91Ó°ÊÓ

Partial derivatives are like regular derivatives, but they're applied to functions of multiple variables.
Instead of measuring how a single variable changes, partial derivatives tell us how a function changes when one particular variable in a multi-variable function varies, while all other variables are held constant.
To compute the partial derivatives, we focus on one variable at a time. For example, with a function like \( z = \frac{1}{2}x - y \), it's understood as a function of both \(x\) and \(y\).

• For \( \frac{\partial z}{\partial x} \), we treat \(y\) as a constant and differentiate with respect to \(x\), giving us the expression \( \frac{1}{2} \).
• For \( \frac{\partial z}{\partial y} \), \(x\) is the constant, and differentiation with respect to \(y\) results in \(-1\).

These derivatives provide us with important detail: they allow us to determine the slope of the tangent line to the surface in the directions of the \(x\) and \(y\) axes.

A normal vector is a vector that is perpendicular to a surface.
For a given surface, the normal vector is a crucial part in defining the orientation of that surface in space.
In the context of tangent planes, the normal vector is derived from the gradient of the function that defines the surface. The gradient is a vector that points in the direction of the greatest rate of increase of the function.
Given a function \(z = \frac{1}{2}x - y\), the normal vector \(n = (A, B, C)\) is found using:

• \(A = \frac{\partial z}{\partial x} = \frac{1}{2}\)
• \(B = \frac{\partial z}{\partial y} = -1\)
• \(C = 1\)

This gives us the normal vector \((\frac{1}{2}, -1, 1)\).
This vector tells us not just how the surface is oriented at a specific point but, when used in conjunction with a point on the surface, it helps to construct the equation of the tangent plane.

To understand the equation of the surface, visualize how a geometric figure can be described using mathematical expressions.
Usually, a surface can be represented by an equation involving \(x\), \(y\), and \(z\) coordinates.
Rewriting the original question equation \((x+z)/(y-z)=2\) as \(z = \frac{1}{2}x - y\) helps us frame this surface in terms of a more direct relationship between these variables.
This simplified form is crucial in the process of finding the tangent plane equations which we use to get a linear approximation of the surface near a point.The equation of the tangent plane at a specific point, here \((4, 2, 0)\), uses the point itself and the normal vector to assert a planarity over a local region of the surface.
By incorporating the normal vector components into the plane equation \((\frac{1}{2}(x - 4) - (y - 2) + z = 0)\), we can simplify to get \(\frac{1}{2}x + y + z = 4\).
This simplified equation represents the flat plane that is just tangent to the surface precisely at our chosen point.