Theory and Examples.Extreme Values on Parametrized Curves
Chapter 14, Problem 61E(choose chapter or problem)
Problem 61E
Theory and Examples.
Extreme Values on Parametrized Curves To find the extreme values of a function ƒ(x, y) on a curve x = x(t), y = y(t), we treat ƒ as a function of the single variable t and use the Chain Rule to find where df/dt is zero. As in any other single-variable case, the extreme values of ƒ are then found among the values at the
a. critical points (points where dƒ/ dt 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.
Functions:
Curves:
i) The semicircle
ii) The quarter circle
Use the parametric equations x = 2 cos t, y = 2 sin t .
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