Упр.780 ГДЗ Алимов 10-11 класс (Алгебра)

1) f(x) =3x+2;
2) f(x) =5x+7;
3) f(x) =3×2-5x;
4) f(x) =-3×2+2.

1. $$f(x)=3x+2$$

   $$\Delta f=f(x+h)-f(x)=3(x+h)+2-(3x+2)=3h$$

   $$\frac{\Delta f}{h}=\frac{3h}{h}=3$$

   $$f'(x)=\lim_{h\to 0}3=3$$

2. $$f(x)=5x+7$$

   $$\Delta f=f(x+h)-f(x)=5(x+h)+7-(5x+7)=5h$$

   $$\frac{\Delta f}{h}=\frac{5h}{h}=5$$

   $$f'(x)=\lim_{h\to 0}5=5$$

3. $$f(x)=3x^2-5x$$

   $$\Delta f=f(x+h)-f(x)=3(x+h)^2-5(x+h)-(3x^2-5x)$$

   $$ \Delta f=3x^2+6xh+3h^2-5x-5h-3x^2+5x=6xh+3h^2-5h $$

   $$\frac{\Delta f}{h}=\frac{6xh+3h^2-5h}{h}=6x+3h-5$$

   $$f'(x)=\lim_{h\to 0}(6x+3h-5)=6x-5$$

4. $$f(x)=-3x^2+2$$

   $$\Delta f=f(x+h)-f(x)=-3(x+h)^2+2-(-3x^2+2)$$

   $$ \Delta f=-3x^2-6xh-3h^2+2+3x^2-2=-6xh-3h^2 $$

   $$\frac{\Delta f}{h}=\frac{-6xh-3h^2}{h}=-6x-3h$$

   $$f'(x)=\lim_{h\to 0}(-6x-3h)=-6x$$

Ответ

1) $$3$$; 2) $$5$$; 3) $$6x-5$$; 4) $$-6x$$.