高数笔记

高数笔记

1. 常用

1. $x$充分大时，有重要关系：
   $e^{\alpha x}>>x^{\beta }>>l{n}^{\gamma }x$
   

2. 海涅定理（归结原则）:
   $\underset{x\to x_0}{\lim }f(x)=A$ 存在 $\iff \underset{n\to \infty }{\lim }f(x_n)=A$ 存在,
   对任何$x\to x_0$
   

3. $(uv)’=u^{\prime }v+v^{\prime }u$逆用所制造的辅助函数:

• ① $[f(x)f(x)]’=[f^2(x){]}^{\prime }=2f(x)f^{\prime }(x)$                见 $f(x)f^{\prime }(x)$ $\Rightarrow$ $F(x)=f^2(x)$
  

• ② $[f(x)f^{\prime }(x){]}^{\prime }=[f^{\prime }(x){]}^2+f(x)f^{\prime \prime }(x)$                 $\Rightarrow$ $F(x)=f(x)f^{\prime \prime }(x)$
  

• ③ $[f(x)e^{\phi (x)}{]}^{\prime }=f^{\prime }(x)e^{\phi (x)}+f(x)e^{\phi (x)}{\phi }^{\prime }(x)$      $\Rightarrow$ $F(x)=f(x)e^{\phi (x)}$
  

4. 奇偶函数的加减乘除：

• ① 奇$\pm$奇=奇                      偶$\pm$偶=偶                       奇$\pm$偶=非奇非偶

• ② 奇$\times /÷$奇=奇                 偶$\times /÷$偶=偶                 奇$\times /÷$偶=奇

5. 复合函数的奇偶性：（单调性）

                                                   $\Rightarrow$内偶则偶，内奇同外

                                                   $\Rightarrow$同增异减

6. ①和差公式

• $\sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$
  

• $\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta$
  

• $\tan (\alpha \pm \beta )=\frac{\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }$
  

• $\cot (\alpha \pm \beta )=\frac{\cot \alpha \mp 1}{\cot \beta \pm \cot \alpha }$
  

   ②倍角公式

• $\sin 2\alpha =2\sin \alpha \cos \alpha$
  

• $\cos 2\alpha ={\cos }^2\alpha -{\sin }^2\alpha =1-2{\sin }^2\alpha =2{\cos }^2\alpha -1$
  

   ③降幂公式

• ${\sin }^2\alpha =\frac{1}{2}(1-\cos 2\alpha )$                ${\cos }^2\alpha =\frac{1}{2}(1+\cos 2\alpha )$
  

   ④半角公式

• $\sin \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{2}}$                        $\cos \frac{\alpha }{2}=\pm \sqrt{\frac{1+\cos \alpha }{2}}$
  

• $\tan \frac{\alpha }{2}=\frac{1-\cos \alpha }{\sin \alpha }=\frac{\sin \alpha }{1+\cos \alpha }=\pm \sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }}$
  

• $\cot \frac{\alpha }{2}=\frac{\sin \alpha }{1-\cos \alpha }=\frac{1+\cos \alpha }{\sin \alpha }=\pm \sqrt{\frac{1+\cos \alpha }{1-\cos \alpha }}$

2. 常用十大不等式

1. $|a\pm b|\le |a|+|b|$,        $||a|-|b||\le |a-b|$

2. $\sqrt{ab}\le \frac{a+b}{2}\le \sqrt{\frac{a^2+b^2}{2}}$,                $(a,b>0)$

   $\sqrt[3]{abc}\le \frac{a+b+c}{3}\le \sqrt{\frac{a^2+b^2+c^2}{3}}$,        $(a,b,c>0)$

3. 设$a>b>0$，则$\{\begin{array}{ll}当n>0时,& a^n>b^n\\ 当n<0时，& a^n<b^n\end{array}$

4. 设$0<a<x<b$，$0<c<y<d$

   则$\frac{c}{b}<\frac{y}{x}<\frac{d}{a}$

5. $sinx<x<tanx$,        $(0<x<\frac{\pi }{2})$

6. $\sin x<x$,        $(x>0)$

7. $arctanx\le x\le arcsinx$,        $(0\le x\le 1)$

8. $e^x\ge x+1$,        $(\forall x)$

9. $x-1\ge lnx$,        $(x>0)$

10. $\frac{1}{1+x}<ln(1+\frac{1}{x})<\frac{1}{x}$,        $(x>0)$