# Practice Exam 2_answers - MATH 2401 Fall 2007 Practice Exam...

This preview shows pages 1–3. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 2401, Fall 2007 Practice Exam 2, Solutions Problem 1 . Calculations. (a) Find the directional derivative of f ( x, y, z ) = xy + yz + zx at P (1 ,- 1 , 1) in the direction of i + 2 j + k Solution: ∇ f = ( y + z ) i + ( x + z ) j + ( y + x ) k , ∇ f (1 ,- 1 , 1) = 2 j . u = √ 6 6 ( i + 2 j + k ), so f u (1 ,- 1 , 1) = ∇ f (1 ,- 1 , 1) • u = 2 3 √ 6 . (b) Find the rate of change of f ( x, y ) = xe y + ye- x along the curve r ( t ) = ( lnt ) i + t ( lnt ) j . Solution: ∇ f = ( e y- ye- x ) i + ( xe y + e- x ) j , ∇ f ( r ( t )) = ( t t- lnt ) i + ( t t lnt + 1 t ) j , df dt = ∇ f ( r ( t )) • r ( t ) = t t ( 1 t + lnt + ( lnt ) 2 ) + 1 t . (c) Find ∂u ∂s for u = x 2- xy , x = scost , y = tsins . Solution: ∂u ∂s = ∂u ∂x ∂x ∂s + ∂u ∂y ∂y ∂s = (2 x- y )( cost )+(- x )( tcoss ) = 2 scos 2 t- t sins cost- st coss cost. (d) Find dy dx if xcos ( xy ) + ycos ( x ) = 2....
View Full Document

{[ snackBarMessage ]}

### Page1 / 5

Practice Exam 2_answers - MATH 2401 Fall 2007 Practice Exam...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online