a6 - ) L ( a,b ) ( x,y ) for all ( x,y ) R 2 . 6. The...

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Math 237 Assignment 6 Due: Friday, Mar 4th * 1. Let f ( x,y ) = ln(1 + x + 2 y ). Use Taylor’s Theorem to show that | R 1 , (0 , 0) ( x,y ) | ≤ 3( x 2 + y 2 ) for x 0, y 0. * 2. Use Taylor’s Theorem to show that if x > 1 and y > 1, then | f ( x,y ) - L (1 , 1) ( x,y ) | ≤ 3 2 [( x - 1) 2 + ( y - 1) 2 ] . * 3. Let f ( x,y ) = x 2 - 2 x + y 3 - xy 2 . Find and classify the critical points of f . 4. Let f ( x,y ) = e - 2 x + y . Use Taylor’s theorem to show that the error in the linear approximation L (1 , 1) ( x,y ) is at most 6 e [( x - 1) 2 + ( y - 1) 2 ] if 0 x 1 and 0 y 1. 5. Consider f : R 2 R defined by f ( x,y ) = 2 x 2 + 3 y 2 . Prove that for any ( a,b ) R 2 we have f ( x,y
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Unformatted text preview: ) L ( a,b ) ( x,y ) for all ( x,y ) R 2 . 6. The magnitude of each second partial derivative of f is less than 2, for all ( x,y ) within 1 / 4 of (0 ,/ 4). Find an upper bound for the error in the linear approximation L (0 ,/ 4) ( x,y ) of f for all ( x,y ) within 1 / 4 of (0 ,/ 4). 7. Find and classify the critical points of f ( x,y ) = ( x + y )( xy + 1). NOTE: Only * questions will be graded...
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