math226spring08mt2s

# math226spring08mt2s - 1 Given f x,y,z = 3 x 2 √ yz a Find...

This preview shows pages 1–2. 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: 1. Given f ( x,y,z ) = 3 x + 2 √ yz, a) Find the maximal increase rate of f at (1 ,- 1 ,- 1) and the direction in which it occurs b) Find the rate of change of f at (1 ,- 1 ,- 1) in the direction of the vector ~ b = h 1 , 2 , 2 i . c) Write the equation of he tangent plane to the surface S : 3 x +2 √ yz = 5 at (1 ,- 1 ,- 1) . Answer. a) f x = 3 ,f y = z √ yz ,f z = y √ zy and ∇ f (1 ,- 1 ,- 1) = h 3 ,- 1 ,- 1 i is the direction of the maximal increase rate and the rate itsel is |∇ f (1 ,- 1 ,- 1) | = q 3 2 + (- 1) 2 + (- 1) 2 = √ 11 . b) The unit vector in ~ b direction is ~u = ~ b | ~ b | = D 1 3 , 2 3 , 2 3 E , the rate of change is ∇ f (1 ,- 1 ,- 1) · ~u = 1- 2 3- 2 3 =- 1 3 . c) The equation is 3( x- 1)- ( y + 1)- ( z + 1) = 0 . 2 . a) Find the work done by the force ~ F ( x,y ) = h 3 e y x 2 ,y i to move an object along the curve C : y = x 3 , ≤ x ≤ 1 . Answer. We parametrize C : x = t,y = t 3 , ≤ t ≤ 1 . We have ~ r ( t ) = h t,t 3 i ,~ r ( t ) = h 1 , 3 t...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

math226spring08mt2s - 1 Given f x,y,z = 3 x 2 √ yz a Find...

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

View Full Document
Ask a homework question - tutors are online