Unformatted text preview: Let f : R 2 → R and g : R → R such that f ( x,y ) = ± x + 1 y 2 ² g ( xy 2 ) , y 6 = 0 . Use the Chain Rule to show that 2 xf xyf y = 2 f . 6. Let f ( x,y ) = ln(1 + x + 2 y ). a) Find the linear approximation L (0 , 0) ( x,y ) of f . b) Use Taylor’s Theorem to show that  R 1 , (0 , 0) ( x,y )  ≤ 7 2 ( x 2 + y 2 ) for x ≥ 0, y ≥ 0. 7. Let f ( x,y,z ) = e xy + z . a) Find the rate of change of f at the point (1 ,1 , 1) in the direction of the vector u = (4 ,2 , 3). b) In what direction from (1 ,1 , 1) does f change most rapidly and what is the maximum rate of change. 8. Consider the function f ( x,y ) = ( x 3 + y 3 x 2 + y 2 , ( x,y ) 6 = (0 , 0) , ( x,y ) = (0 , 0) . a) What is the domain of f ? b) Where is f continuous on its domain? c) Where is f diﬀerentiable on its domain? d) Based on your answer in part c), what can you conclude about the continuity of f x and f y ?...
View
Full
Document
This note was uploaded on 04/18/2010 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.
 Spring '08
 WOLCZUK
 Math

Click to edit the document details