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I. Differentiation – 3. Tangentplan 66 3.25. Opgave ☞ [S] 11.4 Tangent planes and linear approximations Øvelse 9 f(x,y) = x √ y Begrund differentiabilitet om (1,4) og find den lineære approximation. Løsning er kontinuerte om (1,4). når (x,y) ≈ (1,4). fx = √ y, fy = x 2 √ y x √ y ≈ 2 + 2(x − 1) + 1 (y − 4) 4 3.26. Opgave fortsat Øvelse 9 - fortsat ☞ [S] 11.4 Tangent planes and linear approximations Skrives også (1 + ∆x) 4 + ∆y ≈ 2 + 2∆x + 1 4 ∆y Beregn tilnærmelse 0.9 √ 4.4 ≈ 2 + 2(−0.1) + 1 0.4 = 1.9 4
I. Differentiation – 3. Tangentplan 67 3.27. Test differentialet ☞ [S] 11.4 Tangent planes and linear . . . Test Givet z = ln(ax + by). Differentialet er: (a) dz = adx + bdy. (b) dz = a b ax+bydx + ax+bydy. (c) dz = aln(ax + by)dx + bln(ax + by)dy. Løsning Udregningen giver differentialet zx = a ax+by , zy = b ax+by dz = zx dx + zy dy Afkryds den rigtige: (a) (b) (c) 3.28. Udvid til mange variable ☞ [S] 11.4 Tangent planes and linear approx. Definition Omtalen af tangentplan, lineær approximation og differentialer udvides umiddelbart til funktioner af tre eller flere variable. Funktionen w = f(x,y,z) har tangentplan i punktet (a,b,c,d), d = f(a,b,c) med ligning w − d =
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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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