Find the linearizations L(x, y, z) of the functions at the...

Find the linearizations $L(x, y, z)$ of the functions at the given points.
$$
f(x, y, z)=x^{2}+y^{2}+z^{2} \quad \text { at }
$$
$$
\text { a. } (1,1,1) \quad \text { b. }(0,1,0) \quad \text { c. }(1,0,0)
$$

Key Concepts

Partial Derivatives

Partial derivatives involve differentiating a multivariable function with respect to one variable while holding the other variables constant. They provide essential information about the rate of change of the function along each coordinate axis, which is crucial when analyzing and approximating functions in multiple dimensions.

Tangent Plane

The tangent plane is the plane that best approximates a multivariable function at a given point. It is derived from the linearization of the function and represents the first-order approximation of the function's surface near that point. Understanding the tangent plane is essential for visualizing and analyzing the local behavior of functions of several variables.

Multivariable Functions

Multivariable functions are functions that have more than one input variable, such as f(x, y, z). They describe relationships where the output depends on several variables, and understanding their behavior often requires examining how changes in each variable affect the overall function. This concept is fundamental in fields like calculus and optimization.

Linearization

Linearization is the process of approximating a function near a given point by a linear function, typically using the first-order Taylor polynomial. For a function of several variables, this approximation uses the function’s value and its partial derivatives evaluated at the point, yielding an equation that approximates the function’s behavior closely in the vicinity of that point.

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