Question

ДАЮ 50 БАЛІВ! допоможіть

Answer 1

Ответ:

Приращение аргумента :   $\bf \Delta x=x-x_0$  , приращение функции :  

$\bf \Delta y=f(x)-f(x_0)=f(x_0+\Delta x)-f(x_0)$ .

$\bf1)\ \ f(x)=3x\ \ ,\ \ \Delta y=3(x_0+\Delta x)-3x_0=3\cdot \Delta x\\\\2)\ \ f(x)=x^3\ \ ,\ \ \\\\\Delta y=(x_0+\Delta x)^3-x_0^3=x_0^3+3\, x^2_0\, \Delta x+3x_0(\Delta x)^2+(\Delta x)^3-x_0^3=\\\\=3\, x_0^2\, \Delta x+3x_0\, (\Delta x)^2+(\Delta x)^3\\\\3)\ \ f(x)=x^2-x\\\\\Delta y=(x_0+\Delta x)^2-(x_0+\Delta x)-x_0^2-x_0=\\\\=x_0^2+2\, x_0\, \Delta x+(\Delta x)^2-x_0-\Delta x-x_0^2-x_0=2\, x_0\, \Delta x+(\Delta x)^2-\Delta x$  

$\bf 4)\ \ f(x)=x+\dfrac{1}{x}\\\\\Delta y=(x_0+\Delta x)+\dfrac{1}{x_0+\Delta x}-x_0-\dfrac{1}{x_0}=\\\\\\=\Delta x+\dfrac{1}{x_0+\Delta x}-\dfrac{1}{x_0}=\Delta x+\dfrac{x_0-x_0-\Delta x}{x_0(x_0+\Delta x)}=\Delta x-\dfrac{\Delta x}{x_0(x_0+\Delta x)}=\\\\\\=\dfrac{x_0^2\, \Delta x+x_0(\Delta x)^2-\Delta x}{x_0(x_0+\Delta x)}$