Упр.180 ГДЗ Колмогоров 10-11 класс (Алгебра)

Найдём приращение функции по формуле $$\Delta f=f(x_0+\Delta x)-f(x_0).$$

1. $$f(x)=1-3x^2$$

   $$\Delta f=1-3(x_0+\Delta x)^2-(1-3x_0^2)$$
   $$=1-3x_0^2-6x_0\Delta x-3(\Delta x)^2-1+3x_0^2$$
   $$=-6x_0\Delta x-3(\Delta x)^2=-3\Delta x(2x_0+\Delta x).$$

2. $$f(x)=ax+b$$

   $$\Delta f=a(x_0+\Delta x)+b-(ax_0+b)=a\Delta x.$$

3. $$f(x)=2x^2$$

   $$\Delta f=2(x_0+\Delta x)^2-2x_0^2$$
   $$=2x_0^2+4x_0\Delta x+2(\Delta x)^2-2x_0^2$$
   $$=4x_0\Delta x+2(\Delta x)^2.$$

4. $$f(x)=\frac{1}{x}$$

   $$\Delta f=\frac{1}{x_0+\Delta x}-\frac{1}{x_0} =\frac{x_0-(x_0+\Delta x)}{x_0(x_0+\Delta x)} =-\frac{\Delta x}{x_0(x_0+\Delta x)}.$$

Ответ

а) $$\Delta f=-3\Delta x(2x_0+\Delta x)$$;
б) $$\Delta f=a\Delta x$$;
в) $$\Delta f=4x_0\Delta x+2(\Delta x)^2$$;
г) $$\Delta f=-\dfrac{\Delta x}{x_0(x_0+\Delta x)}.$$