91Ó°ÊÓ

Partial derivatives are crucial when dealing with multivariable functions. They represent the rate at which a function changes as one variable changes, while all other variables remain constant. In the context of local linearization, partial derivatives help us understand how a function behaves in the vicinity of a point by providing the slopes of the tangent plane to the function's graph at that point.

Specifically, for a function like
\( f(x, y) = e^{-x} \cos(y) \),
the partial derivative with respect to
\( x \), denoted as
\( f_{x}(x, y) \),
is
\( -e^{-x} \cos(y) \), and
\( f_{y}(x, y) \), the partial derivative with respect to
\( y \),
is
\( -e^{-x} \sin(y) \).
These derivatives tell us how
\( f \) changes in the direction of each axis near a given point, such as (1, 1.5). This information is essential for creating a linear approximation of the function—the heart of local linearization.

Multivariable functions like
\( f(x, y) = e^{-x} \cos(y) \)
introduce more complexity than single-variable functions because they change along multiple axes at once. When we analyze how these functions change near a certain point, we can't simply use a tangent line as we would in single-variable calculus. Instead, we use a tangent plane or linear approximation to capture the behavior of the function in all directions from that point.

To find this linear approximation, we use the function's value and its partial derivatives at the point of interest. This is what forms the local linearization, which can be expressed using the first-order Taylor series expansion for a multivariable function.

The purpose of filling in tables with function values at various points around (1, 1.5) is to visualize how well the linear approximation represents the function in a neighborhood of that point, with the goal that the student understands how a flat plane can serve as a reasonable proxy for the surface created by the function.

In our exercise, the function

\( f(x, y) = e^{-x} \cos(y) \) includes both an exponential component
\( e^{-x} \) and a trigonometric component
\( \cos(y) \).

Exponential functions are known for their rapid growth or decay, and in this case,
\( e^{-x} \) represents a decaying pattern as
\( x \) increases. Trigonometric functions like cosine and sine oscillate between -1 and 1, and their behavior depends on the angle or, in our case, the variable
\( y \).

The combination of these functions can create complex surfaces that are not straightforward to visualize. Local linearization simplifies the visualization by approximating the function locally with a plane, reducing the complexity to understanding a flat surface rather than the hills and valleys of the original function’s graph.