91Ó°ÊÓ

In multivariable calculus, the gradient vector is a mathematical tool that tells us the direction and rate of the steepest ascent of a scalar field. Imagine you're on a hill; the gradient points you directly uphill. This is especially helpful when determining tangent planes of surfaces.To find the gradient vector for a function like our surface, we need to compute the partial derivatives with respect to each variable. For the function \( F(x, y, z) = z - 4x^2 - y^2 \), the gradient vector is composed of:

• The partial derivative with respect to \( x \), \( F_x = -8x \)
• The partial derivative with respect to \( y \), \( F_y = -2y \)
• The partial derivative with respect to \( z \), \( F_z = 1 \)

At the point \((1, 1, 5)\), substituting in these values gives the gradient \( abla F = (-8, -2, 1) \). This gradient vector serves as a normal vector to the tangent plane, pointing perpendicularly away from the surface at our point of interest.

The point-normal form is a practical approach to defining a plane. This is because any plane can be described by a point located on the plane and a vector which is perpendicular to it, known as a normal vector. In the context of our exercise, the gradient vector links beautifully to this form because it directly gives us the normal vector. For a plane tangent to a surface at a point \((x_0, y_0, z_0)\), we use the formula:\[ n_x(x - x_0) + n_y(y - y_0) + n_z(z - z_0) = 0 \]Where \( (n_x, n_y, n_z) \) is the normal vector. Here, we substitute \( (n_x, n_y, n_z) = (-8, -2, 1) \) and \((x_0, y_0, z_0) = (1, 1, 5)\). This results in the equation of the plane:\[ -8(x - 1) - 2(y - 1) + 1(z - 5) = 0 \]The point-normal form helps us derive this very relevant equation of our tangent plane in a very structured manner.

Partial derivatives are a fundamental concept in calculus, particularly when dealing with functions of multiple variables. They measure how a function changes as one of its input variables changes, keeping all other inputs constant.For the function \( F(x, y, z) = z - 4x^2 - y^2 \), calculating partial derivatives involves treating each variable as the sole variable of interest at different times:

• \( F_x = \frac{\partial F}{\partial x} = -8x \), which shows how \( F \) changes as \( x \) changes.
• \( F_y = \frac{\partial F}{\partial y} = -2y \), illustrating \( F \)'s dependence on \( y \).
• \( F_z = \frac{\partial F}{\partial z} = 1 \), indicating how \( F \) shifts with \( z \).

These derivatives collectively contribute to forming our gradient vector \( abla F = (-8, -2, 1) \), which as we've seen, plays a key role in defining the tangent plane. Understanding partial derivatives allows us to analyze and navigate multi-variable functions effectively, such as finding where the tangent plane touches the surface.