To solve the second question, we use the concept of linear approximation or differentials. Here's the step-by-step logic behind the question:

Problem Setup

The goal is to approximate $f(2.1, 2.9)$ using the tangent plane to the function $f(x, y)$ at the point $(2, 3)$.

1. Tangent Plane Equation: The tangent plane to $z = f(x, y)$ at a point $(x_0, y_0)$ is given by: $z \approx f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$ where:

   • $f_x(x_0, y_0)$: Partial derivative of $f$ with respect to $x$ at $(x_0, y_0)$,
   • $f_y(x_0, y_0)$: Partial derivative of $f$ with respect to $y$ at $(x_0, y_0)$.

2. Why Linear Approximation Works: When $(x, y)$ is close to $(x_0, y_0)$, the function $f(x, y)$ can be well-approximated by its tangent plane. This is because the higher-order terms of the Taylor expansion are negligible for small changes in $x$ and $y$.

3. Steps for Linear Approximation:

   • Compute $f(2, 3)$.
   • Find $f_x(x, y)$ and $f_y(x, y)$, then evaluate them at $(2, 3)$.
   • Substitute into the tangent plane formula: $f(2.1, 2.9) \approx f(2, 3) + f_x(2, 3)(2.1 - 2) + f_y(2, 3)(2.9 - 3)$

4. Approximating $f(2.1, 2.9)$: After finding all values from above steps, plug $x = 2.1$ and $y = 2.9$ into the tangent plane equation to get the approximation.

Summary of Logic

The linear approximation method uses the tangent plane of the function at a known point $(x_0, y_0)$ to estimate the value of the function at nearby points. This works effectively for small deviations, as higher-order terms become insignificant.

Would you like me to compute the actual approximation for this specific problem?

Additional Questions to Explore

1. How do you compute partial derivatives $f_x$ and $f_y$ for this specific function?
2. Why is the tangent plane the best linear approximation near a point?
3. What are the limitations of using the tangent plane for approximations?
4. How would the approximation differ if $(2.1, 2.9)$ were farther from $(2, 3)$?
5. How can the concept of total differentials simplify such approximations?

Tip:

Always verify the accuracy of the linear approximation by comparing it with the actual function value for small deviations.