91Ó°ÊÓ

When dealing with surfaces in calculus, partial derivatives are like a tool that helps us understand how the surface changes in different directions.
Given a function of two variables, such as

• a surface defined by \( z = f(x, y) \),
• the partial derivative with respect to \( x \) is noted as \( f_x \), and shows us how \( z \) changes if we change \( x \) while keeping \( y \) constant.
• Similarly, the partial derivative with respect to \( y \), noted as \( f_y \), tells us how \( z \) changes if \( y \) is altered and \( x \) is held constant.

For the surface \( z = y \cos(x-y) \):

• The partial derivative \( f_x \) is \(-y\sin(x-y)\), capturing the rate of change of \( z \) with \( x \).
• \( f_y \) is \( \cos(x-y) - y\sin(x-y) \), showing how \( z \) changes with \( y \).

Partial derivatives are critical for forming tangent planes because they reveal the slope of the surface in different directions.

The tangent plane is a flat surface that just 'touches' a curved surface at a specific point. This concept is crucial because it acts as a simple approximation of the surface near that point.
To find the equation of a tangent plane, especially for a surface defined as \( z = f(x, y) \), the general formula is used:

• \( (z - z_0) = f_x(x_0, y_0) (x - x_0) + f_y(x_0, y_0) (y - y_0) \)

Here, \( (x_0, y_0, z_0) \) represents the point of tangency. The partial derivatives \( f_x(x_0, y_0) \) and \( f_y(x_0, y_0) \) determine the slopes in the \( x \) and \( y \) directions, respectively.
In our specific problem, substituting the point \( (2, 2, 2) \) and the evaluated derivatives, we get the simplified tangent plane equation:

• \( z = y \)

This equation suggests that, at the point \( (2, 2, 2) \), the surface is perfectly horizontal in the context of \( x \).

In mathematics, a surface is a two-dimensional shape or figure within a three-dimensional space. A surface is pivotal in understanding how different variables interact and affect each other.
Specifically, the surface given by \( z = y \cos(x-y) \) is formed in such a way that each point on the surface is determined by corresponding \( x \) and \( y \) values.
By comprehending this relationship:

• we can visualize the surface as a 2D canvas bent or curved across a 3D space.
• At every point, the surface will have a tangent plane that provides a flat approximation of it.

The smooth nature of these surfaces allows for the application of differential calculus to analyze and approximate how surfaces behave under varying circumstances.

Calculus is a broad field of mathematics involving change and motion. Its core encompasses derivatives, integrals, limits, and infinite series.
In the context of tangent planes, calculus allows us to:

• compute partial derivatives, which are crucial in understanding how surfaces change locally.
• Use them to produce equations, like tangent planes, that simplify complex surfaces.

Tangent planes fall under the umbrella of differential calculus, which helps us deal with rates of change and slopes. Through the systematic steps of calculus, we can break down complicated surfaces into understandable local behaviors, making calculus an invaluable tool in mathematical analysis.