91Ó°ÊÓ

A parametric surface is a type of surface in three-dimensional space where the coordinates

• x
• y
• z

are expressed as functions of two parameters, usually denoted as \(u\) and \(v\). For instance, in the given problem, these equations are specified as \(x = u^2\), \(y = v^2\), and \(z = u + v\).
Parametric equations are beneficial when modeling complex surfaces because they let us use two simple parameters to navigate any point on the surface.Reducing complex shapes into simple, navigable forms can simplify the process of finding related attributes such as tangent planes.
The primary task is to find the values of the parameters \(u\) and \(v\) such that the given parametric equations coincide with known points on the surface. In this exercise, by substituting values and solving these relations, parameters \(u = 1\) and \(v = 2\) were determined.

Partial derivatives are a fundamental concept in calculus, especially in dealing with functions of multiple variables. They measure the change in a function concerning one variable while keeping other variables constant.
In this exercise, we need the partial derivatives of our parametric equations with respect to the parameters \(u\) and \(v\).

• The partial derivatives of the coordinates with respect to \(u\) are:
  • \( \frac{\partial x}{\partial u} = 2u \)
  • \( \frac{\partial y}{\partial u} = 0 \)
  • \( \frac{\partial z}{\partial u} = 1 \)
• Those with respect to \(v\) are:
  • \( \frac{\partial x}{\partial v} = 0 \)
  • \( \frac{\partial y}{\partial v} = 2v \)
  • \( \frac{\partial z}{\partial v} = 1 \)

These calculations provide us with the rate of change of each coordinate of the surface with respect to one parameter at a time.

The equation of a tangent plane to a surface at a given point provides a flat approximation of the surface near that point. For a parametric surface, this requires the calculated partial derivatives and the known point on the surface.
In this problem, given the partial derivatives and point \((1, 4, 3)\), we can write a formula for the tangent plane using:
\[(x - x_0) \frac{\partial x}{\partial u} + (y - y_0) \frac{\partial y}{\partial u} + (z - z_0) \frac{\partial z}{\partial u} + (x - x_0) \frac{\partial x}{\partial v} + (y - y_0) \frac{\partial y}{\partial v} + (z - z_0) \frac{\partial z}{\partial v} = 0\]
This formula accounts for the rate of change in \(x\), \(y\), and \(z\) concerning changes in \(u\) and \(v\).
Substituting with the given values, simplifying, and rearranging, we derive the equation for the tangent plane:\[2x + 4y + 2z = 25\]
which gives us a precise linear equation representing the plane tangent to our parametric surface at the specified point.