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IX. Opgaver – 3. Januar 2004 574 Løsning. 1) De partielle afledede er hvorfra gradienten beregnes fx = x x 2 + y 2 , fy = y x 2 + y 2 , ∇f(3,4) = (fx(3,4),fy(3,4)) = ( 3 4 , ) . 5 5 2) Den største retningsafledede er ([S] 11.6 Theorem 15) længden af gradientvektoren |∇f(3,4)| = |( 3 5 4 , )| = 5 9 16 + = 1 . 25 25 Opgave 3. Lad a betegne et vilkårligt reelt tal, og lad A betegne matricen ⎡ 3 0 ⎤ 0 A = ⎣ 1 a 1 ⎦ . 0 0 3 1) Gør rede for, at (−1,0,1) er en egenvektor for A, og angiv den tilhørende egenværdi. 2) Betragt matricen ⎡ 3 0 ⎤ 0 B = ⎣ 1 1 1 ⎦ 0 0 3
IX. Opgaver – 3. Januar 2004 575 y 1 0 1 (3,4) ∇f f(x,y) = 5 (fremkommet af matricen A fra Sp. 1) ved at sætte a = 1) Det oplyses, at λ = 3 er en egenværdi for B. Angiv det tilhørende egenrum E3. x
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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Forord "Slides" til forelæsningern
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I. Differentiation - 1. Kontinuitet
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I. Differentiation - 2. Partielle a
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I. Differentiation - 3. Tangentplan
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I. Differentiation - 6. Maksimum/mi
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I. Differentiation - 7. Lagrangemet
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II. Integration - 1. Dobbelt integr
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III. Potensrækker - 2. Talfølger
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III. Potensrækker - 3. Potensrækk
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III. Potensrækker - 4. Taylorpolyn
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1. Grafiske/numeriske metoder IV Di
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IV. Differentialligninger - 1. Graf
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1. Vektorer og matricer V Matricer
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V. Matricer - 1. Vektorer og matric
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V. Matricer - 2. Lineære afbildnin
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1. Egenvektorer VI Egenvektorer og
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VI. Egenvektorer og diagonalisering
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VIII. Appendiks - 1. Polære koordi
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