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IX. Opgaver – 4. August 2004 588 y 2) Restvektoren er egentlig og vinkelret på U. 1 0 1 (3,4) ∇f f(x,y) = 5 Ortogonal projektion på underrum v − projU(v) = (6,4,8) − (5,6,7) = (1, −2,1) x
IX. Opgaver – 4. August 2004 589 Opgave 6. Det oplyses, at funktionen f(x,y) = x 2 + y 2 antager et minimum under bibetingelsen g(x,y) = 0, hvor g(x,y) = xy − 5. 1) Angiv samtlige de punkter, hvori dette minimum antages. 2) Beregn minimumsværdien. Løsning. 1) De partielle afledede er Lagrange ligningerne er I dette tilfælde fx = 2x, fy = 2y, gx = y, gy = x. fx = λgx, fy = λgy, g = k. 2x = λy, 2y = λx, xy = 5. x,y er ikke nul og har samme fortegn. Det følger, at λ = 2, x = y . Mulige minimumspunkter er da (x,y) = ±( √ 5, √ 5) Da funktionsværdien i disse punkter er ens, er dette de to søgte punkter. 2) Minimumsværdien er f( √ 5, √ 5) = 10 . Opgave 7. Bestem den funktion y(x) (for x > 0), der opfylder og begyndelsesbetingelsen y(1) = 1. y ′ = − 2y x + x2
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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 - 4. Kædereglen
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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 - 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 - 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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