Solution: Find the absolute maximum and minimum values of
Chapter 13, Problem 64E(choose chapter or problem)
Extreme Values on Parametrized Curves 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.
Functions:
a. \(f(x, y)=x^{2}+y^{2}\)
b. \(g(x, y)=1 /\left(x^{2}+y^{2}\right)\)
Curves:
i) The line \(x=t, y=2-2 t\)
ii) The line segment \(x=t, y=2-2 t, 0 \leq t \leq 1\)
Equation Transcription:
Text Transcription:
f(x,y)
x=x(t),y=y(t)
f
t
df/dt
f
df/dt
f(x,y)=x^2+y^2
g(x,y)=1/x^2+y^2
x=t, y=2-2t
x=t, y=2-2t, 0 leq t leq 1
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