Assignment 3 Solutions

Assignment 3 Solutions - Department of Mathematics MATHS...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Department of Mathematics MATHS 208 Assignment 3 Solutions 1. (6 marks) a) The equation x 3- xy + 4 y 2 + xyz- z 3 = 11 defines a function z = f ( x,y ) implicitly. Use implicit differentiation to find the value of z x when ( x,y,z ) = (1 , 2 , 2) . Solution: F = x 3- xy + 4 y 2 + xyz- z 3- 11 = 0 , F x = 3 x 2- y + yz, at (1 , 2 , 2) F x = 3- 2- 4 =- 3 , F z = xy- 3 z 2 , at (1 , 2 , 2) F z = 2- 12 =- 10 , z x =- F x F z =- 3 10 . b) If f ( x,y,z ) = y 2 ze x + y- sin( xyz ) = 0, find z x and y z . Solution: F x = y 2 ze x + y- yz cos( xyz ) , F y = (2 y + y 2 ) ze x + y- xz cos( xyz ) , F z = y 2 e x + y- xy cos( xyz ) , z x =- F x F z =- y 2 ze x + y- yz cos( xyz ) y 2 e x + y- xy cos( xyz ) y z =- F z F y =- y 2 e x + y- xy cos( xyz ) (2 y + y 2 ) ze x + y- xz cos( xyz ) 2. (5 marks) Given w = x 2 y + y 2 z + z 2 x, verify that w x + w y + w z = ( x + y + z ) 2 . Solution: w = x 2 y + y 2 z + z 2 x. Then w x = 2 xy + z 2 . w y = x 2 + 2 yz. w z = y 2 + 2 xz. Therefore w x + w y + w z = x 2 + y 2 + z 2 + 2( xy + yz + xz ) = ( x + y + z ) 2 . 3. (5 marks) Given f ( x,y,z ) = 3 x 2 + xy- 2 y 2- yz + z 2 , find the rate of change of f ( x,y,z ) at the point (1 ,- 2 ,- 1) in the direction of the vector (2 ,- 2 ,- 1) ....
View Full Document

Page1 / 3

Assignment 3 Solutions - Department of Mathematics MATHS...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online