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I. Differentiation – 7. Lagrangemetoden 148 Definition Bestem ekstremumspunkter for en funktion f(x,y,z) under begræsningen g(x,y,z) = k. I ligningen 1 ∇f(x0,y0,z0) = λ∇g(x0,y0,z0) kaldes den ubekendte λ for en Lagrange multiplikator. Ligningen udtrykker at niveaufladen for f i (x0,y0,z0) tangerer begræsningsfladen g(x,y,z) = k. 7.14. Lagrange multiplikator metode ☞ [S] 11.8 Lagrange multipliers Metode Bestem ekstremumspunkter for en funktion f(x,y,z) under begræsningen g(x,y,z) = k. (a) Find x,y,z,λ så ∇f(x,y,z) = λ∇g(x,y,z) g(x,y,z) = k (b) Bestem funktionsværdierne i punkterne fra (a). Maksimum og minimum er blandt disse. 7.15. Lagrange multiplikator metode ☞ [S] 11.8 Lagrange multipliers Ligninger
I. Differentiation – 7. Lagrangemetoden 149 Lagranges ligningssystem for bestemmelse af ekstremumspunkter for en funktion f(x,y,z) under begræsningen g(x,y,z) = k. fx(x,y,z) = λgx(x,y,z) fy(x,y,z) = λgy(x,y,z) fz(x,y,z) = λgz(x,y,z) g(x,y,z) = k 7.16. Kassefabrikant ☞ [S] 11.8 Lagrange multipliers Eksempel 1 En kasse uden låg laves af 12m 2 krydsfiner. Bestem kantlængder der giver størst rumfang. Kantlængder x,y,z. Volumen V = xyz. Overfladeareal g = 2xz + 2yz + xy = 12. Beregn Vx = yz,Vy = xz,Vz = xy gx = y + 2z,gy = x + 2z,gz = 2x + 2y 7.17. Kassefabrikant ☞ [S] 11.8 Lagrange multipliers
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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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