Understanding the Tangent Plane to a Surface Without Calculus

This article explores the process of determining the tangent plane to a given surface without relying on advanced calculus techniques. We will walk through the steps to find the tangent plane to the surface f(x, y, z) 0 at a specified point. Let's consider the unique problem: finding the tangent plane to the surface (xz - yz^3 - yz^2 2) at the point (2, -1, 1).

Given Surface and Point

Our surface is described by the equation:

f(x, y, z)  yz^3 - yz^2 - xz - 2

We want to find the tangent plane to this surface at the point (2, -1, 1).

Transforming the Surface Equation

To determine the tangent plane, we first need to transform the equation around the given point. We do this by letting:

Substituting these into the original equation, we get:

f(x, y, z)  f(2, -1, 1)   f_x(2, -1, 1)X   f_y(2, -1, 1)Y   f_z(2, -1, 1)Z   higher order terms

Since the point (2, -1, 1) is on the surface, we know that (f(2, -1, 1) 0).

Expanding the Equation

Let's expand and simplify the equation, keeping only the linear terms in (X), (Y), and (Z):

f(x, y, z)  - Y Z^3 - Y Z^2 - X Z   2 Y Z^2   Y Z^3 - 2 Z^2 - X Z - 2

After simplifying, we get:

f(x, y, z)  -X - 3Z - 2Z^2 - XZ - YZ   Z^3 - 2YZ^2 - YZ^3

Since the constant term is zero (as (f(2, -1, 1) 0)), we are left with the linear terms:

f(x, y, z)  -X - 3Z

Deriving the Tangent Plane Equation

The best linear approximation to the surface at a point is obtained by setting the linear terms equal to zero:

0  -X - 3Z

Substituting back for (X), (Y), and (Z) in terms of (x), (y), and (z), we get:

0  -(x - 2) - 3(z - 1)

Simplifying this, we obtain:

x - 2 - 3z   3  0

Which simplifies to:

x - 3z  5

Final Answer

The equation of the tangent plane to the surface (xz - yz^3 - yz^2 2) at the point (2, -1, 1) is:

x - 3z 5

Conclusion

In this article, we have demonstrated a method to determine the tangent plane to a surface without using calculus. This approach involves transforming the equation around the given point and retaining only the linear terms.