Equation of Tangent Plane to a Surface23

Find the equation of the tangent plane to the function $f(x, y) = \sqrt{36 - x^2 - y^2}$ at the point $(2, 4, 4)$.

When dealing with functions of two variables, visualizing the behavior of the function surface in three-dimensional space can be quite illuminating. In this problem, you are asked to find the equation of the tangent plane to a specific surface defined by the function $f(x, y) = \sqrt{36 - x^2 - y^2}$ at the given point $(2, 4, 4)$. The concept of a tangent plane in multivariable calculus extends the idea of a tangent line from single-variable calculus. The tangent plane, at a given point on a surface, locally approximates that surface just as a tangent line does to a curve at a point.

To find the equation of a tangent plane to a surface defined by a function of two variables, $f(x, y)$, you typically use the point-slope form of a plane and involve partial derivatives of the function. The partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at a point give the slopes of the tangent plane in the x and y directions, respectively. These slopes form the components of the gradient vector, which directly informs the equation of the tangent plane. In essence, the gradient represents the slope of the surface at a point, providing the best linear approximation of the surface around that point.

Conceptually, this problem involves understanding not just the mechanics of taking derivatives, but also interpreting the geometric and spatial implications of these derivatives on the surface. A fundamental part of this analysis is confirming whether the point lies on the surface, and ensuring continuity and differentiability around the point of tangency to validate the existence of a unique tangent plane. Such a problem further reinforces the utility of linear approximations in analyzing and understanding complex surfaces in multivariable calculus.

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