flerdimlos090826

flerdimlos090826 - LUNDS TEKNISKA HOGSKOLA MATEMATIK...

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

View Full Document Right Arrow Icon
LUNDS TEKNISKA H ¨ OGSKOLA TENTAMENSSKRIVNING MATEMATIK Flerdimensionell analys 2009-08-26 kl 8–13 L ¨ OSNINGSF ¨ ORSLAG 1. ZZ D xy d x d y = Z 1 0 ±Z x - 2 x xy d y ² d x = Z 1 0 - 3 2 x 3 d x = - 3 8 . 2. a) Se l¨ aroboken. b) Se l¨aroboken. c) L˚ at g ( x,y,z ) = x 2 - y 2 - 1 3 z 3 . Vi f˚ ar att grad g = (2 x, - 2 y, - z 2 ) och vi vill hitta punkter ( x,y,z ), s˚ adana att grad g ¨ ar parallell med planets normalvektor (1 , 1 , 1). Ekvationssystemet (2 x, - 2 y, - z 2 ) = c (1 , 1 , 1) ,g = 0 har l¨ osningen ( x,y,z ) = ( - 9 / 2 , 9 / 2 , - 3), vilket ¨ ar den s¨ okta tangeringspunkten. F¨or att planet x + y + z = d ska g˚ a genom denna punkt kr¨ avs d = - 3. 3. Derivering ger f 0 x = (2 xy - 3 x 2 y ) e - 3 x - y = xy (2 - 3 x ) e - 3 x - y och f 0 y = ( x 2 - x 2 y ) e - 3 x - y = x 2 (1 - y ) e - 3 x - y . Den enda l¨osningen till ekvationssystemet f 0 x = f 0 y = 0 som ligger i det inre av definitionsm¨ angden ¨ ar (2 / 3 , 1), vilket allts˚ a ¨ ar den enda station¨ ara punkten. Vi har att
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/07/2011 for the course FMA 430 taught by Professor Tomaspersson during the Spring '11 term at Lund.

Page1 / 2

flerdimlos090826 - LUNDS TEKNISKA HOGSKOLA MATEMATIK...

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