Задание 18.5

Question

Для функции $f(x)=x^{2}-x$ и точки $x_{0}$ найдите $\frac{\Delta f}{\Delta x}$ и $\lim _{\Delta x \rightarrow 0} \frac{\Delta f}{\Delta x}$.

Gimli · Accepted Answer

$$f(x)=x^{2}-x$$
$$\Delta f=\left(x^{2}-x\right)-\left(x_{0}^{2}-x_{0}\right)=$$
$$=\left(x_{0}+\Delta x\right)^{2}-\left(x_{0}+\Delta x\right)-x_{0}^{2}+x_{0}=$$
$$=x_{0}^{2}+2 x_{0} \Delta x+\Delta x^{2}-x_{0}+\Delta x-x_{0}+x_{0}=$$
$$=2x_{0} \Delta x+\Delta x^{2}-\Delta x$$
$$\frac{\Delta f}{\Delta x}=\frac{2 x_{0} \Delta x+\Delta x^{2}-\Delta x}{\Delta x}=2 x_{0}+\Delta x-1$$
$$\lim _{\Delta x \rightarrow 0} \frac{\Delta x}{\Delta x}=\lim _{\Delta x \rightarrow 0}\left(2 x_{0}+\Delta x-1\right)=2 x_{0}-1$$

Задание 18.5

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Задание 18.5

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