MIT18_024S11_Pset6

# MIT18_024S11_Pset6 - f x (0 , y ) = y for any y and f ( x,...

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PSET 6 - DUE MARCH 17 1. 8.22: 14 (5 points) 2. 8.24: 12 (5 points) 3. Let f ( x, y )= xy g ( u ) du where g : R R is a strictly positive continuous function. 0 Find f : R 2 R 2 in terms of g . Consider a level set { ( x, y ) R 2 | f ( x, y )= c } . Prove that for a ﬁxed c = 0 there are exactly two level curves in the set. Moreover, prove they are precisely the graph of the function h ( x )= b/x for exactly one b R . (Do not try to determine b in terms of g ! Just prove it exists and is unique!) Parameterize one curve on a level set and prove that f is orthogonal to the level set at each point on the curve. (6 points) 4. Let f : R 2 R by ± x 2 2 y 2 xy 2 ( x, y ) =(0 , 0) (1) f ( x, y )= x + y 0 ( x, y )=(0 , 0) Prove

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Unformatted text preview: f x (0 , y ) = y for any y and f ( x, 0) = x y for any x . Prove 2 f = 2 f yx xy . (6 points) 4. C20:5 (4 points) 5. C20:6 (4 points) 1 MIT OpenCourseWare http://ocw.mit.edu 18.024 Multivariable Calculus with Theory Spring 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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## This note was uploaded on 01/18/2012 for the course MATH 18.024 taught by Professor Christinebreiner during the Spring '11 term at MIT.

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MIT18_024S11_Pset6 - f x (0 , y ) = y for any y and f ( x,...

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