Tangent Plane to a Circular Paraboloid at a Point

Find the equation of a tangent plane to the circular paraboloid $z = x^2 + y^2$ at the point (-1, -1, 2).

When we talk about finding the equation of a tangent plane to a surface at a specific point, we are essentially exploring the broader concept of tangential approximations in multivariable calculus. At its core, the tangent plane approximation is an extension of the tangent line concept from single-variable calculus to functions of two variables, where the ‘line’ tangentially touching a curve now becomes a ‘plane’ tangentially touching a surface. For surfaces like a circular paraboloid, which is represented by the equation $z = x^2 + y^2$, this involves not only differentiability at a point but also the ability to compute partial derivatives with respect to each variable. To determine the tangent plane, we must first find the partial derivatives of the function with respect to x and y at the given point. These partial derivatives form the components of the normal vector to the plane. This normal vector is critical because it provides a direction perpendicular to the surface at the tangent point. The equation of the tangent plane can then be derived by using these components along with the coordinates of the point provided. By understanding and applying these concepts, students develop an appreciation for how planes can locally approximate surfaces, which is a fundamental technique in linearization and differential calculus.

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