Linearization of a Function of Three Variables at a Point

Find the linearization $L(x, y, z)$ of a function of three variables at the point (2, 1, 0).

The linearization of a multivariable function is a technique used to approximate the function near a given point using the first-order Taylor expansion. This process involves replacing the original nonlinear function with a linear function that has the same value and gradient as the nonlinear function at the given point. This is particularly useful when functions are too complex to analyze or compute directly in their entirety, making it easier to study their behavior around specific points.

The linearization will involve computing the partial derivatives of the function with respect to each of its variables evaluated at the given point. These represent the slopes of the tangent plane to the surface defined by the function at that point. You then form the linear equation by taking these partial derivatives into account to define how the function changes as you move away from the point in each coordinate direction. This linear approximation serves as a simplified model that reflects the original function closely around the vicinity of the point of tangency, can be particularly useful in various applied fields such as physics and engineering where calculations need to be simplified for complex models.

Linearizations also form the foundation for many advanced techniques in calculus and numerical methods used to find approximate solutions to systems of equations. Understanding how to perform linearization at points gives foundational insights into how multivariable calculus extends the concepts of tangent lines and linear approximation from one to multiple dimensions.

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