91Ó°ÊÓ

Parameterization is a method used to represent a curve by expressing the coordinates of the points on the curve as functions of a parameter, usually denoted by \( t \). In this exercise, we parameterize the ellipse using trigonometric functions. The boundary of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is represented by:

• \( x = a \cos t \)
• \( y = b \sin t \)

where \( 0 \leq t \leq 2\pi \). This allows us to explore the properties of the ellipse by using the parameter \( t \), transforming the problem into studying a single variable function of \( t \). This simplification is particularly useful when dealing with more complex shapes or boundaries.

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the occurring variables. A key identity used here is \( \cos^2 t + \sin^2 t = 1 \). This identity helps simplify expressions resulting from the parameterization process.

When substituting the parameterization into the function \( f(x, y) = x^2 + y^2 \), we use this identity to express the function solely in terms of \( \sin^2 t \):

• Original substitution results in \( a^2 \cos^2 t + b^2 \sin^2 t \).
• The identity simplifies this to \( a^2 - (a^2 - b^2) \sin^2 t \).

This simplification significantly eases the calculating of extremum values upon evaluation.

An ellipse is a set of points in a plane such that the sum of the distances from two fixed points (foci) is constant. It is represented by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively.

In the context of this problem, \( a > b \), meaning that the ellipse stretches more along the \( x \)-axis than the \( y \)-axis. Understanding the geometric structure of ellipses helps in simplifying and solving problems, especially when parameterizing and calculating points of interest like maximum or minimum values. The relationship between \( a \) and \( b \) affects the bounds within which the function values are calculated.

Extremum values refer to the maximum and minimum values of a function within a certain interval or domain. To find these values for the function \( f(x, y) = x^2 + y^2 \) on the ellipse, we analyze the simplified expression \( f(x, y) = a^2 - (a^2 - b^2) \sin^2 t \).

• The maximum of this function occurs when \( \sin^2 t = 1 \), resulting in \( f(x, y) = a^2 - (a^2 - b^2) = b^2 \).
• The minimum happens when \( \sin^2 t = 0 \), which simplifies to \( f(x, y) = a^2 \).

By evaluating \( \sin^2 t \) across its possible range \( [0, 1] \), we can clearly see how parameterization and trigonometric identities assist in identifying these critical extremum values on the ellipse's boundary.