A change of $0.9$ in $x$ and $-0.1$ in $y$ creates what change in $z$

Question

Suppose that $z$ is a linear function of $x$ and $y$ with slope $2$ in the $x$ direction and slope $3$ in the $y$ direction.

(a) A change of $0.9$ in $x$ and $-0.1$ in $y$ produces what change in $z$?

I've been at this homework question for a while now, my textbook doesn't mention anything about this kind of question.

Answer 1

Guide:

Suppose $z = 2x+3y$

$$z+\Delta z = 2(x+\Delta x) + 3(y + \Delta y)$$

Answer 2

Given $z=2x+3y$, you can use the total differential:

$$dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy \rightarrow dz = 2\cdot dx+3\cdot dy$$

Thus:

$$dz = 2\cdot 0.9 + 3\cdot -0.1$$

Answer 3

A change of $0.9$ in $x$ and $-0.1$ in $y$ produces $2(0.9)+3(-0.1)$ which is $1.5$ total change.

Remember, $$\Delta Z= (dx/dt)\Delta x +(dy/dt)\Delta y$$