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To find the linearization \( L(x, y) \) of the function \( f(x, y) = \frac{1 + y}{1 + x} \) at the point \( (1, 3) \) and use it to approximate \( f(0.9, 3.2) \), let’s follow these steps:

### Step 1: Calculate \( f(1, 3) \)
Evaluate \( f \) at the point \( (1, 3) \):
\[
f(1, 3) = \frac{1 + 3}{1 + 1} = \frac{4}{2} = 2
\]

### Step 2: Find Partial Derivatives \( f_x \) and \( f_y \)
1. **Partial derivative with respect to \( x \):**
   \[
   f_x(x, y) = \frac{\partial}{\partial x} \left( \frac{1 + y}{1 + x} \right) = -\frac{1 + y}{(1 + x)^2}
   \]
   Evaluating at \( (1, 3) \):
   \[
   f_x(1, 3) = -\frac{1 + 3}{(1 + 1)^2} = -\frac{4}{4} = -1
   \]

2. **Partial derivative with respect to \( y \):**
   \[
   f_y(x, y) = \frac{\partial}{\partial y} \left( \frac{1 + y}{1 + x} \right) = \frac{1}{1 + x}
   \]
   Evaluating at \( (1, 3) \):
   \[
   f_y(1, 3) = \frac{1}{1 + 1} = \frac{1}{2}
   \]

### Step 3: Form the Linearization \( L(x, y) \)
The linearization formula is:
\[
L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)
\]
where \( (a, b) = (1, 3) \).

Substitute the values:
\[
L(x, y) = 2 - 1(x - 1) + \frac{1}{2}(y - 3)
\]
Simplify:
\[
L(x, y) = 2 - (x - 1) + \frac{1}{2}(y - 3) = 3 - x + \frac{y}{2}
\]

### Step 4: Approximate \( f(0.9, 3.2) \)
Now, substitute \( x = 0.9 \) and \( y = 3.2 \) into the linearization:
\[
L(0.9, 3.2) = 3 - 0.9 + \frac{3.2}{2}
\]
Calculate each term:
\[
= 3 - 0.9 + 1.6 = 3.7
\]

### Final Answer
The approximate value of \( f(0.9, 3.2) \) is **3.7**.

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Would you like further details on any step, or have any questions?

Here are some related questions:
1. How do partial derivatives relate to linearization?
2. What is the geometric interpretation of a linearization?
3. How accurate is linearization for points close to \( (a, b) \)?
4. Can linearization be used for functions with three or more variables?
5. How does linearization relate to the concept of tangent planes?

**Tip:** When linearizing, always ensure that the chosen point \( (a, b) \) is within the domain of differentiability for accurate approximation.

Find the linearization L(x, y) of the function f(x, y) = (1 + y) / (1 + x) at the point (1, 3) and use it to approximate the value of f(0.9, 3.2).

This solution provides a step-by-step process to find the linearization of the function f(x, y) = (1 + y) / (1 + x) at the point (1, 3) and uses it to approximate the value of f(0.9, 3.2). Topics covered include partial derivatives and linear approximation, suitable for college-level calculus students.

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