PS4_answers

PS4_answers - Math 237 A2. α = −β 2 . A3. a) Hf (2, 3)...

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: Math 237 A2. α = −β 2 . A3. a) Hf (2, 3) = 9 −6 . −6 4 Problem Set 4 Answers b) La (2, 3) = f (2, 3) + fx (2, 3)(x − 2) + fy (2, 3)(y − 3) = 1 + 3(x − 2) − 2(y − 3). 9 P2,(2,3) (x, y ) = 1 + 3(x − 2) − 2(y − 3) + 2 (x − 2)2 − 6(x − 2)(y − 3) + 2(y − 3)2 c) The level curves of f are straight lines. 1 A4. i) P2,(1,0) (x, y ) = ln 2 + 1 (x − 1) + 1 y − 1 (x − 1)2 − 4 (x − 1)y + 1 y 2 . 2 2 8 8 3 ii) P2,(1,1) (x, y ) = 1 + 2(x − 1) − (y − 1) − 2 (x − 1)2 − 2(x − 1)(y − 1) + (y − 1)2 A5. i) P2,(π,π) (x, y ) = (x − π )2 − (y − π )2 . 2 2 A8. uxx + uyy = D1 f + D2 f , 2 2 uxx − uyy = D1 f · cos 2θ − D1 D2 f · sin 2θ − D2 D1 f · sin 2θ − D2 f · cos 2θ. 12 12 uxy = 2 D1 f · sin 2θ − D1 D2 sin2 θ + D2 D1 cos2 θ − 2 D2 f sin 2θ. A9. a) 4(u2 + v 2 )[(fx )2 + (fy )2 ], b) 4(u2 + v 2 )[fxx + fyy ]: f differentiable. 19 . 2 B1. The only function is f (x, y ) = 1 x2 + 2xy − 3 y 2 − 3x + 3y + 2 2 B2. b) c, c) u(x, t) = Ae−(x+ct) 2 1 ...
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.

Ask a homework question - tutors are online