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IX. Opgaver – 4. August 2004 582 = 1 2 ex2 Dette giver den fuldstændige løsning ([DL] 1.5, 1.6 samt rettelser fra Us. 5) . y(x) = Ce A(x) + B(x)e A(x) = Ce −x2 = Ce −x2 + 1 2 ex2e −x2 + 1 2 , hvor C er en arbitrær konstant. 2) I den partikulære løsning bestemmes C ved y(0) = 1 2 . y(0) = Ce 0 + 1 1 = 2 2 Det ses, at den søgte partikulære løsning er konstant 4. August 2004 y(x) = 1 2 . ⇒ C = 0 . Opgave 1. Lad D betegne den del af cirkelskiven x 2 + y 2 ≤ 4, der ligger i 1. kvadrant (tegn!). 1) Beskriv D i polære koordinater.
IX. Opgaver – 4. August 2004 583 2) Beregn dobbeltintegralet Løsning. 1) Området tegnes D er i polære koordinater y 1 D 1 x 2 + y 2 + 2 dA. (1,2) (2,2) (1,1) 0 1 y = x Kvartcirklen D = {(x,y)|0 ≤ x,0 ≤ y,x 2 + y 2 ≤ 4} x = r cos θ, y = r sin θ 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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II. Integration - 2. Itereret integ
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II. Integration - 3. Generelle omr
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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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IV. Differentialligninger - 3. Gene
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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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V. Matricer - 3. Lineære ligninger
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V. Matricer - 4. Determinanter 419
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V. Matricer - 4. Determinanter = (
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1. Egenvektorer VI Egenvektorer og
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VI. Egenvektorer og diagonalisering
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VII. Skalarprodukt og projektion -
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VIII. Appendiks - 1. Polære koordi
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