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I. Differentiation – 5. Gradient 94 Heraf f.eks. z Duf(x0) = g ′ (0) ≈ 0.5(0.9/0.7 + 1.1/0.8) ≈ 1.33 5.11. Retningsafledt, formel ☞ [S] 11.6 Directional derivatives and the . . . 3 Sætning For en differentiabel funktion f(x,y) er den retningsafledede punktet (x,y) i retning af en enhedsvektor u = (a,b) givet ved Duf(x,y) = fx(x,y)a + fy(x,y)b Bevis Funktionen g(h) = f(x0 + ha,y0 + hb) har afledt g ′ f(x0 + ah,y0 + bh) − f(x0,y0) (0) = lim h→0 Konklusion fra kædereglen g ′ (0) = fx(x0,y0)a + fy(x0,y0)b h 1 h = Duf(x0,y0)
I. Differentiation – 5. Gradient 95 5.12. Retningsafledt udregnet ☞ [S] 11.6 Directional derivatives and the . . . Eksempel - gentaget Funktionen f(x,y) = x 2 + y har partielle afledede fx(x,y) = 2x, fy(x,y) = 1 Den retningsafledede i punktet (1,1) i retningen u = ( 3 4 5 , 5 ) beregnes Duf(1,1) = fx(1,1) 3 4 + fy(1,1) 5 5 = 2 · 3 4 + 1 · 5 5 = 2 5.13. Retningsafledt og vinkel ☞ [S] 11.6 Directional derivatives and the . . . Bemærkning Hvis enhedsvektoren u danner en vinkel på θ med x-aksen, så er u = (cos θ,sin θ) og den retningsafledede kan skrives 6 Duf(x,y) = fx(x,y)cos θ + fy(x,y)sin θ
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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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Litteratur • Arnold, Ordinary dif
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