91Ó°ÊÓ

In multivariable calculus, a **tangent hyperplane** is akin to the tangent line we are familiar with from single-variable calculus. However, as we move to multiple dimensions, the concept extends into higher dimensions too.

A tangent hyperplane is simply a flat surface that just "touches" a given surface without intersecting it, at least on an infinitesimally small scale. It provides the best linear approximation of the surface at a specific point.

The tangent hyperplane at a point \(\mathbf{p} = (a, b, c)\) on a surface defined by a function \(f(x, y, z)\) can be expressed as:

• \(w = f(a, b, c) + f_x(a, b, c)(x-a) + f_y(a, b, c)(y-b) + f_z(a, b, c)(z-c)\)

This equation uses partial derivatives to capture how the surface behaves as each input variable changes, which is essential for creating the hyperplane.

**Partial derivatives** are the building blocks for understanding how multivariable functions change. When we compute a partial derivative of a function like \(f(x, y, z)\), we find how the function changes concerning one variable while keeping the other variables constant.

In our example:

• \(f_x(x, y, z) = 6x + z^2\) assesses changes along the \(x\)-axis.
• \(f_y(x, y, z) = -4y\) focuses on changes along the \(y\)-axis.
• \(f_z(x, y, z) = 2xz\) relates to changes along the \(z\)-axis.

When evaluating these at a specific point, such as \((1, 2, -1)\), they reveal how the surface bends or slopes in multiple directions at that pinpoint location, which is crucial for defining the tangent hyperplane.

The **surface equation** in multivariable calculus describes a 3D object using a function of several variables. In this case, \(f(x, y, z) = 3x^2 - 2y^2 + xz^2\) represents a surface in a 3D space.

Understanding how this surface looks at various points requires using principles like partial derivatives. At the point \(\mathbf{p} = (1, 2, -1)\), we evaluate not only its function value but also its first-order partial derivatives.

The calculation of these values provides information essential for higher concept evaluations, such as forming a tangent hyperplane that approximates our surface locally at point \(\mathbf{p}\). This gives insight into the behavior and shape of the surface.

**Function evaluation** involves computing the value of a function at a specific point to understand the characteristics of a surface at that location. For the function \((f(x, y, z) = 3x^2 - 2y^2 + xz^2)\), computing it at \(\mathbf{p} = (1, 2, -1)\) tells us the height or depth of the surface at that exact point.

From our calculations, \(f(1, 2, -1) = 3 - 8 + 1 = -4\). This result is critical because it determines where the tangent hyperplane will be anchored in a vertical sense.

By combining this height with slopes determined by the partial derivatives, we form a tangent hyperplane equation to closely approximate the behavior of the surface around that point.