tut6 - f ( x, y ) = xy ( x 2-y 2 ) x 2 + y 2 , ( x, y ) 6 =...

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MATH 237 Tutorial 6 Wednesday, June 21st, 2006 Taylor Polynomials, Taylor’s Theorem and Critical points 1) Let f ( x, y ) = y e 1 - cos x + sin( xy ). Find the Taylor polynomials P 1 , a ( x, y ) and P 2 , a ( x, y ) where a = (0 , 0). 2) Let g : R 2 R satisfy the hypotheses of Taylor’s Theorem in the disc B ( a ) which is centred at a = ( a, b ) and which has radius 0 . 1. Suppose that | g xx ( x, y ) | < 0 . 1, | g xy ( x, y ) | < 0 . 05 and | g yy ( x, y ) | < 0 . 2 for all points ( x, y ) B ( a ). Show that the absolute value of the error involved in approximating the function g in B ( a ) by its linear approximation is less than 0 . 002. 3) Let f ( x, y ) = 1 + sin x + cos y . a) Calculate the linear approximation L a ( x, y ) to f at a = (0 , 0). b) Use Taylor’s Theorem to show that the error | L (0 , 0) ( x, y ) - f ( x, y ) | ≤ x 2 + y 2 2 . 4) Let
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Unformatted text preview: f ( x, y ) = xy ( x 2-y 2 ) x 2 + y 2 , ( x, y ) 6 = (0 , 0) and f (0 , 0) = 0. Show that f x (0 , 0) = f y (0 , 0) = 0. Show that f xy (0 , 0) =-1 while f yx (0 , 0) = 1. What can you conclude about the continuity of f xy and f yx at (0 , 0)? 6) a) Consider the function f : R 2 R dened by f ( x, y ) = xy 2-2 x-1 2 x 2-2 3 y 3 . Find all critical points of f . b) Suppose that (1 , 2) is a critical point of a function g : R 2 R , and that g (1 , 2) = 3, Hg (1 , 2) = 1 5 5 9 . Write down the second degree Taylor polyno-mial of g at the point (1 , 2)....
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