Question

Find the equation of the tangent plane to the given surface at the indicated point. \(x^{2}+y^{2}-z^{2}=4 ;(2,1,1)\)

Short answer

The equation of the tangent plane is \(4x + 2y - 2z - 8 = 0\).

Step-by-step solution

Identify the Surface Equation

The given surface is a quadratic surface represented by the equation \(x^{2} + y^{2} - z^{2} = 4\). We need to find the equation of the tangent plane to this surface at the point (2,1,1).

Use Gradient for Tangent Plane

To find the tangent plane, we use the gradient vector of the function \(f(x, y, z) = x^{2} + y^{2} - z^{2} - 4\). This vector is perpendicular to the surface at any point \((x, y, z)\).

Compute Partial Derivatives

Find the partial derivatives: \(f_x = 2x\), \(f_y = 2y\), \(f_z = -2z\). These partial derivatives will form the components of the gradient vector.

Evaluate the Gradient at the Given Point

Substitute the point (2,1,1) into the partial derivatives to find the gradient at this point. \(f_x(2,1,1) = 2(2) = 4\), \(f_y(2,1,1) = 2(1) = 2\), \(f_z(2,1,1) = -2(1) = -2\). Thus, the gradient vector at (2,1,1) is \(\langle 4, 2, -2 \rangle\).

Write the Tangent Plane Equation

The equation of the tangent plane to the surface at point \((2, 1, 1)\) is given by the formula: \( f_x(x_0,y_0,z_0)(x-x_0) + f_y(x_0,y_0,z_0)(y-y_0) + f_z(x_0,y_0,z_0)(z-z_0) = 0\). Substitute: \(4(x-2) + 2(y-1) - 2(z-1) = 0\).

Simplify the Equation

Simplify the equation: \(4x - 8 + 2y - 2 - 2z + 2 = 0\), leading to \(4x + 2y - 2z - 8 = 0\).

Key concepts

Gradient Vector

A gradient vector is a critical concept in calculus and multivariable functions. It provides the direction of the steepest ascent from a point in a multivariable function. The gradient vector is composed of partial derivatives, which express how a function changes as each variable changes individually. For the function \( f(x, y, z) \), the gradient vector \( abla f \) is represented as \( \langle f_x, f_y, f_z \rangle \), where \( f_x \), \( f_y \), and \( f_z \) are the partial derivatives with respect to \( x \), \( y \), and \( z \), respectively.
This vector is always perpendicular, or normal, to the surface defined by \( f \) at any given point. In the context of the tangent plane, this perpendicularity is useful because it allows us to define a plane that 'touches' the surface at a specific point, creating a linear approximation of the surface there.
In the exercise, the gradient vector was calculated at the point \((2,1,1)\) as \( \langle 4, 2, -2 \rangle \), meaning that the tangent plane at this point will be normal to this vector.

Partial Derivatives

Partial derivatives extend the concept of a derivative to functions of several variables. They measure the rate of change of a function with respect to one variable while keeping others constant. For a function \( f(x, y, z) \), its partial derivatives are denoted as \( f_x \), \( f_y \), and \( f_z \). These represent how \( f \) changes as \( x \), \( y \), or \( z \) change individually.
The calculation of a partial derivative is akin to treating all other variables as constants and differentiating with respect to the variable of interest. For instance, to find \( f_x \), one differs \( f \) with respect to \( x \) while treating \( y \) and \( z \) as constants.
In our original exercise, the partial derivatives were calculated for the given function \( f(x, y, z) = x^2 + y^2 - z^2 - 4 \) as follows: \( f_x = 2x \), \( f_y = 2y \), and \( f_z = -2z \). These derivatives form the gradient vector used to find the tangent plane.

Surface Equation

A surface equation represents a geometric surface in three-dimensional space. It is often given as \( f(x, y, z) = 0 \) and defines a set of points \((x, y, z)\) that satisfy the equation. In the exercise, the surface equation is \( x^2 + y^2 - z^2 = 4 \), identifying a specific type of quadratic surface.
Understanding the surface equation is crucial when determining the tangent plane at a given point on the surface. By finding partial derivatives of this equation, one can compute the gradient vector, which helps in establishing the tangent plane equation.
The surface equation tells us which points lie on this surface, while the tangent plane equation provides a linear approximation of the surface at a specific point. By touching the surface only at one point, it ensures the best local linear approximation.

Quadratic Surface

A quadratic surface is a second-degree algebraic surface in three-dimensional space. It is specified by a polynomial equation in which the highest degree is two. Common types of quadratic surfaces include spheres, ellipsoids, hyperboloids, and paraboloids.
In the original exercise, the surface \( x^2 + y^2 - z^2 = 4 \) is a quadratic surface known as a hyperboloid. This equality tells us that the surface is a hyperboloid centered at the origin with a translated location to account for its equation's constancy, in this case, the value 4.
Quadratic surfaces have distinctive properties that make them important in mathematics and engineering. Knowing these properties can help define tangent planes and more, providing insight into the geometric shape of intersections and other spatial elements. Understanding the nature of quadratic surfaces is essential for solving problems that involve three-dimensional geometry and surfaces.