《普林斯顿微积分读本》个人读书笔记

太过爱你忘了你带给我的痛 2023-07-05 05:24 1阅读 0赞

最近重温了一下大一没学好的微积分，于是挑了一本豆瓣评价较高的《普林斯顿微积分读本》，这本书讲的确实通俗易懂，让读者加深对微积分的理解的同时也讲述了一些解题的方法，所以我在这里记下了我认为重要的一些内容。美中不足的是这本书只讲述了单变量微积分有关的部分。

> 参考资料: 《普林斯顿微积分读本》人民邮电出版社

这篇博客只记录我在阅读过程中感到重要的地方，只是一个读书笔记而已

### 解题技巧: 导数伪装的极限 ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70]![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 1]

### 一个好奇函数: 可导而导数不连续的函数 ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 2]![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 3]![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 4]

### 双曲函数 ###

原书9.7节  
![在这里插入图片描述][20200218132714257.png]  
![在这里插入图片描述][20200218132725505.png]  
![在这里插入图片描述][20200218132739935.png]

### 反函数的导数 ###

原书10.1节

##### 反函数导数公式 #####

感觉这个公式挺重要的，同时也不是那么好理解，有点绕  
![在这里插入图片描述][20200219150206440.png]  
写成这样可能要好点  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 5]

##### 反正弦函数的导数 #####

![在这里插入图片描述][20200221193550700.png]  
![在这里插入图片描述][20200221193751380.png]

##### 反余弦函数的导数 #####

![在这里插入图片描述][20200221194013151.png]  
![在这里插入图片描述][20200221194026306.png]

##### 函数图像 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 6]

##### 反正切函数的导数 #####

![在这里插入图片描述][20200221195133803.png]

##### 反正割函数的导数 #####

![在这里插入图片描述][20200221195222492.png]  
![在这里插入图片描述][20200221195308980.png]

##### 反余割与反余切函数及其导数 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 7]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 8]

##### 反双曲函数及其导数 #####

![在这里插入图片描述][20200221203602631.png]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 9]

![在这里插入图片描述][20200221203649449.png]  
![在这里插入图片描述][20200221203701373.png]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 10]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 11]

### 导数与图像 ###

##### 罗尔定理 #####

![在这里插入图片描述][20200224001103360.png]

##### 中值定理 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 12]

![在这里插入图片描述][20200224011611259.png]  
函数拐点: 函数凹凸性变化的点  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 13]  
举例: x=0是 x 3 x^3 x3的拐点, 但不是 x 4 x^4 x4的拐点，即使在x=0处二阶导数都是0

### 解题技巧: 绘制函数图像的步骤 ###

(1)对称性  
(2)y轴截距  
(3)x轴截距  
(4)定义域  
(5)垂直渐近线  
(6)函数的正负  
(7)水平渐近线  
(8)导数的正负  
(9)最大值和最小值  
(10)二次导数的正负  
(11)拐点

### 最优化和线性化 ###

##### 牛顿法 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 14]

牛顿法失灵的四种情况:  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 15]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 16]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 17]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 18]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 19]

### 第十四章 洛比达法则 ###

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 20]

##### 洛比达法则大致证明 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 21]

##### 重要的极限 #####

![在这里插入图片描述][20200224172948257.png]

### 定积分 ###

##### 函数的平均值 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 22]

##### 积分中值定理 #####

![在这里插入图片描述][20200229110059606.png]

### 微积分基本定理 ###

##### 微积分第一定理 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 23]  
![在这里插入图片描述][20200229112559115.png]

##### 微积分第二定理 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 24]

##### 用微积分第一定理解决问题 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 25]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 26]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 27]

##### 上下限都是函数 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 28]

##### 导数伪装的极限 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 29]

##### 求解不定积分 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 30]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 31]

### 积分的方法 ###

原书第18章

##### 替代法 #####

![在这里插入图片描述][20200229173135799.png]  
![在这里插入图片描述][20200229173500772.png]

##### 分部积分法 #####

![在这里插入图片描述][20200301100207674.png]

##### 部分分式法 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 32]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 33]

##### 三角函数积分 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 34]

##### 技巧综述 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 35]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 36]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 37]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 38]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 39]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 40]

### 反常积分 ###

原书第20章  
用极限的思想去求解  
![在这里插入图片描述][20200304102907574.png]  
![在这里插入图片描述][20200304102926192.png]  
![在这里插入图片描述][20200304103236114.png]

##### 比较判别法 #####

![在这里插入图片描述][20200304121606391.png]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 41]

##### P判别法 #####

![在这里插入图片描述][20200304120756678.png]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 42]

##### 绝对收敛判别法 #####

![在这里插入图片描述][20200304121243364.png]

##### 无穷处指数函数的行为 #####

![在这里插入图片描述][20200310163521292.png]  
提示: 利用这个函数进行比较判决(借助P判别法)

##### 无穷处对数函数的行为 #####

![在这里插入图片描述][20200310163619539.png]  
用法同上，但是有一个特别的例子:  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 43]

### 第22/23章 级数的敛散性 ###

##### 几何级数 #####

当公比的绝对值<1时收敛，否则发散

##### 第n项判别法 #####

若级数通项不趋于0，则级数发散；  
级数通项趋于0，需要进一步判断

##### 比式判别 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 44]

##### 根式判别 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 45]

##### 积分判别 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 46]

##### 其他判别方法 #####

类似于积分的敛散性，有:

*  比较判别法
 *  极限比较判别
 *  p判别

##### 交错级数判别法 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 47]

### 24. 泰勒级数 ###

##### 泰勒近似定理 #####

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 48]

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 49]  
c 的取值与积分中值定理有关，c 介于a和x之间，用来反映误差, 且一般是求不出来的

##### 泰勒级数 #####

![在这里插入图片描述][20200314163136112.png]  
是一个无穷级数  
若想要证明一个函数在某些x处等于它的泰勒级数，应该尝试证明，当 N N N趋于无穷时，余项 R N R\_N RN趋于0

##### 麦克劳林级数 #####

麦克劳林级数就是 a=0 的泰勒级数  
![在这里插入图片描述][2020031414531466.png]  
![在这里插入图片描述][20200314174621477.png]  
![在这里插入图片描述][20200314174638257.png]  
![在这里插入图片描述][20200314174658759.png]  
![在这里插入图片描述][20200314205320538.png]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2JhaWR1XzQxNzQzMTk1_size_16_color_FFFFFF_t_70 50]

##### 使用泰勒近似作估算 #####

根据余项 R N ( x ) R\_N(x) RN(x)估算误差的上限，选择合适的 N N N后代回泰勒展开式

### 幂级数 ###

##### 幂级数的收敛 #####

*  收敛半径: 利用比式判别法或根式判别法，由级数收敛的条件即可算得收敛区域、收敛半径
 *  端点通常需要单独讨论

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