8003\u8bd5\u6740\u624b\u950f\uff09**\uff1a\u5728\u9047\u5230\u590d\u6742\u7684\u6d1b\u5fc5\u8fbe\u6cd5\u5219\u90fd\u6c42\u4e0d\u51fa\u7684\u6781\u9650\u65f6\uff0c\u628a\u51fd\u6570\u7528\u6cf0\u52d2\u516c\u5f0f\u5c55\u5f00\u6210\u591a\u9879\u5f0f\uff0c\u5f80\u5f80\u80fd\u77ac\u95f4\u770b\u6e05\u8c01\u5927\u8c01\u5c0f\uff0c\u76f4\u63a5\u7ea6\u5206\u5f97\u51fa\u7ed3\u679c\u3002\n\n### \u603b\u7ed3\n\u6cf0\u52d2\u516c\u5f0f\u5c31\u50cf\u662f\u6570\u5b66\u754c\u7684**\u201c\u964d\u7ef4\u6253\u51fb\u201d\u5de5\u5177**\u3002\u5b83\u544a\u8bc9\u6211\u4eec\uff1a**\u4efb\u4f55\u4e00\u6761\u5e73\u6ed1\u7684\u590d\u6742\u66f2\u7ebf\uff0c\u90fd\u53ef\u4ee5\u901a\u8fc7\u65e0\u6b62\u5883\u5730\u53e0\u52a0\u7b80\u5355\u7684\u591a\u9879\u5f0f\uff08\u76f4\u7ebf\u3001\u629b\u7269\u7ebf\u3001\u4e09\u6b21\u66f2\u7ebf...\uff09\u6765\u5b8c\u7f8e\u590d\u523b\u3002**"
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}
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]
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},
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{
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"username": "abigwc1",
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"password": "$2b$12$QPXTdvkGJzRJwBKsM69fme7FLyCPg2UetsMxIWuKaH2CZDRsUGlcq",
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"id": 1778291515772,
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"createdAt": "2026-05-09T09:51:55.772279",
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"history": [
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{
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"id": 1778292088055,
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"timestamp": "2026-05-09T10:01:28.055200",
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"summary": "\u7cfb\u7edf\u90fd\u4f1a\u9ed8\u9ed8\u628a\u4f60\u7684\u8349\u7a3f\u5b58\u4e0b\u6765\uff0c\u5e76\u4e14\u643a\u5e26\u6765\u6e90\u8def\u7ebf\u3002\u5f53\u4f60\u5b8c\u6210\u6ce8\u518c\uff08\u6ce8\u518c\u5b8c\u4f1a\u81ea\u52a8\u8df3\u56de\u767b\u5f55\u8ba9\u4f60\u767b\u5165\uff09\u6216\u76f4\u63a5\u767b...",
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"markdown": "\u7cfb\u7edf\u90fd\u4f1a\u9ed8\u9ed8\u628a\u4f60\u7684\u8349\u7a3f\u5b58\u4e0b\u6765\uff0c\u5e76\u4e14\u643a\u5e26\u6765\u6e90\u8def\u7ebf\u3002\u5f53\u4f60\u5b8c\u6210\u6ce8\u518c\uff08\u6ce8\u518c\u5b8c\u4f1a\u81ea\u52a8\u8df3\u56de\u767b\u5f55\u8ba9\u4f60\u767b\u5165\uff09\u6216\u76f4\u63a5\u767b\u5f55\u6210\u529f\u540e\uff0c\u9875\u9762\u90fd\u4f1a\u7cbe\u51c6\u65e0\u7f1d\u5730\u5c06\u4f60\u9001\u56de\u7f16\u8f91\u9875\uff0c\u4e00\u5207\u539f\u5c01\u4e0d\u52a8"
|
|
}
|
|
],
|
|
"last_login": "2026-05-09T10:41:02.911602"
|
|
},
|
|
{
|
|
"username": "admin",
|
|
"password": "$2b$12$mIzQ9cAoiZ0IsmrYy/cXqOyA3Lh1ngHqZVuD.SB/5t5F6Wlz9BUa2",
|
|
"raw_password": "50283279",
|
|
"id": 1778294424851,
|
|
"createdAt": "2026-05-09T10:40:24.851834",
|
|
"register_ip": "127.0.0.1",
|
|
"last_login": "2026-05-09T11:07:18.541802",
|
|
"history": []
|
|
}
|
|
] |