Theory and Examples.Extreme Values on Parametrized Curves

Chapter 14, Problem 61E

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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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