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Unformatted text preview: κ =  r ( t ) × r 00 ( t )   r ( t )  3 . Proof. The key point is to remember that the parametric equations of the ellipse can be chosen to be x = 3 cos t,y = 2 sin t or x = 3 sin t,y = 2 cos t . Consider the vector function r ( t ) = < 3 cos t, 2 sin t, >,t ∈ R As (3 cos t ) 2 9 + (2 sin t ) 4 = 1 and z = 0, then r ( t ) is the right vector function representing the ellipse on the xyplane. The point (0 , 2) corresponds to r ( π/ 2). r ( t ) = <3 sin t, 2 cos t, >, then r ( π/ 2) = <3 , , > . r 00 ( t ) = <3 cos t,2 sin t, >, then r 00 ( π/ 2) = < ,2 , > . As  r ( π/ 2)  = 3, and r ( π/ 2) × r 00 ( π/ 2) = 6 k then κ = 6 3 3 = 2 9 . ±...
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 Spring '11
 Y
 Calculus, 20 minutes, 3k, Parametric equation, 6k

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