完成:高数一元积分学Part1

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# 谏学高数者十思疏
学高数者诚能见等价则思加减不能替将有洛则思代换以化简念复合则思勿漏层而求导惧积分则思不定以加C乐微分则思dx而莫忘忧定积则思牛莱而相减虑换元则思积分上下限惧级数则思判别勿用错项所加则思无因忽以谬导拐所及则思无因x而漏y。总此十思宏兹九章简能而任之择善而从之则牛顿尽其谋莱氏竭其力泰勒播其惠柯西效其忠。文理争驰学生无事可以尽数分之乐可以养高代之寿。
# 逐句解析
##### 学高数者,诚能见等价,则思加减不能替;
等价无穷小的替换不能出现在加减法中。如果一定需要进行加减法,请改用**泰勒展开**。
等价无穷小的替换不能出现在加减法中。如果一定需要进行加减法,请改用**泰勒展开**,并留意泰勒展开的程度,你需要根据皮亚诺余项来判断需要展开到多少阶
##### 将有洛,则思代换以化简;
准备用洛必达的时候,先考虑能不能用等价代换来化简式子,减小求导的压力;
##### 念复合,则思勿漏层而求导;
@ -25,19 +26,22 @@ flowchart TD
B --> |"是"| D["$$\lim\limits_{n\to\infty}a_n$$ 是正项级数?"]
D --> |"否"| E["$$\sum\limits_{n=1}^\infty |a_n|$$ 收敛?"]
D --> |"是"| F["比较判别法;<br>比值/根值判别法"]
E --> |"否"| G["$$\sum\limits_{n=1}^\infty a_n$$ 为交错级数?"]
E --> |"否"| L["使用比值/根值判别法判别的$$\sum\limits_{n=1}^\infty |a_n|$$ 发散?"]
L --> |"否"| G["$$\sum\limits_{n=1}^\infty a_n$$ 为交错级数?"]
L --> |"是"| M["$$\sum\limits_{n=1}^\infty a_n$$ 发散"]
E --> |"是"| H["$$\sum\limits_{n=1}^\infty a_n$$ <b>绝对收敛</b>"]
G --> |"否"| I["利用级数的运算性质,<br>如部分和,或其他判别法"]
G --> |"是"| J["级数每一项的<br>绝对值单调递减?"]
J --> |"否"| I
J --> |"是"| K["莱布尼兹判别法:$$\sum\limits_{n=1}^\infty a_n$$ <b>条件收敛</b>"]
```
>[!error] 非正项级数需要写明“绝对收敛”和“条件收敛”!
##### 项所加,则思无因忽以谬导;
在求导的时候,观察好每一项!不要漏掉了什么
##### 拐所及则思无因x而漏y。
>[!error] 考试必考点!拐点是一个二维的点,零点、驻点等是一个一维的 $x$ 值!
>[!error] 考试必考点!拐点是一个二维的点!!而零点、驻点等是一个一维的 $x$ 值!
##### 总此十思,宏兹九章,简能而任之,择善而从之,则牛顿尽其谋,莱氏竭其力,泰勒播其惠,柯西效其忠。文理争驰,学生无事,可以尽春节之乐,可以养寒假之寿。
只要易错点不错,你就放心去考吧,我们已经帮你找好关系了——找的是牛顿、莱布尼兹、泰勒和柯西,我说服了他们往你的大脑注入一点他们的智慧,考试当天生效。
<span class="emphasize">最后提前祝大家过个好年!</span>

@ -685,27 +685,21 @@ $\displaystyle\int x \ln x\mathrm dx = \frac{x^2}{2}\ln x - \frac{x^2}{4} + C$
>[!abstract] 练习
>尝试推导上述公式
# Trivia
魔法六边形
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# 碎碎念
### 魔法六边形
左边全是正,正弦/正切/正割
右边全是余,余弦/余切/余割
![[魔法六边形.svg]]
你可以绕着这个六边形旋转来得到三角函数的商:
![[魔法六边形-2.svg]]
除此之外还有 $\sin x=\frac{\cos x}{\cot x},\cos x=\frac{\cot x}{\csc x}$...有时能够取得一些意想不到的效果。
穿过六边形的中心 $1$,即为取倒数:
$\sin x=\frac{1}{\csc x}$$\cos x=\frac{1}{\sec x}$
![[魔法六边形-3.svg]]
防止在过度紧张的时候将正割/余割的倒数记错。
对于每一个小倒三角 $\nabla$,三角形上边的两个角的平方和等于下面的角,用这种方法可以快速记忆三角函数平方和公式 $\tan^2 x+1=\sec^2 x, 1+\cot^2 x=\csc^2x$
![[魔法六边形-5.svg]]
![[谏学高数者十思疏]]

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After

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<text x="-179.5" y="-475.8" style="font-style:italic;">sin</text>
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After

Width:  |  Height:  |  Size: 1.3 KiB

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