Section03_其他知识点

凹凸性

定义

1. 对于 $y = f(x),(x\in \mathbb{D})$，若 $\forall\ x_{1}, x_{2} \in \mathbb{D}$ 且 $x_{1}\neq x_{2}$ 有$\displaystyle{f\bigg(\frac{x_{1}+x_{2}}{2}\bigg) < \frac{f(x_{1})+f(x_{2})}{2}}$ 称 $f(x)$ 在 $\mathbb{D}$ 上为凹函数
2. 对于 $y = f(x),(x\in \mathbb{D})$，若 $\forall\ x_{1}, x_{2} \in \mathbb{D}$ 且 $x_{1}\neq x_{2}$ 有$\displaystyle{f\bigg(\frac{x_{1}+x_{2}}{2}\bigg) > \frac{f(x_{1})+f(x_{2})}{2}}$ 称 $f(x)$ 在 $\mathbb{D}$ 上为凸函数

Notes 若 $x=x_{0}$ 两侧凹凸性不同，$(x_{0},f(x_{0}))$ 为拐点

判别法

• Th 设$f(x)\in C[a,b]$，$(a,b)$内二阶可导
  1. $f''(x)>0\quad (a<x<b)$，$f(x)$ 在 $(a,b)$ 上为凹函数
  2. $f''(x)<0\quad (a<x<b)$，$f(x)$ 在 $(a,b)$ 上为凸函数

例题

• 例1 $f''(1) = 0, f'''(1)=2$，$(1,f(1))$是否为拐点？ $$\begin{array}{ll} \because & f''(1) = 0, f'''(1) = 2 \\ \therefore & \displaystyle{\lim_{x\to 1}\frac{f(x)-f''(1)}{x-1} = 2 > 0} \\ \therefore & \exists\ \delta > 0, 0<\vert x- 1\vert <\delta, \frac{f''(x)-f''(1)}{x-1} > 0 \\ \therefore & \begin{cases} f''(x) = f''(x) - f''(1) < 0,\quad x\in(1-\delta, 1) \\ f''(x) = f''(x) - f''(1) > 0,\quad x\in(1, 1+\delta) \\ \end{cases} \\ \therefore & (1, f(1)) \text{两侧的凹凸性不同，其为拐点} \\ \end{array}$$

渐近线

定义

1. 水平渐近线 若 $\displaystyle{\lim_{x\to \infty}f(x) = A}$，称 $y=A$ 为水平渐近线。最多仅有两条，且与斜渐近线互斥
2. 铅直渐近线 若$\begin{cases}\displaystyle{\lim_{x\to a}f(x) = \infty}\\\displaystyle{f(a-0) = \infty}\\ \displaystyle{f(a+0) =\infty}\end{cases}$ 满足其一，则$x=a$为铅直渐近线。铅直渐近线，仅可能存在于间断点处
3. 斜渐近线 若$\displaystyle{\lim_{x\to \infty}\frac{f(x)}{x}=a,\quad(a\neq 0, \infty)}$, $\displaystyle{\lim_{x\to\infty}[f(x)-ax]=b}$，称 $y=ax+b$ 为斜渐近线

例题

• 例1 $\displaystyle{y=\frac{x^{2}+3x+2}{x-1}}$，求渐近线 $$\begin{array}{ll} \because & \displaystyle{\lim_{x\to \infty}y = \infty} \\ \therefore & \text{不存在在水平渐近线} \\ \because & \displaystyle{\lim_{x\to 1}} y = \infty \\ \therefore & x = 1 \text{为铅直渐近线} \\ \because & \displaystyle{\lim_{x\to\infty}\frac{y}{x} = \lim_{x\to\infty}\frac{2x^{2}+3x+2}{x^{2}-x}=2} \\ & \displaystyle{\lim_{x\to \infty}(y-2x) = \lim_{x\to \infty}\frac{5x+2}{x-1} = 5}\\ \therefore & y=2x+5 \text{为斜渐近线} \\ \end{array}$$
• 例2 $\displaystyle{y= \frac{x^{2}-x-2}{x^{2}-1}e^{\frac{1}{x}}}$ $$\begin{array}{ll} \because & \displaystyle{\lim_{x\to \infty}y = 1} \\ \therefore & y=1\text{为水平渐近线} \\ \because & \displaystyle{\lim_{x\to -1}} y = \lim_{x\to -1}\frac{(x+1)(x-2)}{(x+1)(x-1)}e^{\frac{1}{x}} = \frac{3}{2e}\ne \infty \\ & \displaystyle{\lim_{x\to 1}} y = \lim_{x\to 1}\frac{(x+1)(x-2)}{(x+1)(x-1)}e^{\frac{1}{x}} = + \infty \\ \therefore & x = 1 \text{为铅直渐近线} \\ \because & \displaystyle{\lim_{x\to 0^{+}}y = +\infty,\ \lim_{x\to0^{-}}y=0} \\ \therefore & x= 0 \text{为铅直渐近线} \\ \end{array}$$

• 例3 $y=f(x) = \sqrt{x^{2}-4x+7}+x$ $$\begin{array}{ll} & \displaystyle y=f(x) = \frac{-4x+7}{\sqrt{x^{2}-4x+7}-x} \\ \because & \displaystyle{\lim_{x\to +\infty}y = \infty, \lim_{x\to -\infty} y = 2} \\ \therefore & y=2 \text{为水平渐近线} \\ \because & f(x)\text{不存在间断点} \\ \therefore & \text{不存在铅直渐近线} \\ \because & \displaystyle{\lim_{x\to +\infty}\frac{y}{x} = 2, \lim_{x\to -\infty}\frac{y}{x}= -\infty} \\ & \displaystyle{\lim_{x\to +\infty}(y-2x) = -2} \\ \therefore & y=2x-2 \text{为斜渐近线} \\ \end{array}$$

  弧微分

  

• $\text{L: }y=f(x)$: $ds = \sqrt{(dx)^{2}+(dy)^{2}} = \sqrt{1+(\frac{dy}{dx})^{2}}\cdot dx$ 即 $ds=\sqrt{1+[f'(x)]^{2}}\cdot dx$

• $\text{L: }\begin{cases}x=\phi(t)\\y=\psi(t)\end{cases}$: $ds = \sqrt{(dx)^{2}+(dy)^{2}} = \sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}}\cdot dt$ 即$ds = \sqrt{[\phi'(t)]^{2}+[\psi'(t)]^{2}}\cdot dt$