Further 22 cos 2 2 2 2 cos 2 2 2 cos 1 and

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Unformatted text preview: − 3), then 2 + sin 2 − cos 2 ( + )2 + cos + 4 + 9 . Further, 22 cos ( + )2 2 + + 2 2 + cos 2 − 2 ( + )2 + cos + 1 and thus (π 2π 2π ) = − 8√2 − 4 . Further, ( )= ( ) and 9 1 (π 2π 2π ) = − 8√2 − 4 . Thus the equation of the tangent plane is 9 4 1 √+ 29 ( − π) − 4 1 √+ 82 9 ( − 2π ) − 4 1 √+ 82 9 0 = 0 (= 2π ), so ( − 2π ) = 0 which can be re-written as 4 1 √+ 29 4 1 √+ 82 9 − − 4 1 √+ 82 9 4π π =√− 3 22 E Prove that any tangent plane to the surface dinate axes form a tetrahedron of a constant volume. = 3 :) ( > 0) and the coor- S Let ( 0 0 0 ) be an arbitrary point on the surface. Notice that the equation doesn’t change if we change signs of two variables at the same time, hence it’s enough to assume that 0 0 0 > 0. The tangent plane is 0 0( − 0) + 0 0( − 0) + 0 0( − 0) =0 ⇐ 00 + 00 + 00 =3 3 Further, the -intercept is obtained from this equation by setting = = 0, so it is 33 = 3 0 . Similarly, the -intercept is 3 0 and the -intercept is 3 0 . 00 Final...
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This document was uploaded on 02/10/2014.

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