人工智能教程 - 数学基础课程1.2 - 数学分析(二)-8

等值面,偏导数,切平面逼近

• Function of 1 variable .f(x)=sin(x)
• Function of 2 variables:
  given(x, y)$\rightarrow$get a number f(x, y)
  Example $f(x,y)=x^2+y^2$
  f(x, y)=temperature at point (x, y)
• or …3 or more parameters!

How to visualize f of 2 variables?
$\rightarrow$gragh: z=f(x, y)

Ex:
$f(x,y)=1-x^2-y^2$
$\rightarrow$in y-z plane: $x=0,z=1-y^2$
$\rightarrow$in x-z plane: $y=0,z=1-x^2
$\rightarrow$in x-y plane: $z=0,1-x^2-y^2=0$
$x^2+y^2=1$ (unit circle)

等高线图(Coutour plot)

Shows all the points where f(x, y)=some fixed value(constant)
chosen at regular values
$\Leftrightarrow$ we slice the graph by horizontal plane z=c

f(x, y)=1为等高线(level curve)

偏导数(partial derivatives)

1)function of 1 variable f(x)

$f'(x)=\frac{df}{dx}$

$\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$

逼近表达式(Approximation formula)

$x_0\rightarrow f(x_0)$

$f(x)\approx f(x_0)+f'(x_0).(x-x_0)$

2)function of 2 variable

$\frac{\partial f}{\partial x}(x_0,y_0)=\lim_{\Delta x\rightarrow 0}\frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}$

$\frac{\partial f}{\partial y}(x_0,y_0)=\lim_{\Delta y\rightarrow 0}\frac{f(x_0,y_0+\Delta y)-f(x_0,y_0)}{\Delta y}$

物理上:$\frac{\partial f}{\partial x}=f_x$
treat y as constant
treat x as variable

Ex:
$f(x ,y)=x^3y+y^2$

$\frac{\partial f}{\partial x}=x^3+2y$

来源：CSDN

作者：KuFun人工智能

链接：https://blog.csdn.net/fsdaewrq/article/details/104106166