├── .gitignore ├── LICENSE ├── README.md ├── requirements.txt ├── 概率论与数理统计 ├── TASK3_概率论.ipynb ├── TASK4_数理统计.ipynb └── images │ ├── 1.png │ ├── 10.png │ ├── 11.png │ ├── 12.svg │ ├── 13.svg │ ├── 14.svg │ ├── 15.png │ ├── 16.png │ ├── 17.png │ ├── 18.png │ ├── 19.png │ ├── 2.png │ ├── 20.png │ ├── 3.png │ ├── 4.png │ ├── 5.png │ ├── 6.png │ ├── 7.png │ ├── 8.png │ ├── 9.png │ ├── QRCode.jpg │ └── 一维多维知识点.drawio ├── 线性代数 ├── TASK2_线性代数.ipynb └── figures │ ├── 2-1.png │ ├── 2-2.png │ ├── 2-3.jpeg │ ├── 2-4.png │ ├── 2-5.jpg │ ├── 2-6.jpg │ ├── 2-7.png │ └── 2-8.png └── 高等数学 ├── TASK1_高等数学.ipynb └── figures ├── 1-10.png ├── 1-11.png ├── 1-12.jpg ├── 1-13.jpg ├── 1-14.jpg ├── 1-15.jpg ├── 1-16.jpg ├── 1-17.jpg ├── 1-18.png ├── 1-2.png ├── 1-4.png ├── 1-5.png ├── 1-6.jpg ├── 1-7.jpg ├── 1-8.jpg ├── 1-9.jpg ├── 1.jpg ├── fig2.0.1.jpg ├── fig2.0.3.jpg ├── fig2.1.jpg └── fig2.jpg /.gitignore: -------------------------------------------------------------------------------- 1 | # Byte-compiled / optimized / DLL files 2 | __pycache__/ 3 | *.py[cod] 4 | *$py.class 5 | 6 | # C extensions 7 | *.so 8 | 9 | # Distribution / packaging 10 | .Python 11 | build/ 12 | develop-eggs/ 13 | dist/ 14 | downloads/ 15 | eggs/ 16 | .eggs/ 17 | lib/ 18 | lib64/ 19 | parts/ 20 | sdist/ 21 | var/ 22 | wheels/ 23 | pip-wheel-metadata/ 24 | share/python-wheels/ 25 | *.egg-info/ 26 | .installed.cfg 27 | *.egg 28 | MANIFEST 29 | 30 | # PyInstaller 31 | # Usually these files are written by a python script from a template 32 | # before PyInstaller builds the exe, so as to inject date/other infos into it. 33 | *.manifest 34 | *.spec 35 | 36 | # Installer logs 37 | pip-log.txt 38 | pip-delete-this-directory.txt 39 | 40 | # Unit test / coverage reports 41 | htmlcov/ 42 | .tox/ 43 | .nox/ 44 | .coverage 45 | .coverage.* 46 | .cache 47 | nosetests.xml 48 | coverage.xml 49 | *.cover 50 | *.py,cover 51 | .hypothesis/ 52 | .pytest_cache/ 53 | 54 | # Translations 55 | *.mo 56 | *.pot 57 | 58 | # Django stuff: 59 | *.log 60 | local_settings.py 61 | db.sqlite3 62 | db.sqlite3-journal 63 | 64 | # Flask stuff: 65 | instance/ 66 | .webassets-cache 67 | 68 | # Scrapy stuff: 69 | .scrapy 70 | 71 | # Sphinx documentation 72 | docs/_build/ 73 | 74 | # PyBuilder 75 | target/ 76 | 77 | # Jupyter Notebook 78 | .ipynb_checkpoints 79 | 80 | # IPython 81 | profile_default/ 82 | ipython_config.py 83 | 84 | # pyenv 85 | .python-version 86 | 87 | # pipenv 88 | # According to pypa/pipenv#598, it is recommended to include Pipfile.lock in version control. 89 | # However, in case of collaboration, if having platform-specific dependencies or dependencies 90 | # having no cross-platform support, pipenv may install dependencies that don't work, or not 91 | # install all needed dependencies. 92 | #Pipfile.lock 93 | 94 | # PEP 582; used by e.g. github.com/David-OConnor/pyflow 95 | __pypackages__/ 96 | 97 | # Celery stuff 98 | celerybeat-schedule 99 | celerybeat.pid 100 | 101 | # SageMath parsed files 102 | *.sage.py 103 | 104 | # Environments 105 | .env 106 | .venv 107 | env/ 108 | venv/ 109 | ENV/ 110 | env.bak/ 111 | venv.bak/ 112 | 113 | # Spyder project settings 114 | .spyderproject 115 | .spyproject 116 | 117 | # Rope project settings 118 | .ropeproject 119 | 120 | # mkdocs documentation 121 | /site 122 | 123 | # mypy 124 | .mypy_cache/ 125 | .dmypy.json 126 | dmypy.json 127 | 128 | # Pyre type checker 129 | .pyre/ 130 | 131 | # idea 132 | *.idea*/ 133 | 134 | # virtual_documents 135 | *virtual_documents*/ 136 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | Apache License 2 | Version 2.0, January 2004 3 | http://www.apache.org/licenses/ 4 | 5 | TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION 6 | 7 | 1. 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We also recommend that a 185 | file or class name and description of purpose be included on the 186 | same "printed page" as the copyright notice for easier 187 | identification within third-party archives. 188 | 189 | Copyright [yyyy] [name of copyright owner] 190 | 191 | Licensed under the Apache License, Version 2.0 (the "License"); 192 | you may not use this file except in compliance with the License. 193 | You may obtain a copy of the License at 194 | 195 | http://www.apache.org/licenses/LICENSE-2.0 196 | 197 | Unless required by applicable law or agreed to in writing, software 198 | distributed under the License is distributed on an "AS IS" BASIS, 199 | WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 200 | See the License for the specific language governing permissions and 201 | limitations under the License. 202 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # 🇬🇲0️⃣1️⃣0️⃣ - 数学建模的基石 2 | > 英文名：\_\_init_Modeling\_\_ 3 | 4 | 👋这门课里有非常多你熟悉的老朋友，同时也是令你头疼的最佳损友. 在这里，我们将帮你从编程的角度，重新学习微积分、线性代数、概率论、统计学······ 5 | 6 | 有许多朋友一直在吐槽，我学了一系列模型，基础知识能够娓娓道来，结果到比赛来发现学习和实践的内容难以衔接，压根就不会求解😥 7 | 8 | 这样的困难我们也曾遇到过，在我们学习的理论和真实实践之间存在一座看不见的桥梁，因此，在这门课中，我们将带领你最熟悉的高等数学出发，（如果你忘了也没事）我们会从实战的角度出发，只讲最有用的知识，一步步带领你从理论到Python实现。结合实践讲解每一步代码的逻辑，让你彻底了解你曾经学过的数学工具，并从根源上消除理论与实践的鸿沟! 9 | 10 | 并且❗❗❗在此后所有的课程中，我们始终使用Python进行实践，期间不需要再学习其他的数学工具，通过这门课大家不仅可以熟练上手Python，还能用Python打出漂亮的成绩！ 11 | 12 | ## 🎯课程目标 13 | 回顾数学建模中最基本的数学知识，建立起理论-实战的翻译思维，打破**说啥啥都懂，用啥啥不会**的困境！ 14 | 15 | ## 📆课程安排 16 | 1. 动手学**高等数学 ➡️ [Sympy、Scipy、Matplotlib]**【3天】 17 | 2. 动手学**线性代数 ➡️ [Numpy]**【3天】 18 | 3. 动手学**概率论 ➡️ [Sympy、Scipy、Matplotlib]**【4天】 19 | 4. 动手学**数理统计 ➡️ [Sympy、Scipy、Matplotlib]**【4天】 20 | 21 | ## 🛠️我们将用到的工具 22 |

23 | 24 |

25 | 26 | ## 🧑‍💻贡献者 27 | ### 🧑‍🔧课程开发 28 | | 昵称 | 内容 | 联系方式 | 29 | | ------ | ---------------- | --------------------------------------------- | 30 | | 李祖贤 | 概率论与数理统计、随机过程 |1028851587@qq.com| 31 | | 刘斯豪 | 线性代数 |amihua@mail2.gdut.edu.cn| 32 | | 郑伯斌 | 高等数学 |20182232011@m.scnu.cn| 33 | 34 | ### 🏃‍♀️课程运营 35 | 36 | ### 🧩改进优化 37 | 38 |

📫 欢迎沟通鸭！

39 | 40 | - 📧 **邮箱**：gitmodel@163.com 41 | - 📺 **bilibili**：[GitModel](https://space.bilibili.com/1051016998) 42 | 43 |

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44 |

45 | 46 | 47 | 48 |

49 | 50 | **我们非常期待有兴趣的你一同加入，我们共同维护国内第一个数学建模开源社区！** 51 | 52 | -------------------------------------------------------------------------------- /requirements.txt: -------------------------------------------------------------------------------- 1 | sympy 2 | scipy 3 | matplotlib 4 | scikit-image 5 | pandas 6 | baostock -------------------------------------------------------------------------------- /概率论与数理统计/images/1.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Git-Model/__init_Modeling__/29d23a92a18d08e67569371759310d7d6c596172/概率论与数理统计/images/1.png -------------------------------------------------------------------------------- /概率论与数理统计/images/10.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Git-Model/__init_Modeling__/29d23a92a18d08e67569371759310d7d6c596172/概率论与数理统计/images/10.png -------------------------------------------------------------------------------- /概率论与数理统计/images/11.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Git-Model/__init_Modeling__/29d23a92a18d08e67569371759310d7d6c596172/概率论与数理统计/images/11.png -------------------------------------------------------------------------------- /概率论与数理统计/images/12.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 |
样本点、样本空间
样本点、样本空间
随机事件
随机事件
随机事件的概率
随机事件的概率
条件概率
条件概率条件概率的三大公式
乘法公式
乘法公式
全概率公式
全概率公式
贝叶斯公式
贝叶斯公式
随机变量
随机变量
密度函数
密度函数
分布列
分布列
分布函数
分布函数
数字特征
数字特征
均匀分布
均匀分布
正态分布
正态分布
指数分布
指数分布
连续型随机变量
连续型随机变量
0-1分布/伯努利分布
0-1分布/伯努利分布
二项分布
二项分布
泊松分布
泊松分布
离散型随机变量
离散型随机变量
一维随机变量
一维随机变量
数学期望/均值
数学期望/均值
方差/标准差
方差/标准差
分位数与中位数
分位数与中位数
数字特征
数字特征
随机向量
随机向量
联合密度函数
联合密度函数
联合分布列
联合分布列
联合分布函数
联合分布函数
边际分布
边际分布
多维随机变量
多维随机变量条件分布
条件密度函数/分布列/分布函数
条件密度函数/分布列/分布函数
连续版本
全概率公式
连续版本 全概率公式
连续版本
贝叶斯公式
连续版本 贝叶斯公式
数字特征
数字特征
多维均匀分布
多维均匀分布
多维正态分布
多维正态分布
多维指数分布
多维指数分布
连续型随机变量
连续型随机变量
多项分布
多项分布
多维泊松分布
多维泊松分布
离散型随机变量
离散型随机变量
数学期望向量
数学期望向量
协方差协方差矩阵
协方差协方差矩阵
数字特征
数字特征
相关系数/相关系数矩阵
相关系数/相关系数矩阵Text is not SVG - cannot display -------------------------------------------------------------------------------- /概率论与数理统计/images/13.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 |
总体
总体
个体1
个体1
个体2
个体2
个体N
个体N
样本
样本
样品1
样品1
样品2
样品2
样品n
样品n
抽样
抽样Text is not SVG - cannot display -------------------------------------------------------------------------------- /概率论与数理统计/images/14.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 |
总体
总体
样本1:
x1,x2,...,x100
样本1:...
样本2:
x1,x2,...,x100
样本2:...
样本1000:
x1,x2,...,x100
样本1000:...
抽样
抽样
统计量：mean_1
统计量：mean_1
统计量：mean_2
统计量：mean_2
统计量：mean_1000
统计量：mean_1000
计算统计量
计算统计量
计算统计量
计算统计量
计算统计量
计算统计量Text is not SVG - cannot display -------------------------------------------------------------------------------- /概率论与数理统计/images/15.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Git-Model/__init_Modeling__/29d23a92a18d08e67569371759310d7d6c596172/概率论与数理统计/images/15.png -------------------------------------------------------------------------------- /概率论与数理统计/images/16.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Git-Model/__init_Modeling__/29d23a92a18d08e67569371759310d7d6c596172/概率论与数理统计/images/16.png -------------------------------------------------------------------------------- /概率论与数理统计/images/17.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/Git-Model/__init_Modeling__/29d23a92a18d08e67569371759310d7d6c596172/概率论与数理统计/images/17.png 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| { 14 | "cell_type": "markdown", 15 | "id": "36194748", 16 | "metadata": {}, 17 | "source": [ 18 | "
\n", 19 | "❓ GitModel 公司发明了一种专用于数值计算的平板电脑 GitNum，考虑到员工生产效率的问题，GitNum 的生产成本是与生产台数相关的。函数关系为 \n", 20 | "\n", 21 | "$$C(n)=1250n(2-e^{-(n-5000)^2})$$\n", 22 | " \n", 23 | "请问如何确定这批 GitNum 的生产台数使得每台的平均生产成本最小使得经济效益最大呢?\n", 24 | "
" 25 | ] 26 | }, 27 | { 28 | "cell_type": "markdown", 29 | "id": "cae3bd5d", 30 | "metadata": {}, 31 | "source": [ 32 | "**分析：** \n", 33 | "  求平均生产成本最小，本质上就是求平均成本函数 $\\overline{C}(n)=\\dfrac{C(n)}{n}$ 的最小值。事实上，我们在初中就经常接触类似的最值问题，只不过所谓的目标函数往往是最简单的二次函数。而在高中之后，我们学习了求导工具，发现对任意可导的函数，都可以探究其导数异号零点来确定其极值点。 \n", 34 | "  诚然，仅仅是导数零点并不足以说明其就是极值点。如下图的函数 $y=x^3$，我们发现 $x=0$ 是其导数零点，但在图像上，仅仅只是函数的一个“拐点”。" 35 | ] 36 | }, 37 | { 38 | "cell_type": "markdown", 39 | "id": "f7f843d1", 40 | "metadata": {}, 41 | "source": [ 42 | "\n", 43 | "\n", 44 | "
图 1 : $y=x^3$ 及其导数图像
" 45 | ] 46 | }, 47 | { 48 | "cell_type": "markdown", 49 | "id": "646dd213", 50 | "metadata": {}, 51 | "source": [ 52 | "  这是因为导函数 $y'=3x^2$ 在导数零点 $x=0$ 处仅仅是“切过” $x$ 轴而非“穿过”$x$轴! 而分析学告诉我们，导函数 $f$ 在零点 $x_0$ 处 “穿过” $x$轴的图像，其本质就是 $f$ 在零点处附近是单调的，亦即二阶导函数（导数的导数） $f''(x_0)>0$ 或 $f''(x_0)<0$。" 53 | ] 54 | }, 55 | { 56 | "cell_type": "markdown", 57 | "id": "21258a46", 58 | "metadata": {}, 59 | "source": [ 60 | "\n", 61 | "\n", 62 | "
表 1 : 极值点函导数性态
" 63 | ] 64 | }, 65 | { 66 | "cell_type": "markdown", 67 | "id": "de069089", 68 | "metadata": {}, 69 | "source": [ 70 | "  回到问题上来，求 $\\overline{C}(n)$ 的极值点。首先，我们需要求出导函数 $\\overline{C}'(n)$ 的零点 $n_0$，如果 $\\overline{C}''(n_0)>0$，那么 $n_0$ 即为 $\\overline{C}(n)$ 的极小值点。" 71 | ] 72 | }, 73 | { 74 | "cell_type": "markdown", 75 | "id": "cda80e3c", 76 | "metadata": {}, 77 | "source": [ 78 | "代码如下：" 79 | ] 80 | }, 81 | { 82 | "cell_type": "markdown", 83 | "id": "c64442f1", 84 | "metadata": {}, 85 | "source": [ 86 | "
\n", 87 | "注：运行如下代码，需要提前安装sympy包，执行如下命令：pip install sympy\n", 88 | "
" 89 | ] 90 | }, 91 | { 92 | "cell_type": "code", 93 | "execution_count": 1, 94 | "id": "07a13dde", 95 | "metadata": { 96 | "ExecuteTime": { 97 | "end_time": "2022-06-03T07:11:41.255705Z", 98 | "start_time": "2022-06-03T07:11:36.453769Z" 99 | } 100 | }, 101 | "outputs": [ 102 | { 103 | "data": { 104 | "text/latex": [ 105 | "$\\displaystyle - 1250 \\cdot \\left(10000 - 2 n\\right) e^{- \\left(n - 5000\\right)^{2}}$" 106 | ], 107 | "text/plain": [ 108 | "-1250*(10000 - 2*n)*exp(-(n - 5000)**2)" 109 | ] 110 | }, 111 | "execution_count": 1, 112 | "metadata": {}, 113 | "output_type": "execute_result" 114 | } 115 | ], 116 | "source": [ 117 | "from sympy import *\n", 118 | "\n", 119 | "n = symbols('n')\n", 120 | "y = 1250*(2-exp(-(n-5000)**2))\n", 121 | "\n", 122 | "# 求一阶导数\n", 123 | "func1 = diff(y,n)\n", 124 | "func1" 125 | ] 126 | }, 127 | { 128 | "cell_type": "code", 129 | "execution_count": 2, 130 | "id": "8d65dd5d", 131 | "metadata": { 132 | "ExecuteTime": { 133 | "end_time": "2022-06-03T07:11:41.271596Z", 134 | "start_time": "2022-06-03T07:11:41.256636Z" 135 | } 136 | }, 137 | "outputs": [ 138 | { 139 | "name": "stdout", 140 | "output_type": "stream", 141 | "text": [ 142 | "该函数驻点为 [5000]\n" 143 | ] 144 | } 145 | ], 146 | "source": [ 147 | "# 计算驻点\n", 148 | "stag = solve(diff(y,n),n)\n", 149 | "print(\"该函数驻点为\",stag)" 150 | ] 151 | }, 152 | { 153 | "cell_type": "code", 154 | "execution_count": 3, 155 | "id": "447bfa09", 156 | "metadata": { 157 | "ExecuteTime": { 158 | "end_time": "2022-06-03T07:11:41.303591Z", 159 | "start_time": "2022-06-03T07:11:41.272594Z" 160 | } 161 | }, 162 | "outputs": [ 163 | { 164 | "data": { 165 | "text/latex": [ 166 | "$\\displaystyle 2500 \\cdot \\left(1 - 2 \\left(n - 5000\\right)^{2}\\right) e^{- \\left(n - 5000\\right)^{2}}$" 167 | ], 168 | "text/plain": [ 169 | "2500*(1 - 2*(n - 5000)**2)*exp(-(n - 5000)**2)" 170 | ] 171 | }, 172 | "execution_count": 3, 173 | "metadata": {}, 174 | "output_type": "execute_result" 175 | } 176 | ], 177 | "source": [ 178 | "# 计算二阶导数\n", 179 | "func2 = diff(y, n, 2)\n", 180 | "func2" 181 | ] 182 | }, 183 | { 184 | "cell_type": "code", 185 | "execution_count": 4, 186 | "id": "97ef8204", 187 | "metadata": { 188 | "ExecuteTime": { 189 | "end_time": "2022-06-03T07:11:41.319060Z", 190 | "start_time": "2022-06-03T07:11:41.305099Z" 191 | } 192 | }, 193 | "outputs": [ 194 | { 195 | "name": "stdout", 196 | "output_type": "stream", 197 | "text": [ 198 | "2500.00000000000\n" 199 | ] 200 | }, 201 | { 202 | "data": { 203 | "text/latex": [ 204 | "$\\displaystyle \\text{True}$" 205 | ], 206 | "text/plain": [ 207 | "True" 208 | ] 209 | }, 210 | "execution_count": 4, 211 | "metadata": {}, 212 | "output_type": "execute_result" 213 | } 214 | ], 215 | "source": [ 216 | "# 计算驻点的二阶导数值，验证正负\n", 217 | "print(func2.evalf(subs = {n:5000}))\n", 218 | "func2.evalf(subs = {n:5000}) > 0" 219 | ] 220 | }, 221 | { 222 | "cell_type": "code", 223 | "execution_count": 5, 224 | "id": "c1153255", 225 | "metadata": { 226 | "ExecuteTime": { 227 | "end_time": "2022-06-03T07:11:41.335018Z", 228 | "start_time": "2022-06-03T07:11:41.320057Z" 229 | } 230 | }, 231 | "outputs": [ 232 | { 233 | "data": { 234 | "text/latex": [ 235 | "$\\displaystyle 1250.0$" 236 | ], 237 | "text/plain": [ 238 | "1250.00000000000" 239 | ] 240 | }, 241 | "execution_count": 5, 242 | "metadata": {}, 243 | "output_type": "execute_result" 244 | } 245 | ], 246 | "source": [ 247 | "# 函数的极小值\n", 248 | "y.evalf(subs = {n:5000})" 249 | ] 250 | }, 251 | { 252 | "cell_type": "markdown", 253 | "id": "448f2097", 254 | "metadata": {}, 255 | "source": [ 256 | "  经验证，$n_0=5000$ 确实是 $\\overline{C}(n)$ 的极小值点，即第一批 GitNum 计划生产台数为 $5000$ 台时，平均成本最小为 $1250$ 元/台。" 257 | ] 258 | }, 259 | { 260 | "cell_type": "markdown", 261 | "id": "5689cf79", 262 | "metadata": {}, 263 | "source": [ 264 | "
\n", 265 | " 抽象提炼 \n", 266 | " \n", 267 | "  在数学建模中，优化问题渗透到各个方面，小到最优参数的确定，大到最优策略的规划。每一个优化问题都可以用如下标准形式给出：\n", 268 | "$$\n", 269 | "\\begin{array}{ll}\n", 270 | "\\max & f(x) \\\\\n", 271 | "\\text{s.t.} & g_i(x) \\geqslant 0,i=1,2,\\cdots ,n \\\\\n", 272 | "& h_j(x)=0, j=1,2,\\cdots,m\n", 273 | "\\end{array} \\tag{1.1}\n", 274 | "$$\n", 275 | "其中 $f(x)$ 即是我们需要求最值的函数，$x$ 是我们用来“决策”的变量。$g_i(x) \\geqslant 0, h_j(x)=0$ 分别是决策变量 $x$ 满足的条件，即决策变量 $x$ 的约束。当决策变量 $x$ 没有约束时，称为无约束优化问题。\n", 276 | "
" 277 | ] 278 | }, 279 | { 280 | "cell_type": "markdown", 281 | "id": "26a567be", 282 | "metadata": {}, 283 | "source": [ 284 | "  上文提到的GitNum生产台数问题是一个最简单的无约束优化问题——目标函数有**明确表达式，二阶以上可导，定义域离散程度不高，往往生产一批产品的数量范围是足够大，以至于可以将离散的整数视为连续的**。对于这样的简单优化问题，利用数学分析的知识可以做到精确的分析。我们给出优化领域最经典也是最实用的判别条件：" 285 | ] 286 | }, 287 | { 288 | "cell_type": "markdown", 289 | "id": "c4a64109", 290 | "metadata": {}, 291 | "source": [ 292 | "
\n", 293 | " 命题1：[一阶最优化必要条件] \n", 294 | "设 $f(x)$ 是 $\\mathbf{R}$ 上的可导函数，若 $x^*$ 是 $f$ 的极值点，则\n", 295 | "$$\n", 296 | "f'(x^*)=0 \\tag{1.2}\n", 297 | "$$\n", 298 | "其中 $f'$ 是 $f$ 的一阶导数。\n", 299 | "
" 300 | ] 301 | }, 302 | { 303 | "cell_type": "markdown", 304 | "id": "c90fa67a", 305 | "metadata": {}, 306 | "source": [ 307 | "
\n", 308 | " 命题2：[二阶最优化必要条件] \n", 309 | "设 $f(x)$ 是 $\\mathbf{R}$ 上的可导函数，若 $x^*$ 是 $f$ 的极值点，设$f(x)$在 $\\mathbf{R}$ 上二阶可导，若 $x^*$ 是 $f$ 的极大（小）值点，则\n", 310 | "$$\n", 311 | "f'(x^*)=0 \\\\\n", 312 | "f''(x^*)\\leqslant0 ( f''(x^*) \\geqslant 0 ) \\tag{1.3}\n", 313 | "$$\n", 314 | "其中 $f''$ 是 $f$ 的二阶导数。\n", 315 | "
" 316 | ] 317 | }, 318 | { 319 | "cell_type": "markdown", 320 | "id": "db24c2fa", 321 | "metadata": {}, 322 | "source": [ 323 | "
\n", 324 | " 命题3：[二阶最优化充分条件] \n", 325 | "设 $f(x)$ 是 $\\mathbf{R}$ 上的二阶可导函数，若 $x^*$ 满足\n", 326 | "$$\n", 327 | "f'(x^*)=0 \\\\\n", 328 | "f''(x^*)<0 (f''(x^*)>0)\n", 329 | "$$\n", 330 | "则 $x^*$ 是 $f$ 的极大$($小$)$值点。\n", 331 | "
" 332 | ] 333 | }, 334 | { 335 | "cell_type": "markdown", 336 | "id": "2fa455d1", 337 | "metadata": {}, 338 | "source": [ 339 | "
\n", 340 | "❓ GitModel 公司通过 GitNum 的发售赚到第一桶金后，马不停蹄地投入到 GitNum 的改良研制中，经市场调查后，GitModel 公司发现根据用户需求可推出轻便版以及高性能版的 GitNum。三种 GitNum如下表所示：\n", 341 | "\n", 342 | "\n", 343 | " \n", 344 | "在这一批生产中，GitModel 公司应该制定怎么样的生产计划，如何分配三种 GitNum 的生产台数才能使得利润与成本的比值最大化？\n", 345 | "
" 346 | ] 347 | }, 348 | { 349 | "cell_type": "markdown", 350 | "id": "ab4470ce", 351 | "metadata": {}, 352 | "source": [ 353 | "**分析：** \n", 354 | "  我们把三种 GitNum 的生产台数按上表顺序标记序号 $n_i,i=1,2,3$，分别用 $R,Q,C$ 来代表收入、利润和成本。问题旨在求出利润/成本 $r=\\dfrac{Q}{C}$ 的最大值，而我们需要同时考虑三种 GitNum 的生产台数，也就是说，需要求极值的函数 $r$，该函数是 $n_1,n_2,n_3$ 的多元函数 ! " 355 | ] 356 | }, 357 | { 358 | "cell_type": "markdown", 359 | "id": "5f65c2ac", 360 | "metadata": {}, 361 | "source": [ 362 | "  我们逐步将各种函数列出来，可得到目标函数 $\\dfrac{Q}{C}$，则 \n", 363 | "  总收入函数为\n", 364 | "$$\n", 365 | "R(n_1,n_2,n_3)=2000n_1+3000n_2+4500n_3\n", 366 | "$$\n", 367 | "  总成本函数为\n", 368 | "$$\n", 369 | "C(n_1,n_2,n_3)=750n_1(2-e^{-(n_1-6000)^2})+1250n_2(2-e^{-(n_2-5000)^2})+2000n_3(2-e^{-(n_3-3000)^2})\n", 370 | "$$\n", 371 | "  总利润函数为\n", 372 | "$$\n", 373 | "Q(n_1,n_2,n_3)=R(n_1,n_2,n_3)-C(n_1,n_2,n_3)\n", 374 | "$$" 375 | ] 376 | }, 377 | { 378 | "cell_type": "markdown", 379 | "id": "09e060c1", 380 | "metadata": {}, 381 | "source": [ 382 | "  所以，总利润与总成本比值为\n", 383 | "$$\n", 384 | "\\begin{align}\n", 385 | "r(n_1,n_2,n_3)\n", 386 | "& = \\dfrac{R(n_1,n_2,n_3) - C(n_1,n_2,n_3)}{C(n_1,n_2,n_3)} \\tag{1.4} \\\\\n", 387 | "& = \\dfrac{R(n_1,n_2,n_3)}{C(n_1,n_2,n_3)} - 1 \\tag{1.5} \\\\\n", 388 | "& = \\dfrac{2000n_1+3000n_2+4500n_3}{750n_1(2-e^{-(n_1-6000)^2}) + 1250n_2(2-e^{-(n_2-5000)^2}) + 2000n_3(2-e^{-(n_3-3000)^2})} - 1 \\tag{1.6} \n", 389 | "\\end{align}\n", 390 | "$$" 391 | ] 392 | }, 393 | { 394 | "cell_type": "markdown", 395 | "id": "dc39e290", 396 | "metadata": {}, 397 | "source": [ 398 | "  优化问题即为\n", 399 | "$$\n", 400 | "\\max \\limits_{n_1,n_2,n_3 \\geqslant 1} r(n_1,n_2,n_3)\n", 401 | "$$\n", 402 | "\n", 403 | "  依样画葫芦，该问题不过是第一个问题的变量 Plus 版，我们仍然可以采用研究导数以及二阶导数性态的方式，去求出极大值组合 $(n_1^*,n_2^*,n_3^*)$。" 404 | ] 405 | }, 406 | { 407 | "cell_type": "markdown", 408 | "id": "f0cd25c2", 409 | "metadata": {}, 410 | "source": [ 411 | "  **那有的同学可能就会问了：一元函数分析跟多元函数分析有什么区别呢?**\n", 412 | "\n", 413 | "  在多元函数 $f$ 中，我们只能对单个变量 $x$ 求偏导数 $f_x$，这远远不足以刻画导数的全部信息。所以，梯度（全导数）以及 Hesse 矩阵的概念替代了一元函数中导数以及二阶导数。" 414 | ] 415 | }, 416 | { 417 | "cell_type": "markdown", 418 | "id": "8d9988f6", 419 | "metadata": {}, 420 | "source": [ 421 | "
\n", 422 | " 定义1：[Hesse 矩阵] \n", 423 | "设 $f:\\mathbf{R}^n\\to R$ 是 $n$ 元函数，$x_i,i=1,2,\\cdots,n$ 是变量。\n", 424 | " \n", 425 | "记 $f$ 对变量 $x_i$ 方向的偏导数为 $f'_i$，称所有偏导数组成的列向量 $[f_1',f_2',\\cdots,f_n']^T := \\nabla f$ 为函数 $f$ 的全导数，亦称**梯度**。\n", 426 | "\t\n", 427 | "记偏导数 $f_i'$ 对变量 $x_j$ 方向的二阶偏导数为 $f''_{ij}$ 称 $n\\times n$ 方阵\n", 428 | "$$\n", 429 | "\\begin{bmatrix}\n", 430 | "f''_{11} & f''_{12} & \\cdots & f''_{1n} \\\\\n", 431 | "f''_{21} & f''_{22}&\\cdots & f''_{1n} \\\\\n", 432 | "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n", 433 | "f''_{n1} & f''_{n2} & \\cdots & f''_{nn} \\\\\n", 434 | "\\end{bmatrix} \n", 435 | ":= \\nabla^2 f\n", 436 | "$$ \n", 437 | "为函数 $f$ 的 Hesse 矩阵。\n", 438 | "
" 439 | ] 440 | }, 441 | { 442 | "cell_type": "markdown", 443 | "id": "fb36c1c8", 444 | "metadata": {}, 445 | "source": [ 446 | "  在推广最优化条件之前，我们先从直观上进行条件的表述，如下图所示：\n", 447 | "\n", 448 | "\n", 449 | "\n", 450 | "" 451 | ] 452 | }, 453 | { 454 | "cell_type": "markdown", 455 | "id": "4e771b31", 456 | "metadata": {}, 457 | "source": [ 458 | "  从上图我们不难猜测： " 459 | ] 460 | }, 461 | { 462 | "cell_type": "markdown", 463 | "id": "cedc0843", 464 | "metadata": {}, 465 | "source": [ 466 | "" 467 | ] 468 | }, 469 | { 470 | "cell_type": "markdown", 471 | "id": "82d1ca09", 472 | "metadata": {}, 473 | "source": [ 474 | "  事实上，通过 Taylor 展开，我们可验证猜想的正确性，理论证明不难而繁，但我们只保留最精华实用的命题，其中最优化条件对应的多元函数版本则如下所述：" 475 | ] 476 | }, 477 | { 478 | "cell_type": "markdown", 479 | "id": "2357324d", 480 | "metadata": {}, 481 | "source": [ 482 | "
\n", 483 | " 命题4：[多元无约束问题一阶最优化必要条件] \n", 484 | "设 $f(x)$ 是 $\\mathbf{R}^n$ 上的 $n$ 元可导函数，若 $x^*$ 是 $f$ 的极值点，则\n", 485 | "$$\n", 486 | "\\nabla f(x^*)=0 \\tag{1.7}\n", 487 | "$$\n", 488 | "其中 $\\nabla f$ 是 $f$ 的梯度。\n", 489 | "
" 490 | ] 491 | }, 492 | { 493 | "cell_type": "markdown", 494 | "id": "f5310dde", 495 | "metadata": {}, 496 | "source": [ 497 | "
\n", 498 | " 命题5：[多元无约束问题二阶最优化必要条件] \n", 499 | "设 $f(x)$ 是 $\\mathbf{R}^n$ 上的 $n$ 元二阶可导函数，若 $x^*$ 是 $f$ 的极大（小）值点，则\n", 500 | "$$\n", 501 | "\\nabla f(x^*)=0 \\\\ \n", 502 | "\\nabla^2 f(x^*)\\text{是半负定} ( \\nabla^2 f(x^*)\\text{是半正定} ) \\tag{1.8} \n", 503 | "$$\n", 504 | "其中 $\\nabla^2 f$ 是 $f$ 的 Hesse 矩阵。\n", 505 | "
" 506 | ] 507 | }, 508 | { 509 | "cell_type": "markdown", 510 | "id": "1c318411", 511 | "metadata": {}, 512 | "source": [ 513 | "
\n", 514 | " 命题6：[多元无约束问题二阶最优化充分条件] \n", 515 | "设 $f(x)$ 是 $\\mathbf{R}^n$ 上的 $n$ 元二阶可导函数，若 $x^*$ 满足\n", 516 | "$$\n", 517 | "\\nabla f(x^*)=0 \\\\\n", 518 | "\\nabla^2 f(x^*) \\ \\text{负定} \\ (\\nabla^2 f(x^*)\\ \\text{正定})\n", 519 | "$$\n", 520 | "则 $x^*$ 是 $f$ 的极大（小）值点。\n", 521 | "
" 522 | ] 523 | }, 524 | { 525 | "cell_type": "markdown", 526 | "id": "5c3ca493", 527 | "metadata": {}, 528 | "source": [ 529 | "  而代数学的知识告诉我们，**验证一个对称矩阵是否正（负）定只需要 check 其所有特征值是否大（小）于0。**\n", 530 | "\n", 531 | "  所以，要制定最佳的生产计划 $(n_1^*,n_2^*,n_3^*)$ 使得利润与成本比值最大，即求解问题(2)，只需求出三元方程组\n", 532 | "$$\n", 533 | "\\nabla r = 0\n", 534 | "$$\n", 535 | "的解集，再挑出解集中使得 $\\nabla^2 r$ 负定的解，这就是我们的最佳生产计划 $(n_1^*,n_2^*,n_3^*)$ !" 536 | ] 537 | }, 538 | { 539 | "cell_type": "markdown", 540 | "id": "02e17141", 541 | "metadata": {}, 542 | "source": [ 543 | "  最优化条件给出了这样一个基本事实：不管函数 $f$ 表达式复杂与否，只要二阶以上可导，我们可以找出满足 Hesse 矩阵 $\\nabla ^2f$ 半负定（半正定）的梯度零点，极大（小）值点必定存在其中! " 544 | ] 545 | }, 546 | { 547 | "cell_type": "markdown", 548 | "id": "d46610fc", 549 | "metadata": {}, 550 | "source": [ 551 | "
\n", 552 | "❓ 这时喜欢思考的同学就会问了。老师，既然我们求极值要通过求解梯度方程 $\\nabla f=0$，那么变量一旦多起来，求解方程组的难度也会迅速上升，并且还存在方程无法求解的问题（比如超越方程 $\\ln x+x-3=0$)，这个时候该怎么办呢?\n", 553 | "
" 554 | ] 555 | }, 556 | { 557 | "cell_type": "markdown", 558 | "id": "f05a342a", 559 | "metadata": {}, 560 | "source": [ 561 | "  虽然抽象的理论只能提供许多“理论上可行”的解决方式，但俗话说的好，车到山前必有路，方程求解不出来，我们能不能找到近似解呢？计算机说：💃退!💃退! 💃退!" 562 | ] 563 | }, 564 | { 565 | "cell_type": "markdown", 566 | "id": "6e1773e3", 567 | "metadata": {}, 568 | "source": [ 569 | "  其实，在现实生活中，我们遇到的许多优化问题需要解的方程都是超越方程，早已超出人力计算的范畴，而我们衍生的许多数值计算方法就是为了求出这类方程的近似解的。在优化问题中，一个最经典的求极值算法便是针对优化条件(1)发明的**梯度下降法**，该方法是通过迭代的方式找到一列**梯度递减的点**，当点的**梯度下降的足够接近0**时，便可认为该点即是极值的候选之一。 " 570 | ] 571 | }, 572 | { 573 | "cell_type": "markdown", 574 | "id": "b5b4f24a", 575 | "metadata": {}, 576 | "source": [ 577 | "  下面我们介绍如何使用`scipy`求解多元函数的极值" 578 | ] 579 | }, 580 | { 581 | "cell_type": "markdown", 582 | "id": "e60e4afc", 583 | "metadata": {}, 584 | "source": [ 585 | "
\n", 586 | "注：运行如下代码，需要提前安装scipy包，执行如下命令：pip install scipy\n", 587 | "
" 588 | ] 589 | }, 590 | { 591 | "cell_type": "code", 592 | "execution_count": 6, 593 | "id": "a8ae5439", 594 | "metadata": { 595 | "ExecuteTime": { 596 | "end_time": "2022-06-03T07:11:43.629821Z", 597 | "start_time": "2022-06-03T07:11:41.336015Z" 598 | } 599 | }, 600 | "outputs": [], 601 | "source": [ 602 | "import scipy.optimize as opt\n", 603 | "from scipy import fmin\n", 604 | "import numpy as np" 605 | ] 606 | }, 607 | { 608 | "cell_type": "code", 609 | "execution_count": 7, 610 | "id": "5b32bfbd", 611 | "metadata": { 612 | "ExecuteTime": { 613 | "end_time": "2022-06-03T07:11:43.739148Z", 614 | "start_time": "2022-06-03T07:11:43.630818Z" 615 | } 616 | }, 617 | "outputs": [ 618 | { 619 | "data": { 620 | "text/plain": [ 621 | " fun: 0.126003403677694\n", 622 | " hess_inv: <3x3 LbfgsInvHessProduct with dtype=float64>\n", 623 | " jac: array([ 6.24122979e-04, 3.71258582e-04, -8.03801974e-06])\n", 624 | " message: 'CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL'\n", 625 | " nfev: 64\n", 626 | " nit: 15\n", 627 | " njev: 16\n", 628 | " status: 0\n", 629 | " success: True\n", 630 | " x: array([ 1. , 1. , 123.5759835])" 631 | ] 632 | }, 633 | "execution_count": 7, 634 | "metadata": {}, 635 | "output_type": "execute_result" 636 | } 637 | ], 638 | "source": [ 639 | "# C(n)\n", 640 | "def func0(cost, x, a):\n", 641 | " return cost*x*(2 - exp(-(x - a)**2))\n", 642 | "\n", 643 | "# 总利润与总成本比值\n", 644 | "func = lambda x: (2000*x[0] + 3000*x[1] + 4500*x[2]) / (func0(750, x[0], 6000) + func0(1250, x[1], 5000) + func0(2000, x[2], 3000)) - 1 \n", 645 | "\n", 646 | "# 设定n的范围\n", 647 | "bnds = ((1, 10000), (1, 10000), (1, 10000))\n", 648 | "# 计算多元函数func的极值\n", 649 | "res = opt.minimize(fun=func, x0=np.array([2, 1, 1]), bounds=bnds)\n", 650 | "res" 651 | ] 652 | }, 653 | { 654 | "cell_type": "markdown", 655 | "id": "dbb221de", 656 | "metadata": {}, 657 | "source": [ 658 | "## 2 以插值知识解决数据处理问题" 659 | ] 660 | }, 661 | { 662 | "cell_type": "markdown", 663 | "id": "c9856a8c", 664 | "metadata": {}, 665 | "source": [ 666 | "
\n", 667 | "❓ GitModel 公司工作室刚刚建完，准备安装宽带以及路由，基于传输数据要求，GitModel 公司需要的宽带运营商网速越快越好，由于网速是一个随时间变化的非固定量，简单衡量平均网速也并非一个好的评价手段, GitModel 公司准备收集在同一个地点各个运营商的宽带一天内的网速数据以做接下来的建模分析，出于人力考虑，网速监视每小时汇报一次数据。以下是 A 运营商的宽带24小时网速情况（见下表）：\n", 668 | "
\n", 669 | "\n", 670 | "" 671 | ] 672 | }, 673 | { 674 | "cell_type": "markdown", 675 | "id": "ff58a2f9", 676 | "metadata": {}, 677 | "source": [ 678 | "
\n", 679 | "  然而，应该注意到离散的数据点无法完整体现网速的整体性质，我们需要通过仅有的数据点还原宽带一天内的实时网速。以 A 运营商为例，请问 GitModel 公司应如何“模拟”出网速曲线呢？\n", 680 | "
" 681 | ] 682 | }, 683 | { 684 | "cell_type": "markdown", 685 | "id": "e49aabd8", 686 | "metadata": {}, 687 | "source": [ 688 | "  该问题转换为数学话版，就是需要找出一条经过上表24个点 $(t_i,s_i),i=1,2,\\cdots,24$ 的曲线 $s=s(t)$，那我们自然会问，要如何构造合适的 $s(t)$ 呢？又应该怎么说明构造的 $s(t)$ 能有较好的模拟效果呢？" 689 | ] 690 | }, 691 | { 692 | "cell_type": "markdown", 693 | "id": "851dec82", 694 | "metadata": {}, 695 | "source": [ 696 | "**我们接着往下分析：** \n", 697 | "  由于宽带用户千千万万，大基数导致了使用宽带的用户数关于时间 $t$ 的变化是近似连续的，从而网速 $s(t)$ 可以假设为 $t$ 的连续函数，更甚之，可以假设为斜率 $s'(t) 以及曲率 s''(t)$ 都是连续的！我们将 $s(t)$ 的满足条件罗列如下： \n", 698 | "- $s(t)$ 过24个数据点 $s_i=s(t_i)$，其中 $ i=1,2,\\cdots,24$\n", 699 | "- $s(t)$ 是连续的\n", 700 | "- $s'(t)$ 是连续的\n", 701 | "- $s''(t)$ 是连续的" 702 | ] 703 | }, 704 | { 705 | "cell_type": "markdown", 706 | "id": "a847f167", 707 | "metadata": {}, 708 | "source": [ 709 | "  $s(t)$ 应该满足的条件暂且如此，我们再进行思考：有什么构造 $s(t)$ 方式是比较简单的呢？\n", 710 | "\n", 711 | "  在高中，我们就已经学过了，给定一组数据点可通过**最小二乘法**来拟合出一条回归直线，两点一直线的数学直觉告诉我们，能经过24个数据点的直线几乎不存在，那么我们可否放宽条件，构造出经过24个数据点的折线呢？这显然是可以的！过数据点的条件可由分段函数来解决。" 712 | ] 713 | }, 714 | { 715 | "cell_type": "markdown", 716 | "id": "b1d2a5ca", 717 | "metadata": {}, 718 | "source": [ 719 | "  用数学的语言，即是分别在每一个区间 $[t_i,t_{i+1}],i=1,2,\\cdots,23$ 上，以 $(t_i,s_i)$ 和 $(t_{i+1},s_{i+1})$ 为两端构造线段 $s_{i}(t)=k_it+b_i,t\\in [t_i,t_{i+1}],i=1,2,\\cdots,23$，其中 $k_i,b_i$ 为参数，确定 $k_i$ 和 $b_i$ 对我们来说也是小菜一碟的。具体构造如下图所示：\n", 720 | "\n", 721 | "" 722 | ] 723 | }, 724 | { 725 | "cell_type": "markdown", 726 | "id": "5868f384", 727 | "metadata": {}, 728 | "source": [ 729 | "  对比 $s(t)$ 要满足的条件，折线的构造方式显然满足了前两个条件，我们再结合上图进行思考：折线不满足 $s'(t)$ 连续，是因为在数据点 $(t_i,s_i)$ 处不一定可导，即左右导数不相同。以此反推，我们希望构造出来的分段函数 $s_i(t)$ 们在“连接处” $(t_{i},s_i)$ 和 $(t_{i+1},s_{i+1})$ 都应该有导数以及二阶导数相等。现在，我们正式将条件写为数学形式：\n", 730 | "\n", 731 | "$$\n", 732 | "\\begin{align}\n", 733 | "\\text{过点：} & s_{i}(t_i)=s_i,s_{23}(t_{24})=s_{24} \\quad i= 1,2,\\cdots,23 \\tag{2.1} \\\\\n", 734 | "\\text{分段连接：} & s_{i}(t_{i+1})=s_{i+1}(t_{i+1}) \\quad i=1,2,\\cdots,22 \\tag{2.2} \\\\\n", 735 | "\\text{斜率相等：} & s_{i}'(t_{i+1})=s_{i+1}'(t_{i+1}) \\quad i=1,2,\\cdots,22 \\tag{2.3} \\\\\n", 736 | "\\text{曲率相等：} & s_{i}''(t_{i+1})=s_{i+1}''(t_{i+1}) \\quad i=1,2,\\cdots,22 \\tag{2.4} \n", 737 | "\\end{align}\n", 738 | "$$" 739 | ] 740 | }, 741 | { 742 | "cell_type": "markdown", 743 | "id": "1faa4cb2", 744 | "metadata": {}, 745 | "source": [ 746 | "  那么，既然折线（分段）一次函数不满足导数相等，从求导的难度上，我们自然会考虑分段二次函数怎么样呢？Unfortunately，分段二次函数满足了导数相等但二阶导数不相等，而按图索骥，我们知道分段三次函数即能满足我们对 $s(t)$ 的所有期待！\n", 747 | "\n", 748 | "$$\n", 749 | "\\begin{aligned}\n", 750 | "\\text{分段直线构造} & \\rightarrow \\text{解决} s(t) \\text{经过24个数据点且连续} \\\\\n", 751 | "\\text{分段抛物线构造} & \\rightarrow \\text{解决} s'(t) \\text{连续} \\\\\n", 752 | "\\text{分段三次曲线构造} & \\rightarrow \\text{解决} s''(t) \\text{连续} \\\\\n", 753 | "\\end{aligned}\n", 754 | "$$" 755 | ] 756 | }, 757 | { 758 | "cell_type": "markdown", 759 | "id": "90da2df6", 760 | "metadata": {}, 761 | "source": [ 762 | "  构造出来的分段三次曲线如下： \n", 763 | "$$\n", 764 | "s_i(t) = s_{i,0} + s_{i,1}t + s_{i,2}t^2 + s_{i,3}t^3 \\quad i=1,2,\\cdots, 23 \\tag{2.5}\n", 765 | "$$\n", 766 | "\n", 767 | "  上式(2.5)应该满足公式(2.1)、公式(2.2)、公式(2.3)、公式(2.4)，一共是90条方程，其中未知数有92个，事实上，仍需确定的只有起点 $(t_1,s_1)$ 与 $(t_{24},s_{24})$ 的二阶函数值 $s_1''(t_1)$ 和 $s_{23}''(t_{24})$。" 768 | ] 769 | }, 770 | { 771 | "cell_type": "markdown", 772 | "id": "81c2e079", 773 | "metadata": {}, 774 | "source": [ 775 | "  这里，我们假设 $s_1''(t_1)=s_{23}''(t_{24})=0$，实际含义为在 $0:00 \\sim 1:00$ 处的网速变化应该逐渐变慢，那么便可以通过公式(2.1)、公式(2.2)、公式(2.3)、公式(2.4)与 $s_1''(t_1) = s_{24}''(t_{24}) = 0$ 建立方程组求出 公式(2.5)的所有未知数！ \n", 776 | "\n", 777 | "  好事成双！思维敏捷的同学应该已经发现了：对于多项式函数，无论求多少阶导，都是 $1,t,t^2,t^3$ 的线性组合！而这即告诉我们需要解的方程组是一个线性方程组！\n", 778 | "\n", 779 | "" 780 | ] 781 | }, 782 | { 783 | "cell_type": "markdown", 784 | "id": "a6d03b18", 785 | "metadata": {}, 786 | "source": [ 787 | "
\n", 788 | " 📋 小结：我们将这种构造过数据点函数的过程称为插值，在建模中常常用于数据的“补充”或者“还原”，能有效解决建模中数据量过少的问题。目前最通用的插值方式，即上文提到的分段三次曲线构造，亦称为三次样条插值。由于我们接触的许多数据是具有某种意义上的“连续性”的，而三次样条插值具有的最佳实用性质就是其“摆动极小”，应用也就屡见不鲜。\n", 789 | "
" 790 | ] 791 | }, 792 | { 793 | "cell_type": "markdown", 794 | "id": "a7193fea", 795 | "metadata": {}, 796 | "source": [ 797 | "  从上文我们知道，如果有 $n$ 个数据点，那么三次样条插值的未知数（即多项式系数）一共有 $4n$ 个，而我们的条件所给出的方程只有 $4n-2$ 条，还需要有2个条件诱导2条方程来解未知数。在本问题的最后，我们给出三次样条插值的定义以及根据不同需求诱导的几种三次样条插值。" 798 | ] 799 | }, 800 | { 801 | "cell_type": "markdown", 802 | "id": "3f1aea51", 803 | "metadata": {}, 804 | "source": [ 805 | "
\n", 806 | " 定义2：[插值] \n", 807 | "设 $(x_i,y_i)$ 是 $\\mathbf{R}^2$ 上的 $n$ 个点，$f:\\mathbf{R} \\to \\mathbf{R}$ 是一条满足 $y_i=f(x_i)$ 的函数曲线，则 $f$ 称为 $(x_i,y_i)$ 的插值曲线，其中$i=1,2,\\cdots,n$。\n", 808 | "
" 809 | ] 810 | }, 811 | { 812 | "cell_type": "markdown", 813 | "id": "3f50702b", 814 | "metadata": {}, 815 | "source": [ 816 | "
\n", 817 | " 定义3：[三次样条插值] \n", 818 | "设 $(x_i,y_i)$ 是 $\\mathbf{R}^2$ 上的 $n$ 个点，其中$i=1,2,\\cdots,n$，其三次样条插值函数为\n", 819 | "$$\n", 820 | "f_i(t) = s_{i,0} + s_{i,1}t + s_{i,2}t^2 + s_{i,3}t^3 \\quad i=1,2,\\cdots, n-1\n", 821 | "$$\n", 822 | "满足\n", 823 | "$$\n", 824 | "\\begin{aligned}\n", 825 | "f_{i}(x_i) & =y_i,f_{n-1}(x_{n})=y_{n} \\quad i=1,2,\\cdots,n-1 \\\\\n", 826 | "f_{i}(x_{i+1}) &= y_{i+1}(x_{i+1}) \\quad i=1,2,\\cdots,n-2 \\\\\n", 827 | "f_{i}'(x_{i+1}) &= y_{i+1}'(x_{i+1}) \\quad i=1,2,\\cdots,n-2 \\\\\n", 828 | "f_{i}''(x_{i+1})&=y_{i+1}''(x_{i+1}) \\quad i=1,2,\\cdots,n-2\n", 829 | "\\end{aligned}\n", 830 | "$$\n", 831 | "
" 832 | ] 833 | }, 834 | { 835 | "cell_type": "markdown", 836 | "id": "9d465c4d", 837 | "metadata": {}, 838 | "source": [ 839 | "\n", 840 | "\n", 841 | "其中 $h_i=x_{i+1}-x_i,d_i=\\dfrac{y_{i+1}-y_i}{x_{i+1}-x_i},i=1,2,\\cdots,n-1$" 842 | ] 843 | }, 844 | { 845 | "cell_type": "markdown", 846 | "id": "80d6dd03", 847 | "metadata": {}, 848 | "source": [ 849 | "## 3 以积分知识解决工程问题" 850 | ] 851 | }, 852 | { 853 | "cell_type": "markdown", 854 | "id": "f4999f7b", 855 | "metadata": {}, 856 | "source": [ 857 | "
\n", 858 | "❓ 鉴于 GitModel 公司刚上市，承接了许多数学建模相关业务。有一家工程企业给出了这样一个任务：该企业准备制造一批抽水机，需要得到各种数据来决定抽水机的性能参数。企业首先对圆柱形的储水桶进行了抽水测试，需要求出将桶内的水全部抽出需要做多少功？\n", 859 | "\n", 860 | "储水桶的数据如下：高为5m，底面圆半径为3m：\n", 861 | "\n", 862 | "
" 863 | ] 864 | }, 865 | { 866 | "cell_type": "markdown", 867 | "id": "9820e9fa", 868 | "metadata": {}, 869 | "source": [ 870 | "**分析：** \n", 871 | "  为了将水抽出来，抽水机只需要施加与水相同的重力 $G$ 即可。然而，如图所示，不同水平面上抽水的位移是不同的，这导致我们无法将整桶水看作一个质点，自然无法用高中物理的做功计算求出数值。\n", 872 | "\n", 873 | "
不同平面位移不同
\n", 874 | "\n", 875 | "\n", 876 | "
离散分割
" 877 | ] 878 | }, 879 | { 880 | "cell_type": "markdown", 881 | "id": "440baa98", 882 | "metadata": {}, 883 | "source": [ 884 | "  老师，我只知道高中物理的方法，能不能改进高中物理方法来解决这个问题呢？**整桶水无法看作一个质点，那能否将整桶水看作 $n$ 个单位质点呢？** \n", 885 | "\n", 886 | "  众所周知，水桶中的水是**连续**的整体，如果我们将水分割成 $n$ 个单位，那么求出来的数值只是一个近似，不过当 $n$ 越来越大时，误差就会越来越小！ " 887 | ] 888 | }, 889 | { 890 | "cell_type": "markdown", 891 | "id": "400f88d5", 892 | "metadata": {}, 893 | "source": [ 894 | "  由于在抽水过程中，水的位移只有垂直方向，我们从高到低将水有序分割成 $n$ 个单位，每个单位都是一个小型水柱，长为 $h_i=\\dfrac{5}{n} \\mathrm{m}$，简易计算可知，单位重力为 $G_i=\\rho V_ig=\\rho gr^2\\pi h_i=\\dfrac{441\\pi}{n} \\mathrm{N}$，水柱的重心在体心，那么抽第 $i$ 个单位的水需要的位移是 $x_i=\\dfrac{5i}{n}\\mathrm{m}$。所以，抽水做功的近似值就是对数列 $G_ix_i$ 的前 $n$ 项求和：\n", 895 | "$$\n", 896 | "W\\approx \\sum_{i=1}^nG_ix_i=\\dfrac{441\\pi}{n}\\sum_{i=1}^n\\dfrac{5i}{n}=441\\pi\\dfrac{5n+5}{2n}=1102.5\\pi+\\dfrac{5\\pi}{2n}\n", 897 | "$$\n", 898 | "当 $n$ 越来越大时，误差项 $\\dfrac{5\\pi}{2n}$ 就会越来越小，我们有理由猜测抽水做功 $W$ 就是 $1102.5\\pi\\approx 3436 \\mathrm{J}$！" 899 | ] 900 | }, 901 | { 902 | "cell_type": "markdown", 903 | "id": "7a2b0da8", 904 | "metadata": {}, 905 | "source": [ 906 | "  老师，那只是我们的猜测，有没有更精确的方法呢? \n", 907 | "\n", 908 | "  当然有！事实上，我们能分割水柱是因为每个**单位水柱中所有水的位移**是**近似相同**的，当分割到什么程度时，单位中所有水的位移是相同的呢？分割至质点大小的时候，如下图所示，这时每个单位水柱的位置及其位移可以用距离桶底的高度 $h$ 来完美表达！这便是高中物理常用的微元法，而此时数列求和将变为积分！\n", 909 | "\n", 910 | "" 911 | ] 912 | }, 913 | { 914 | "cell_type": "markdown", 915 | "id": "96db9f3b", 916 | "metadata": {}, 917 | "source": [ 918 | "$$\n", 919 | "\\begin{aligned}\n", 920 | "\\text{单位长度} \\ h_i & \\rightarrow n \\text{越来越大，长度微分为} \\ dh \\\\\n", 921 | "\\text{单位重力} \\ G_i & \\rightarrow n \\text{越来越大，重力微分为} \\ dG=\\rho g r^2\\pi dh=88.2\\pi dh \\\\\n", 922 | "\\text{单位位移} \\ x_i & \\rightarrow n \\text{越来越大，位移函数为} \\ x(h)=5-h \\\\\n", 923 | "\\text{做功总和} \\ \\sum_{i=1}^nG_ix_i & \\rightarrow n \\text{越来越大，做功积分为} \\ \\displaystyle \\int_{0}^5 x(h)dG =\\displaystyle\\int_{0}^5 88.2\\pi (5-h)dh\n", 924 | "\\end{aligned}\n", 925 | "$$" 926 | ] 927 | }, 928 | { 929 | "cell_type": "markdown", 930 | "id": "8e689196", 931 | "metadata": {}, 932 | "source": [ 933 | "利用python进行数值计算：" 934 | ] 935 | }, 936 | { 937 | "cell_type": "code", 938 | "execution_count": 8, 939 | "id": "50e96187", 940 | "metadata": { 941 | "ExecuteTime": { 942 | "end_time": "2022-06-03T07:16:48.741776Z", 943 | "start_time": "2022-06-03T07:16:48.732778Z" 944 | } 945 | }, 946 | "outputs": [ 947 | { 948 | "data": { 949 | "text/plain": [ 950 | "3463.605900582747" 951 | ] 952 | }, 953 | "execution_count": 8, 954 | "metadata": {}, 955 | "output_type": "execute_result" 956 | } 957 | ], 958 | "source": [ 959 | "from scipy import integrate \n", 960 | "from scipy import pi\n", 961 | "\n", 962 | "# 已知函数表达式积分\n", 963 | "def f(h):\n", 964 | " \"\"\"\n", 965 | " 定义函数表达式.\n", 966 | " \"\"\"\n", 967 | " return 88.2 * pi * (5 - h)\n", 968 | "\n", 969 | "\n", 970 | "v, err = integrate.quad(f, 0, 5) # 被积函数与积分区间\n", 971 | "v" 972 | ] 973 | }, 974 | { 975 | "cell_type": "markdown", 976 | "id": "38b83c98", 977 | "metadata": {}, 978 | "source": [ 979 | "
\n", 980 | " 📋 小结：\n", 981 | "本质上，微元法其实就是分析学中积分的应用：将连续变化的函数（变力做功、恒力变位移做功、连续型随机变量期望值等连续物理量）先进行微分再积分得到相关量，这在处理非离散或离散程度小的建模问题中尤为常见。\n", 982 | "
" 983 | ] 984 | }, 985 | { 986 | "cell_type": "markdown", 987 | "id": "65b35f41", 988 | "metadata": {}, 989 | "source": [ 990 | "
\n", 991 | "❓ 紧接着 GitModel 公司又接到另一家工程企业的任务：由于吊桥的年久失修导致一些铁链生锈，从而质量发生了改变，为保证安全，必须重新测量铁链的总体质量，而体积巨大难以拆卸重组的铁链无法直接测量其质量，仅能靠检测每块铁环的质量来估计整体变化的质量，然而逐块检测是一项巨大的工程，需要耗费巨额的时间与人力。\n", 992 | "\n", 993 | "  据专家分析，若以桥梁中心为原点建立空间坐标系，铁链的 $y$ 轴坐标可以近似相同。铁链上每块铁环的密度仅与其所处的横向距离以及海拔 $(x,z)$ 有关，经数值拟合可得到密度 $(\\text{kg}/\\text{m}^3)$ 与位置 $(\\text{m})$ 的函数为 \n", 994 | "$$\n", 995 | "\\rho(x,z) = 7860\\left(1 + 1.5^{2-\\frac{z}{10} - 0.1\\left(\\frac{x}{50}\\right)^2}\\right)\n", 996 | "$$ 及铁链的垂直面曲线方程为\n", 997 | "$$\n", 998 | "z = 30 \\cosh \\dfrac{x}{300}\n", 999 | "$$ \n", 1000 | "铁环是圆柱形的，半径为 $r=0.15 \\text{m}$。GitModel 公司如何通过不直接检测的方式来估计铁链的质量呢？\n", 1001 | " \n", 1002 | "\n", 1003 | "
吊索桥图示
\n", 1004 | "\n", 1005 | "
平面曲线
\n", 1006 | "
" 1007 | ] 1008 | }, 1009 | { 1010 | "cell_type": "markdown", 1011 | "id": "88cb92a8", 1012 | "metadata": {}, 1013 | "source": [ 1014 | "**分析：** \n", 1015 | "  我们发现，这其实也是类似于上文提到的离散程度小的问题（因为铁链的铁环数足够多，使得我们可以近似于连续问题），每块铁环可以当作圆柱形状的微元，其具有密度与位置的函数 $\\rho(x,z)$，长度微分 $ds$ 以及体积微分 $dV$，那么我们同样可以应用积分来求出质量。\n", 1016 | "\n", 1017 | "  那么该问题是否可以照搬上文的方法呢？我们接着往下分析两个问题的不同。应该注意到：第一个问题是位移**在垂直直线上的分布**，而该问题是密度（质量）**在曲线上的分布**，这代表着我们无法通过常规积分的方式来求解，因为我们的坐标轴不是一条直线！" 1018 | ] 1019 | }, 1020 | { 1021 | "cell_type": "markdown", 1022 | "id": "1521bddf", 1023 | "metadata": {}, 1024 | "source": [ 1025 | "  当然，分析学同样提供了在曲线上的积分方式，我们称为曲线积分。本问题就是一个经典的**第一型曲线积分**问题——知道曲线上每个点 $(x,z)$ 上的**分布函数** $\\rho(x,z)$，求曲线的质量 $M=\\displaystyle\\int_L m(x,z)ds$，类似于我们对坐标 $x$ 做微分 $dx$，曲线上的坐标可以用长度来表示，所以我们对曲线的长度做了微分 $ds$ 再进行积分处理。与直线积分有所不同的是，我们无法以曲线为轴，但仍然可以将曲线放在坐标系中，此时由勾股定理得知，我们可以求出长度微分的表达式，如下：\n", 1026 | "$$\n", 1027 | "\\begin{aligned}\n", 1028 | "\\text{单位长度} \\ s_i=\\sqrt{x_i^2+z_i^2} & \\rightarrow \\text{分割越来越细，长度微分为} \\ ds=\\sqrt{(dx)^2+(dz)^2} \\\\\n", 1029 | "\\text{单位体积} \\ V_i=s_ir^2\\pi & \\rightarrow \\text{分割越来越细，体积微分为} \\ dV=r^2\\pi ds \\\\\n", 1030 | "\\text{单位质量} \\ m_i=\\rho(x,z)V_i & \\rightarrow \\text{分割越来越细，质量函数为} \\ \\rho(x,z) dV \\\\\n", 1031 | "\\text{总质量} \\ \\sum_{i=1}^{n}\\rho(x_i,z_i)V_i & \\rightarrow \\text{分割越来越细，质量积分为} \\ \\displaystyle \\int_L \\rho(x,z) dV=\\displaystyle\\int_L \\rho(x,z)r^2\\pi ds\n", 1032 | "\\end{aligned}\n", 1033 | "$$\n", 1034 | "\n", 1035 | "**注：**当 $\\rho(x,z) \\equiv 1$ 时，事实上就是在求曲线的长度。" 1036 | ] 1037 | }, 1038 | { 1039 | "cell_type": "markdown", 1040 | "id": "9487702f", 1041 | "metadata": {}, 1042 | "source": [ 1043 | "  在进行铁链质量数值估算之前，我们先给出第一型曲线积分的计算方式。往往我们计算曲线积分时，需要知道曲线 $L$ 的参数方程 $x=x(t),z=z(t)$ 以及参数 $t$ 的取值范围 $[a,b]$。问题中提到铁链的近似方程为 $z=30\\cosh \\dfrac{x}{300}$，也就是说，铁链曲线的参数方程可用 $x$ 作为参数，其范围是 $[-300,300]$，我们将铁链曲线方程写成如下参数形式：\n", 1044 | "$$\n", 1045 | "\\begin{cases}\n", 1046 | "z(x)=30\\cosh \\dfrac{x}{300}\\\\\n", 1047 | "x(x)=x\\\\\n", 1048 | "\\end{cases} ,x \\in [-300,300]\n", 1049 | "$$ \n", 1050 | "  那么，铁链的质量可由如下曲线积分计算方式给出： " 1051 | ] 1052 | }, 1053 | { 1054 | "cell_type": "markdown", 1055 | "id": "46ae470c", 1056 | "metadata": {}, 1057 | "source": [ 1058 | "
\n", 1059 | " 定义4：[第一型曲线积分的计算方式] \n", 1060 | "设平面曲线 $L$ 的参数方程为 $x=x(t),z=z(t),t\\in [a,b],$ $\\rho(x,z)$ 为定义在 $L$ 上的连续函数，则\n", 1061 | "$$\n", 1062 | "\\int_{L}\\rho(x,z) ds = \\int_{a}^b \\rho(x(t),z(t)) \\sqrt{(x'(t))^2+(z'(t))^2} dt\n", 1063 | "$$ \n", 1064 | "
" 1065 | ] 1066 | }, 1067 | { 1068 | "cell_type": "markdown", 1069 | "id": "65ce8b02", 1070 | "metadata": {}, 1071 | "source": [ 1072 | "  我们代入问题给出的方程 $\\rho(x,z),z(x)$ 以及铁环半径 $r=0.15 \\mathrm{m}^3$，即可得到铁链的总质量计算式为\n", 1073 | "$$\n", 1074 | "11.79\\displaystyle\\int_{-300}^{300}\\left(1+1.5^{2-3\\cosh\\frac{x}{300}-0.1\\left(\\frac{x}{50}\\right)^2}\\right)\\sqrt{100+\\sinh^2\\dfrac{x}{300}}dx\n", 1075 | "$$\n", 1076 | "\n", 1077 | "  下面给出求第一型曲线积分的python代码，铁链质量的估算值为 98635t。" 1078 | ] 1079 | }, 1080 | { 1081 | "cell_type": "code", 1082 | "execution_count": 9, 1083 | "id": "59a2af5a", 1084 | "metadata": { 1085 | "ExecuteTime": { 1086 | "end_time": "2022-06-03T07:34:39.959817Z", 1087 | "start_time": "2022-06-03T07:34:39.950841Z" 1088 | } 1089 | }, 1090 | "outputs": [ 1091 | { 1092 | "name": "stderr", 1093 | "output_type": "stream", 1094 | "text": [ 1095 | "e:\\learningdisk\\learning_more\\datawhale\\01-组队学习\\gitmodel\\venv\\lib\\site-packages\\ipykernel_launcher.py:5: DeprecationWarning: scipy.cosh is deprecated and will be removed in SciPy 2.0.0, use numpy.cosh instead\n", 1096 | " \"\"\"\n", 1097 | "e:\\learningdisk\\learning_more\\datawhale\\01-组队学习\\gitmodel\\venv\\lib\\site-packages\\ipykernel_launcher.py:6: DeprecationWarning: scipy.sinh is deprecated and will be removed in SciPy 2.0.0, use numpy.sinh instead\n", 1098 | " \n", 1099 | "e:\\learningdisk\\learning_more\\datawhale\\01-组队学习\\gitmodel\\venv\\lib\\site-packages\\ipykernel_launcher.py:6: DeprecationWarning: scipy.sqrt is deprecated and will be removed in SciPy 2.0.0, use numpy.lib.scimath.sqrt instead\n", 1100 | " \n" 1101 | ] 1102 | }, 1103 | { 1104 | "data": { 1105 | "text/plain": [ 1106 | "98635.09908278256" 1107 | ] 1108 | }, 1109 | "execution_count": 9, 1110 | "metadata": {}, 1111 | "output_type": "execute_result" 1112 | } 1113 | ], 1114 | "source": [ 1115 | "from scipy import sinh, cosh, sqrt\n", 1116 | "\n", 1117 | "# 第一型曲线积分\n", 1118 | "def f(x):\n", 1119 | " index1 = (2 - 3 * cosh(x/300) - 0.1 * ((x/50)**2))\n", 1120 | " return (1 + 1.5 ** index1) * sqrt(100 + (sinh(x/300))**2)\n", 1121 | "\n", 1122 | "v, err = integrate.quad(f, -300, 300)\n", 1123 | "v * 11.79" 1124 | ] 1125 | }, 1126 | { 1127 | "cell_type": "markdown", 1128 | "id": "42df69db", 1129 | "metadata": {}, 1130 | "source": [ 1131 | "
\n", 1132 | "❓ 有了直线积分和曲线积分，我们自然会问：有没有平面积分以及曲面积分呢？当然！事实上，由于我们还会接触变力曲线做功，这代表在每个点上的分布函数不再是标量函数而是向量函数！分析学还给出了第二类曲线积分以及曲面积分。\n", 1133 | "
\n", 1134 | "" 1135 | ] 1136 | }, 1137 | { 1138 | "cell_type": "markdown", 1139 | "id": "b06110aa", 1140 | "metadata": {}, 1141 | "source": [ 1142 | "  有一位爱钻牛角尖的同学发出了提问——老师，这样的方法还是太“数学”了，并且理想很丰满，现实很骨感！万一**没有函数表达式呢**？又万一有函数表达式但**积分太难算了呢**？\n", 1143 | "\n", 1144 | "  别急，数学家也想到了应用性的问题，理论总是要落地的嘛！所以有了计算机之后，许多数值计算的方法也被发明了出来，其中就包括了**数值积分**的方法——计算机可以通过数值方法对许多**不连续**函数甚至是**无法写出表达式**的函数进行积分！这便很好地解决了积分在实际应用上的问题。这将在我们的**2.0课程中与大家见面**👋" 1145 | ] 1146 | }, 1147 | { 1148 | "cell_type": "markdown", 1149 | "id": "1312e4a8", 1150 | "metadata": {}, 1151 | "source": [ 1152 | "## 实战项目——人口增长问题" 1153 | ] 1154 | }, 1155 | { 1156 | "cell_type": "markdown", 1157 | "id": "76065673", 1158 | "metadata": {}, 1159 | "source": [ 1160 | "
\n", 1161 | "

💼 任务来袭

\n", 1162 | "\n", 1163 | "  GitModel 公司对面试的实习生给出了这样一个问题：搜集 $1950\\sim 2020$ 年间美国人口数据，猜测其满足的函数关系，并综合数据预测美国 $2030$ 年的人口数。\n", 1164 | "\n", 1165 | "  公司希望实习生就以下的开放性问题给出自己的想法，公司要求实习生提交预测的思路、模型、算法以及结果。\n", 1166 | "\n", 1167 | "  面试官给出了如下提示：预测值与真实值存在误差，该问题如何转化为可用上述所学的知识解决的问题呢？ \n", 1168 | "
" 1169 | ] 1170 | }, 1171 | { 1172 | "cell_type": "markdown", 1173 | "id": "95f6c8e8", 1174 | "metadata": {}, 1175 | "source": [ 1176 | "
\n", 1177 | "（1）从你的数据来看，你猜想美国人口与时间的关系应该满足怎样的函数关系（线性、二次或是其他）？这样的函数关系你如何确定？又怎么衡量这个函数具有良好的预测效果？[请在下方Markdown方格中写下你的答案]\n", 1178 | "
" 1179 | ] 1180 | }, 1181 | { 1182 | "cell_type": "markdown", 1183 | "id": "7981e8ee", 1184 | "metadata": {}, 1185 | "source": [ 1186 | "> 双击写下你的想法：" 1187 | ] 1188 | }, 1189 | { 1190 | "cell_type": "markdown", 1191 | "id": "2ca629dd", 1192 | "metadata": {}, 1193 | "source": [ 1194 | "
\n", 1195 | "（2）也许你可以尝试用不同的函数关系进行预测，比较一下哪种函数更贴合数据，但是，越贴合数据的函数是否真的预测的结果越可信？你可以查阅资料去了解该问题，但我们更希望你能解释这个问题的内在原理，如果能给出相应的解决方法那就更好了！[请在下方Markdown方格中写下你的答案]\n", 1196 | "
" 1197 | ] 1198 | }, 1199 | { 1200 | "cell_type": "markdown", 1201 | "id": "7da76694", 1202 | "metadata": {}, 1203 | "source": [ 1204 | "> 双击写下你的想法：\n" 1205 | ] 1206 | }, 1207 | { 1208 | "cell_type": "markdown", 1209 | "id": "2072791d", 1210 | "metadata": {}, 1211 | "source": [ 1212 | "
\n", 1213 | "

Well done!

\n", 1214 | "

恭喜你，完成了《动手学高等数学》的学习，希望在日后的数模竞赛中，通过本课程的学习，你可以更加深刻的理解微积分的强大之处。

\n", 1215 | "
\n", 1216 | "

如有需要领取本次组队学习的答案，敬请关注公众号以及留意直播讲解！

\n", 1217 | "
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