Equation of Tangent Plane to a Surface

Find the equation of the tangent plane to the graph of the function f(x, y) = 2 - x^2 - y^2 at the point $\left(\frac{1}{2}, \frac{1}{2}\right)$.

The problem of finding the equation of a tangent plane to a surface involves understanding the geometry of surfaces in three-dimensional space. The tangent plane at a given point on a surface can be considered as the plane that just touches the surface at that point, without intersecting it. One useful way to approach this problem is to first compute the partial derivatives of the function defining the surface, which provide the gradients in the x and y directions. These gradients are crucial, as they form the components of the normal vector to the surface at the given point.

The normal vector is essential in defining the plane, as a plane in three-dimensional space can be described by a point on the plane and a perpendicular vector (the normal vector). By using these partial derivatives to construct the normal vector, you can then apply the point-normal form of the equation of a plane. This connection between calculus and geometry is key for understanding how surfaces can be approximated near a specific point.

Finding tangent planes is an early topic in multivariable calculus which builds towards more complex topics like linearization, optimization, and integration over surfaces. Understanding how to compute and interpret tangent planes allows students to analyze and visualize more complex geometrical structures and their behaviors near certain points, which is a foundation of higher-dimensional calculus studies.

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