# t1f08s - 1. Suppose that the function f : R 2 R is...

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Suppose that the function is differentiable and is defined by R R f 2 : 2 3 : R R g . ) , ( ) , , ( yz xz z y x = g (a) [4 marks] Let . Find D g and ) , , ( ) , ( z y x v u g = ) , ( ) , ( z x v u . , = = y z x z v v v u u u D z y x z y x 0 0 g ) , ( ) , ( z x v u = yz y x z = 0 (b) [6 marks] If , then ) , ( ) , )( ( yz xz f y x f w = = g o k z w z y w y x w x = + , where k is a constant. Determine the value of k . = = = v f u f z w y w x w w Dw + = v f y u f x v f z u f z y z x z 0 0 . Substituting to the left hand side of the expression we get 0 ) ( ) ( ) ( = + + v f y u f x z v f z y u f z x , so k = 0. 2. (a) [6 marks] In the neighborhood of the point the equation ) 2 , 1 , 1 ( xyz z y x 5 3 3 3 = + + defines z as a function of x and y , that is ) , ( y x f z = . Find the directional derivative (or in another notation) at the point in the direction f u z u ) 1 , 1 ( 0 P of the line from to . 0 P ) 5 , 4 ( 1 P and by implicit differentiation formulas we get 0 5 ) , , ( 3 3 3 = + + = xyz z y x z y x F xy z yz x z F x F x z 5 3 5 3 2 2 = = and xy z xz y z F x F y z 5 3 5 3 2 2 = = Now ) 1 , 1 ( ) 5 12 10 3 , 5 12 10 3 ( ) , ( ) 1 , 1 ( ) 2 , 1 , 1 ( ) 2 , 1 , 1 ( = = = y z x z z and ) 4 , 3 ( 5 1 1 1 0 1 0 = = P P P P u . Hence 5 7 5 4 3 ) 4 , 3 ( 5 1 ) 1 , 1 ( ) 1 , 1 ( = + = = = = u u u z z f (b) [3 marks] Give an example of a particular point 3 R P in the neighborhood of which the equation in part (a) does not define z as a function of x and y . By Implicit Function Theorem the relation will not define z as a function of x and y in the neighborhood of a point if 3 ) , , ( R c b a 0 ) , , ( = c b a z F that is if . ab c 5 3 2 = Setting for example 3 and 5 , 5 = = = b a c , we get the particular point . ) 5 , 5 , 3 ( Other answers possible. 2

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## This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto- Toronto.

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t1f08s - 1. Suppose that the function f : R 2 R is...

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