91Ó°ÊÓ

Understanding the gradient is essential in finding the equation of a tangent plane. The gradient, denoted by \( abla f \), is a vector that contains all the partial derivatives of a function. It provides essential information about the direction and rate of steepest ascent of the function. When dealing with surfaces, the gradient vector is perpendicular to the surface at a given point, turning it into a powerful tool for tangent plane calculations.

To compute the gradient of a function such as \( f(x, y, z) = x^{2} + y^{2} - z^{2} \), consider the partial derivatives with respect to each variable:

• Partial derivative with respect to \( x \): \( \frac{\partial f}{\partial x} = 2x \)
• Partial derivative with respect to \( y \): \( \frac{\partial f}{\partial y} = 2y \)
• Partial derivative with respect to \( z \): \( \frac{\partial f}{\partial z} = -2z \)

The gradient vector is then \( abla f = (2x, 2y, -2z) \), indicating how the function changes in each spatial direction. This vector will help us define the tangent plane at specific points.

Partial derivatives are key to understanding changes in multivariable functions by analyzing how a particular variable affects the outcome while keeping others constant. In the context of tangent planes, partial derivatives form the components of the gradient vector, which further help in constructing the equation of the tangent plane.

For a function like \( f(x, y, z) = x^{2} + y^{2} - z^{2} \), the partial derivatives are calculated as:

• With respect to \( x \), represented as \( \frac{\partial f}{\partial x} \), resulting in \( 2x \).
• With respect to \( y \), represented as \( \frac{\partial f}{\partial y} \), yielding \( 2y \).
• With respect to \( z \), represented as \( \frac{\partial f}{\partial z} \), with value \(-2z \).

These derivatives show the sensitivity of the function to changes in each variable independently, providing a snapshot of the surface's behavior near any given point. Through these derivatives, we can better visualize and compute the tangent planes on surfaces defined by multivariable functions.

The equation of a plane is crucial when discussing tangent planes to surfaces in three-dimensional space. In this context, the equation is typically formed using the gradient, which acts as the normal vector. The general formula for the equation of a plane is \( ax + by + cz = d \), where \( (a, b, c) \) is the normal vector, and \( d \) is a constant.

When finding the tangent plane at a point, the gradient vector at that point provides the necessary coefficients for the plane's equation. For our given surface \( x^{2} + y^{2} - z^{2} = 0 \) at points \( (3, 4, 5) \) and \( (-4, -3, 5) \), we computed:

• \( abla f(3, 4, 5) = (6, 8, -10) \)
• \( abla f(-4, -3, 5) = (-8, -6, -10) \)

These vectors are used to write the tangent plane equations by ensuring the normal vector dot product with any point on the plane provides a consistent value corresponding to the surface shift. Therefore, the equations are:
- \( 6x + 8y - 10z = 38 \) for the plane through \( (3, 4, 5) \)
- \( -8x - 6y - 10z = -86 \) for the plane through \( (-4, -3, 5) \).
These equations represent the planes tangent to the surface at the specified points, characterized by their normal vectors obtained from the gradient.