Linear Approximation Using Tangent Plane at a Point

Find the linear approximation to this multivariable function at the point $(2, 3)$ using the tangent plane, and then use the linear approximation to estimate the value of the function at $(2.1, 2.99)$.

Linear approximation is a fundamental concept in multivariable calculus that allows us to approximate a function using the tangent plane at a specific point. The tangent plane, akin to the tangent line in single-variable calculus, provides the best linear approximation to the function at the given point. Essentially, it captures the behavior of the function in the immediate vicinity of the point, facilitating predictions and estimations with reasonable accuracy when small deviations occur near that point.

The effectiveness of linear approximation stems from the use of partial derivatives, which measure the function's rate of change along the axes of the input variables. At the point of interest, these partial derivatives define the slope of the tangent plane in each direction. By combining these derivatives, the linear approximation formula is derived, offering a simplified representation of the function around the point. This method proves highly useful in contexts where calculating the exact function value is computationally expensive or when a rough estimation suffices for decision-making.

In the context of this problem, the task is to find the linear approximation at the given point $(2, 3)$ and use it to estimate the function's value at a nearby point $(2.1, 2.99)$. This not only reinforces understanding of tangent planes and linear approximations but also illustrates the practical application of these concepts in analyzing how changes in input variables influence the function’s output. It emphasizes the importance of understanding the function’s local behavior and how this insight aids in grasping more complex, non-linear interrelationships in multivariable systems.

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