Engineering Calculus Notes 335

Engineering Calculus Notes 335 - dg dθ = 4sin θ This...

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3.6. EXTREMA 323 Boundary Behavior: The boundary is the circle of radius 2 x 2 + y 2 = 4 which we can parametrize as x = 2cos θ y = 2sin θ so the function restricted to the boundary can be written g ( θ ) = f (2cos θ, 2sin θ ) = 4cos 2 4cos θ + 4sin 2 θ = 4 4cos θ. To Fnd the extrema of this, we can either use common sense (how?) or take the derivative:
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Unformatted text preview: dg dθ = 4sin θ. This vanishes when θ = 0 ,π. The values at these places are g (0) = 4 − 4 = 0 g ( π ) = 4 + 4 = 8 and we see that max x 2 + y 2 ≤ 4 x 2 − 2 x + y 2 = 8 = g ( π ) = f ( − 2 , 0) min x 2 + y 2 ≤ 4 x 2 − 2 x + y 2 = − 1 = f (1 , 0) ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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