what is the linear approximation of f(x,y) = e^(xy) around (1,1)

To find the linear approximation of the function \( f(x, y) = e^{xy} \) around the point \((1,1)\), we can use the formula for the linear approximation:

\[
L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)
\]

where \( (a, b) = (1,1) \), and \( f_x(a, b) \) and \( f_y(a, b) \) are the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \), evaluated at \( (a, b) \).

### Step 1: Evaluate \( f(1,1) \)
\[
f(1, 1) = e^{(1)(1)} = e
\]

### Step 2: Find the partial derivatives

- **Partial derivative with respect to \( x \):**

\[
f_x(x, y) = \frac{\partial}{\partial x} \left(e^{xy}\right) = y e^{xy}
\]

\[
f_x(1, 1) = 1 \cdot e^{(1)(1)} = e
\]

- **Partial derivative with respect to \( y \):**

\[
f_y(x, y) = \frac{\partial}{\partial y} \left(e^{xy}\right) = x e^{xy}
\]

\[
f_y(1, 1) = 1 \cdot e^{(1)(1)} = e
\]

### Step 3: Write the linear approximation

\[
L(x, y) = e + e(x - 1) + e(y - 1)
\]

This simplifies to:

\[
L(x, y) = e \left(1 + (x - 1) + (y - 1)\right)
\]

\[
L(x, y) = e \left(x + y - 1\right)
\]

So, the linear approximation of \( f(x, y) = e^{xy} \) around the point \( (1,1) \) is:

\[
L(x, y) = e(x + y - 1)
\]

Would you like further details or have any questions? Here are five related questions you might find interesting:

1. How does the linear approximation relate to the tangent plane at a given point?
2. What is the error term in a linear approximation, and how can it be estimated?
3. Can you perform a linear approximation for a function of three variables?
4. How does linear approximation differ from a Taylor series expansion?
5. What are some practical applications of linear approximations in real-world problems?

**Tip:** Linear approximations are most accurate when the point of interest is close to the point of approximation, as the approximation error increases with distance.

Learn how to find the linear approximation of the function f(x, y) = e^(xy) around the point (1,1) using partial derivatives and the linear approximation formula. Explore step-by-step solutions and related concepts in calculus.

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