To find the extreme values of a function f(x, y) on a curve...

To find the extreme values of a function $f(x, y)$ on a curve $x=x(t), y=y(t),$ we treat $f$ as a function of the single variable $t$ and use the Chain Rule to find where $d f / d t$ is zero. As in any other single-variable case, the extreme values of $f$ are then found among the values at the a. critical points (points where $d f / d t$ is zero or fails to exist), and b. endpoints of the parameter domain. Find the absolute maximum and minimum values of the following functions on the given curves. Curves: i) The semicircle $x^{2}+y^{2}=4, \quad y \geq 0$ ii) The quarter circle $x^{2}+y^{2}=4, \quad x \geq 0, \quad y \geq 0$ Use the parametric equations $x=2 \cos t, y=2 \sin t$.

To find the extreme values of a function $f(x, y)$ on a curve $x=x(t), y=y(t),$ we treat $f$ as a function of the single variable $t$ and use the Chain Rule to find where $d f / d t$ is zero. As in any other single-variable case, the extreme values of $f$ are then found among the values at the
a. critical points (points where $d f / d t$ is zero or fails to exist), and
b. endpoints of the parameter domain.
Find the absolute maximum and minimum values of the following functions on the given curves.
Curves:
i) The semicircle $x^{2}+y^{2}=4, \quad y \geq 0$
ii) The quarter circle $x^{2}+y^{2}=4, \quad x \geq 0, \quad y \geq 0$
Use the parametric equations $x=2 \cos t, y=2 \sin t$.

Key Concepts

Parametric Equations

Parametric equations represent a curve using a parameter, typically denoted by t, where both x and y are expressed as functions of t. This approach simplifies the analysis of curves in the plane by reducing problems in two variables to problems in one variable, making it easier to apply single-variable calculus techniques.

Chain Rule

The chain rule is a fundamental calculus tool used to differentiate composite functions. When a function is expressed in terms of a parameter, the chain rule allows one to compute the derivative of a function of the curve (e.g., f(x(t), y(t))) with respect to t, by linking the derivatives of the outer function and the inner functions.

Critical Points

Critical points are values of the parameter where the derivative of the function under consideration is zero or undefined. In optimization problems, these points are essential because they are potential candidates for local minima or maxima, and identifying them is a key step in the process of finding extreme values.

Endpoints Consideration

In optimization over a closed interval, such as a segment of a curve defined by parameter t, the endpoints of the interval must be examined in addition to the critical points. This is because the extreme values of a function may occur at the boundaries of the domain, not just where the derivative is zero.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous over a closed and bounded interval, then it must attain both a maximum and a minimum value on that interval. This theorem underpins the strategy of evaluating both critical points and endpoints when searching for absolute extrema in optimization problems.

Transcript

Please give Ace some feedback