a6 - x 1 and 0 y 1. 4. Consider f : R 2 R dened by f ( x, y...

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Math 237 Assignment 6 Due: Friday, Oct 30th 1. Find the 2nd degree Taylor polynomial for the following functions at the given point. a) f ( x, y ) = x 2 y 3 - xy at (1 , 1). b) f ( x, y ) = (1 + x ) y at (0 , 0). 2. Let f ( x, y ) = ln( x + 2 y ). a) Show that for ( x, y ) sufficiently close to (3 , - 1) we have f ( x, y ) ( x - 3) + 2( y + 1). b) Prove that if x 3 and y ≥ - 1, then the error in the approximation in a) satisfies | f ( x, y ) - [( x - 3) + 2( y + 1)] | ≤ 3[( x - 3) 2 + ( y + 1) 2 ] . 3. Let f ( x, y ) = 1 + 2 x + y . a) Find the linear approximation L (0 , 0) ( x, y ) of f . b) Use Taylor’s theorem to show that the error in the linear approximation L (0 , 0) ( x, y ) is at most 3 4 ( x 2 + y 2 ) if 0
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Unformatted text preview: x 1 and 0 y 1. 4. Consider f : R 2 R dened by f ( x, y ) = 2 x 2 + 3 y 2 , and let a R 2 . Prove that f ( x, y ) L a ( x, y ) for all ( x, y ) R 2 . 5. Prove that if f : R 2 R has continuous partial derivatives in some neighborhood N ( a, b ) of ( a, b ), then for all ( x, y ) N ( a, b ) there exists a point ( c, d ) on the line segment from ( a, b ) to ( x, y ) such that f ( x, y ) = f ( a, b ) + f x ( c, d )( x-a ) + f y ( c, d )( y-b ) ....
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