Example 1 (to $1^{\circ}$). Find the partial and total increments of the function $z=x^{2} y$ at the initial values $x=1, y=2$, if $\Delta x=0.1$; $\Delta y=-0.2$.
Solution. We have:

$$
\begin{aligned}
& \Delta_{x} z=(x+\Delta x)^{2} \cdot y-x^{2} y=(1+0.1)^{2} \cdot 2-1^{2} \cdot 2=0.42 \\
& \Delta_{y} z=x^{2}(y+\Delta y)-x^{2} y=1^{2} \cdot(2-0.2)-1^{2} \cdot 2=-0.2 \\
& \Delta z=(x+\Delta x)^{2} \cdot(y+\Delta y)-x^{2} y=(1+0.1)^{2} \cdot(2-0.2)-1^{2} \cdot 2=0.178 \\
& \Delta z \neq \Delta_{x} z+\Delta_{y} z ; 0.178 \neq 0.42-0.2=0.22
\end{aligned}
$$