We are asked to find the Taylor series expansion of $f(x, y) = xy + 3y - 2$ in powers of $(x-1)$ and $(y+2)$, which means we will expand the function around the point $(x_0, y_0) = (1, -2)$.

Step 1: Function at the point $(x_0, y_0) = (1, -2)$

The value of the function at $(x_0, y_0)$: $f(1, -2) = (1)(-2) + 3(-2) - 2 = -2 - 6 - 2 = -10$

Step 2: First-order partial derivatives

Now, we find the first-order partial derivatives of $f(x, y)$.

$f_x(x, y) = \frac{\partial}{\partial x}(xy + 3y - 2) = y$ $f_y(x, y) = \frac{\partial}{\partial y}(xy + 3y - 2) = x + 3$

Evaluating these at $(x_0, y_0) = (1, -2)$:

$f_x(1, -2) = -2$ $f_y(1, -2) = 1 + 3 = 4$

Step 3: Second-order partial derivatives

Next, we find the second-order partial derivatives.

$f_{xx}(x, y) = \frac{\partial}{\partial x}(f_x(x, y)) = \frac{\partial}{\partial x}(y) = 0$ $f_{yy}(x, y) = \frac{\partial}{\partial y}(f_y(x, y)) = \frac{\partial}{\partial y}(x + 3) = 0$ $f_{xy}(x, y) = \frac{\partial}{\partial y}(f_x(x, y)) = \frac{\partial}{\partial y}(y) = 1$

Evaluating these at $(x_0, y_0) = (1, -2)$: $f_{xx}(1, -2) = 0, \quad f_{yy}(1, -2) = 0, \quad f_{xy}(1, -2) = 1$

Step 4: Taylor series expansion

The general form of the Taylor series expansion for $f(x, y)$ around $(x_0, y_0) = (1, -2)$ is:

$f(x, y) \approx f(1, -2) + f_x(1, -2)(x - 1) + f_y(1, -2)(y + 2) + \frac{1}{2}f_{xx}(1, -2)(x - 1)^2 + \frac{1}{2}f_{yy}(1, -2)(y + 2)^2 + f_{xy}(1, -2)(x - 1)(y + 2)$

Substitute the values we computed:

$f(x, y) \approx -10 - 2(x - 1) + 4(y + 2) + 0 + 0 + (x - 1)(y + 2)$

Simplifying the expression:

$f(x, y) \approx -10 - 2(x - 1) + 4(y + 2) + (x - 1)(y + 2)$

This is the Taylor series expansion of the given function around $(x_0, y_0) = (1, -2)$.

Would you like any further clarification or details on the steps?

Here are 5 related questions:

1. What is the general form of a Taylor series for functions of two variables?
2. How do we find higher-order derivatives for multi-variable Taylor expansions?
3. What is the role of mixed partial derivatives in Taylor series expansions?
4. How does the point of expansion affect the resulting Taylor series?
5. Can you explain the physical or geometric meaning of a Taylor series in two variables?

Tip: Always check if the function simplifies at the expansion point, as it can reduce computational complexity in the Taylor series.