ps3-solns-2009

# ps3-solns-2009 - 18.02A pset 3 part II solutions fall 2009...

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: 18.02A pset 3, part II solutions, fall 2009 Problem 1 a) ∇ C =- 2 x 10 4 e- ( x 2 + y 2 ) / 10 4 ,- 4 y 10 4 e- ( x 2 + y 2 ) / 10 4 =- 2 C ( x,y ) 10 4 h x, 2 y i . The radial direction at ( x,y ) is b u = h x,y i p x 2 + y 2 . ⇒ dC ds b u = ∇ C · b u =- 2 C ( x,y ) 10 4 ( x 2 + 2 y 2 ) p x 2 + y 2 . b) The path must have the same slope as the gradient vector at each point. That is, dy dx = 2 y x . In 18.01 you learned to separate variables and integrate: dy y = 2 dx x ⇒ ln y = ln( x 2 ) + C ⇒ y = Kx 2 . Using the starting position to determine K , we get the shark’s path is along y = y x 2 x 2 . Problem 2 a) The surface is the level surface w = z 2 + x 2 y 2 + y 3 + x 2 = 4. The normal is ∇ w = h 2 xy 2 + 2 x, 2 yx 2 + 3 y 2 , 2 z i . At (1 , 1 , 1) we get ∇ w = h 4 , 5 , 2 i ⇒ the equation of the tangent plane is 4 x + 5 y + 2 z = 11. b) We want to minimize the distance squared = f ( x,y,z ) = x 2 + y 2 + z 2 subject to the constraint 4 x + 5 y + 2 z = 11. Lagrange multipliers gives: 2 x = 4 λ, 2 y = 5 λ, 2 z = 2 λ , 4 x + 5 y + 2 z = 11. Substituting for x , y , z in terms of λ in the constraint gives 8 λ + 25 2 λ + 2 λ = 11 ⇒ λ = 22 45 ⇒ the one critical point is 11 45 (4 , 5 , 2) ⇒ minimum distance = 11 45 √ 45 . Problem 3 We want to minimize distance squared = x 2 + ( y- b ) 2 , subject to the constraint y = x 2 ....
View Full Document

## This note was uploaded on 05/14/2010 for the course 18.02A 18.02A taught by Professor Johnbush during the Winter '10 term at MIT.

### Page1 / 3

ps3-solns-2009 - 18.02A pset 3 part II solutions fall 2009...

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

View Full Document
Ask a homework question - tutors are online